Mathematics 137 - Los Angeles Mission College
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|Mathematics 137 |
|Fall 2019 |
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JOURNAL 1
Please use a separate piece of paper to address the following questions using complete sentences.
You may type or neatly handwrite this assignment.
Do not write your journal using one paragraph. The preferred format for this assignment is to number your responses according to the question number.
Before you submit this assignment, please read both sides of this document and verify that you have addressed each question and met the journal requirements described in the syllabus. Also, this journal will not be returned to you.
1. Name: How do you prefer to be addressed in class? How do I pronounce your first and last name (if it’s not obvious how to pronounce them, and especially if I’ve been mispronouncing your name)?
2. Classes: Which classes are you are taking this semester, and what is the total number of units?
3. Work: How many hours do you work each week?
4. Previous math courses: What previous math classes did you complete at the community college level? If you placed into this course, what was the last math class you took, and when and where did you take it?
5. Math 137: As of Spring 2014, students have the option of taking either Math 115/125 or taking Math 137 to satisfy the prerequisite for Math 227 (Statistics). Why did you decide to enroll in this course? How did you hear about it?
6. Math anxiety: Describe your math anxiety level (low, medium, high).
Difficulties with math:
a. Are you a “math person”? Why or why not?
If you do not think you are a “math person,” do you think it is possible to change?
b. Are you comfortable with solving word problems/applications?
7. Academic goals: What is your major or area of concentration, and what is your academic goal (certificate, AA degree, 4-year degree, etc.)?
8. Career goals: What is your career goal?
9. Other interests or hobbies: What other interests or hobbies do you have?
10. Other information: Is there anything else you would like to share with me (being a DSPS or EOPS student, athlete, veteran, parent, etc.)?
11. Peers/study partners/tutors: Do you have any friends taking this class with you? Do you have any study partners or tutors with whom you will regularly meet during the semester?
12. Is this your first time taking Math 137? If the answer is no, why do you think you did not pass the class? And what do you plan to do differently in order to be successful this time around?
Introductory Activity
One of the most important ideas in statistics is that we can learn a lot about a large group (called a population) by studying a small piece of it (called a sample). Consider the population of 268 words in the following passage:
Four score and seven years ago, our fathers brought forth upon this continent a new nation: conceived in liberty, and dedicated to the proposition that all men are created equal.
Now we are engaged in a great civil war, testing whether that nation, or any nation so conceived and so dedicated, can long endure. We are met on a great battlefield of that war.
We have come to dedicate a portion of that field as a final resting place for those who here gave their lives that that nation might live. It is altogether fitting and proper that we should do this.
But, in a larger sense, we cannot dedicate, we cannot consecrate, we cannot hallow this ground. The brave men, living and dead, who struggled here have consecrated it, far above our poor power to add or detract. The world will little note, nor long remember, what we say here, but it can never forget what they did here.
It is for us the living, rather, to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced. It is rather for us to be here dedicated to the great task remaining before us, that from these honored dead we take increased devotion to that cause for which they gave the last full measure of devotion, that we here highly resolve that these dead shall not have died in vain, that this nation, under God, shall have a new birth of freedom, and that government of the people, by the people, for the people, shall not perish from the earth.
(a) Select a sample of ten representative words from this population by circling them in the passage above.
The above passage is, of course, Lincoln’s Gettysburg Address. For this activity we are considering this passage a population of words, and the 10 words you selected are considered a sample from this population. In most studies, we do not have access to the entire population and can only consider results for a sample from that population. The goal is to learn something about a very large population (e.g., all American adults, all American registered voters) by studying a sample. The key is in carefully selecting the sample so that the results in the sample are representative of the larger population (i.e., has the same characteristics).
(b) Record the word and the number of letters in each of the ten words in your sample:
| |1 |2 |3 |4 |5 |
|ID number | | | | | |
|Word | | | | | |
|Word length | | | | | |
Sampling frame:
|001 |Four |
|Anna |$85,000 |
|Bob |$50,000 |
|Cathy |$45,000 |
|Dave |$40,000 |
|Eric |$30,000 |
|Fran |$30,000 |
|Gail |$35,000 |
1. Find the mean and median salary before Bill Gates walked into the diner.
2. Are the two measurements pretty similar? Would they represent a “typical” diner’s?
Now, Bill Gates walks in with annual income of, say, $1 billion, ($1,000,000,000)
3. Find the mean and median salary after Bill Gates walked into the diner.
4. Are the two measurements pretty similar? Would they represent a “typical” diner’s salary?
5. Should we be using the mean or median in each case? Why?
Module 8 Mean and Median (2)
1. As a group, find a data set with at least 10 numbers that has a mean of 13 and also a median of 13.
2. Add two numbers to your data set in number 1, so that the mean and median remain 13.
3. As a group, find a data set with at least 8 numbers where the mean is higher than the median. Find a data set with at least 8 numbers where the mean is lower than the median.
4. Make a data set with at least 6 numbers. Find the mean and median of the data set. Now add one large number to the data set. How do the mean and median change? Which changes more, the mean or the median?
Module 9 Boxplots: Class Example
A data value is an outlier when it is: Greater Than: Q3 + 1.5xIQR
Or Less Than: Q1 – 1.5xIQR
Example: The following data shows the responses of 12 adults when asked “How many hours do you exercise per month?”
[pic]
Module 9 Quartiles and IQR
Activity 1: Calculating Quartiles, Range, Q1, Q2 (median), Q3, and IQR
For each of the following data sets, calculate the range, Q1, Q2, Q3, and IQR for the following sets
1. The following data represents the ages of sixteen randomly selected people who attended a rated G movie.
[pic]
2. The following data represents the inches of rain per year from ten randomly selected cities.
[pic]
3. The following data represents the test scores of eleven randomly selected students.
31, 34, 41, 52, 68, 71, 79, 83, 88, 90, 103, 153
4. The following data represents the weights of nine randomly selected team members from a high school wrestling team.
130, 150, 152, 154, 155, 157, 159, 163, 164, 165, 185
Activity 2: Exploring Quartiles and IQR
1. Make a dot plot of the following data. Divide the dot plot into four groups with an equal amount of dots per group. Find approximate values of the quartiles. Make sure that 25% of the numbers are less than Q1 , 25% of the numbers are between Q1 and Q2 , 25% of the numbers are between Q2 and Q3, and 25% of the numbers are greater than Q3 . Approximate the IQR.
6 , 6 , 7 , 7 , 7 , 7 , 9 , 9 , 11 , 13 , 13 , 13 , 15 , 15 , 15 , 15 , 16 , 16 , 19 , 20 , 21 , 22 , 22 , 22
2. Which data set has more spread? Why? Find approximate values of the quartiles. Approximate the distances betweenQ1 and Q2 , and between Q2 and Q3 for data set A and data set B and compare.
3. Find two data sets with at least 15 different numbers that have the same median and the same range, but different IQR’s. Calculate the medians, ranges, quartiles and IQR to confirm that it works.
Module 9 Exploring Variability about the Median
For each of the following sets of data:
a) Find the Mean (round to one decimal place if needed) Write a sentence describing the meaning of this for the data.
b) Find the Median Write a sentence describing the meaning of this for the data.
c) Find the Range Write a sentence describing the meaning of this for the data.
d) Find the 5 point summary (minimum, maximum, Q1, Q2 (median), Q3) Write a sentence describing the meaning of each value for the data.
e) Find the IQR.
f) Fences and any possible Outliers.
g) Create a boxplot. Then verify your work using Statcrunch.
1. Ages of a person’s 8 cousins:. 4, 12, 4, 6, 9, 8, 13, 36
2. Scores for ten students on a 30 point test: 27, 20, 12, 14, 25, 25, 26, 27, 25, 2
3. Number of units taken by 16 students: 8, 5, 0, 5, 6, 9, 14, 6, 10, 8, 4, 7, 6, 6, 15, 8
Module 9 Constructing boxplots Homework
These are the same data sets from Module 9 Quartiles and IQR - Activity 1 where range Q1, Q2 (median), Q3, and IQR were already calculated. Use additional paper as needed.
For each of the data sets below find:
a. Find the mean (round to tenths if needed). Then write a sentence in context describing the meaning of this for the data.
b. Find the median. Then write a sentence in context describing the meaning of this for the data.
c. Find the range. Then write a sentence in context describing the meaning of this for the data.
d. Find the 5 point summary (min, [pic]).
e. Find the IQR.
f. Compute the fences to identify any possible outliers.
g. Draw the boxplot. Then verify your results using Statcrunch.
h. What is the shape of the boxplot?
i. Write two sentences in context describing the meaning of the IQR for this data. (Hint: Use percentages for one sentence and ‘typical’ for the second sentence.)
1. The following data represents the ages of sixteen randomly selected people who attended a rated G movie.
[pic]
2. The following data represents the inches of rain per year from ten randomly selected cities.
[pic]
3. The following data represents the test scores of eleven randomly selected students.
31, 34, 41, 52, 68, 71, 79, 83, 88, 90, 103, 153
4. The following data represents the weights of nine randomly selected team members from a high school wrestling team.
130, 150, 152, 154, 155, 157, 159, 163, 164, 165, 185
Module 10 Exploring Variability about the Mean
Activity 1: Discover ADM
We will now be looking at a measure of spread from the mean. One measure of spread is often called the Average Distance from the Mean (or ADM for short). Look at the following data sets.
SET A: 2 , 3 , 5 , 5 , 6 , 8 , 10 , 10 , 11 , 13 , 15
SET B: 4 , 4 , 5 , 6 , 6 , 15 , 23 , 24 , 24 , 29
1. Find the mean for each set.
2. Estimate the average distance of the numbers from the mean.
3. Try to find the exact value of the average distance of the numbers from the mean? (This is an actual number and not an estimation.)
4. Compare your computed ADM with what you estimated. How close did you get?
5. Can you recommend a procedure for finding the ADM?
Activity 2 Average Distance from the Mean
1. Calculate the Mean and the ADM for each of the following three data sets. Then answer the following questions.
Set A: 13 , 11 , 4 , 21 , 15 , 8 , 19 , 17
Set B: 11, 8, 12 , 10 , 9 , 13 , 14 , 10
Set C: 19 , 15 , 21 , 23 , 20 , 16 , 17 , 21
a) Which data set had the highest center?
b) Which data set had the lowest center?
c) Which data set has the most spread?
d) Which data set has the least spread?
2. For each data set (A,B and C) above, use the mean and ADM to give an average value for the data set and two values that typical numbers in the data set fall in between.
3. Find a data set with at least 7 different numbers that has a mean of 20 and an ADM of 5.
Module 10 ADM and Standard Deviation
For each of the following sets of data use Statcrunch to:
a) Find the Mean (round to one decimal place if needed) (write a full sentence about what this means for the data)
b) Find the Standard Deviation (write a full sentence about what this means for the data)
c) Find the Quartiles (write a full sentence about what this means for the data)
d) Using the mean, the typical values fall between what two numbers
e) Using the median, the typical values fall between what two numbers
1. Number of hours a student has slept at night during a week: 5, 7, 8, 6, 8, 10, 9
2. Ages of 10 people in a room: 27, 20, 12, 14, 25, 23, 26, 27, 22, 47
3. Number of years 10 employees have worked at a company: 16, 24, 14, 36, 20, 14, 15, 17, 17, 18
4. Shoe sizes of 16 fourth graders: 8, 5, 7, 5, 6, 9, 9, 6, 10, 8, 4, 7, 6, 6, 13, 8
Unit 2 & 3 Homework: How to Write A Summary Analysis of a Distribution
This is a sample structure for your summary that describes the distribution of a quantitative variable.
1. Begin with a statement that describes the data set.
Example: “The data set consists of the reported ages of 285 students who are enrolled in Math 137 during Fall 2015.”
2. Discuss the shape of the distribution. Consider whether the outlier is valid, possibly skewing the graph to the left or right, or if the data point is invalid (for example, if a person’s age in years is 1985, the student probably misread the question, and the value would be excluded). Please discuss both situations.
Example: “Excluding the invalid data point of 1985 years from the distribution, the graph will be symmetric.” “Assuming that the outliers are not omitted from the data set, the graph will be skewed to the right.”
3. Discuss the center of the distribution. You may use one representative value or a small range of values.
Example: “The center of the distribution is 25 years” or “The center of the distribution is between 24 and 25 years.”
4. Discuss the spread of the distribution. Address the overall range and the range of typical values.
Example: “The minimum age is 15 years and maximum age is 70 years, resulting in an overall range of 55 years. The range of typical values is between 19 and 27 years.
5. Discuss the outlier(s) of the distribution.
Example: “The data contains one outlier of 1985 years.”
6. Write a conclusion or answer the question, if there is one.
Example: “Based on the above information, most of the students (85%) enrolled in Math 137 during Fall 2015 appear to be between the ages of 22 and 26 years old.
As you gain more experience in writing paragraphs and calculating percentages, consider incorporating more calculations (focusing your attention on specific bins of the histogram) to strengthen your discussion.
IMPORTANT:
Do not write statements that overgeneralize and draw conclusions for a population instead of the sample (the data you analyzed). Quote the statistics provided. I do not want definitions of the statistical terms.
Assignment due next class meeting:
A researcher is interested in comparing the life expectancy of males and females in the United States. The following box plots and sample statistics describe the life expectancy of the two genders. The data was collected in 1999-2001 by state. Write a short essay analyzing and comparing the two data sets. Include the shapes, any outliers and the best measures of center and spread. Make sure to interpret only the best measure of center (mean or median) and spread (SD or IQR) in the context of this problem and quote the actual statistics. In your conclusion state what you found when you compared the life expectancies between the two genders. Include in your conclusion how this information might be useful in the real world and who would be interested in this information.
[pic]
JOURNAL 2 (To be completed after Exam 1 has been returned)
Please review your Exam 1 and answer the following questions in a paragraph format on a separate sheet of paper. Please be honest. I’m not judging you. It is for you to reflect on your progress.
Paragraph 1
Preparing for Exam 1:
• Approximately how many times have you missed class or been late for class?
• At the end of each class, do you understand the main points of the class activities or labs that we completed?
• Have you been regularly completing the homework assignments? If not, why not?
• If you didn’t understand a topic or concept, did you ask the instructor, go to the TLC Lab, or ask someone else for assistance?
• What did you do to prepare for this exam, and how long did you spend studying for this exam? Did you review the answers to the practice problems posted on my website?
• Before you entered the classroom on exam day, did you feel prepared for the exam?
• Do you read the OLI assignments before the lecture day?
• When working on classwork, do you stay on task in class or do you get distracted by other off topic things (your phone for example)?
• Did you spent all the lecture period on review day going over the questions with others and the instructor or did you find yourself socializing more?
• When working in groups, do you tend to get the answers from your teammates without asking for explanations (or do you try to figure it out on your own and with help from the group when you need it)?
Paragraph 2
Preparing for Exam 2
• What is your plan to prepare for Exam 2? Will you use any different strategies? If so, please describe them.
• If you scored below 60%, what will you do to improve your overall grade in the class? For example did you turn in your homework on time when it was due?
• Recall the Dweck Mindset article (brainology) in terms of test scores. How does a person with a fixed minset react to a low test score? How does a person with a growth mindset react to a low test grade? When you got your test back, how did you react (with a fixed or growth mindset)?
Module 11 Introduction: Class Example
Identify the explanatory and response variables in the following:
1. How the price of a package of meat is related to the weight of the package?
a. Explanatory:_____________________________
b. Response:_______________________________
2. Is the distance you walk related to the calories you burn?
a. Explanatory:_____________________________
b. Response:_______________________________
There are four things we want to know when looking at scatterplots:
1. Direction (positive/negative)
2. Form (linear/non-linear)
3. Strength (measured by r)
4. Outliers
[pic] [pic]
***Let us watch Freakonomics on ice cream and polio ***
Association [pic]Causation!!!
Example:
Think about possible statistical relationships between ice cream consumption and number of drowning deaths for a given period. These two variables have a positive, and potentially high correlation with each other.
Does that mean they are related?_______________ What could be the lurking variable, what could be something that is connected to both?
Use this diagram to describe the lurking variable:
[pic] Countries with fewer TVs have lower life expectancy. So, does television cause longer lives?
Module 11: Writing Linear Equations of the form y = b + mx
To write the linear equation for a line, two ordered points are needed.
Example
Given (2, 4) and (4, 9) write the linear equation.
Step 1. First find the slope, m
m = [pic] = [pic] always write the decimal form for our models
Step 2. Put in point slope form: Using one of the two points provided, say (2, 4):
[pic]
[pic] and solve for x
[pic]
[pic]
Or
[pic]
Alternate Method Step 2. Find the y-intercept, “b”
by using one of the ordered pairs say (2,4):
y = b + mx becomes
4 = b + 2.5(2)
4 = b + 5
-1 = b
Put m and b into the model y = b + mx
final answer: y = -1 +2.5x
*Notice that you might have learned that y = mx + b in algebra but in statistics it is customary to write it in the reverse order y = b + mx. They are the same.
Module 11: Linear Regression Activity 1: Lines of Best Fit
Part 1:
1. When a tree was 14 years old, it measured 67 feet. When it was 24 years old, it measured 80 feet. The ordered pairs (24, 80) and (14, 67) represent this data. Graph these two ordered pairs on one of the grids (#1) of the graph paper (next page). Choose an appropriate scale for your graph (everyone in the group should use the same scale). Draw a line (using a straight edge) through the two points.
2. Write a linear equation for this line in y = b + mx form. Everyone should do this on their own paper. Verify with all group members that the same equation was obtained.
3. A third measurement is given as the following. When the tree was 20 years old, it measured 69 feet. Plot the ordered pairs (24, 80), (14, 67), and (20, 69) on a new grid (#2). Using a straight edge (or stand of pasta), draw a line of what you think is the best fit for these three points once you discuss it as a group.
4. Find the linear equation for this second line in y = b + mx form (use a separate paper if needed). Discuss the process as a group so that everyone gets the same linear equation (but each person should do this on their own paper). Write your equation on the poster paper provided but don’t graph it.
The instructor will assign you to graph a linear equation from another group.
5. Once you receive another groups’ equation, graph the equation on the poster.
Part 2: Answer the following questions after your poster paper has been returned to your group.
6. a) According to your equation: When the tree was first planted, the height was _________.
b) According to the best fit line drawn by the other group: When the tree was first planted, the height was _________.
7. a) According to your equation: For every year the tree ages, the tree will grow an additional ________ feet.
b) According to the best fit line drawn by the other group (Use the graph to estimate this as best as you can.): For every year the tree ages, the tree will grow an additional ________ feet.
8. Does the line on your poster graphed from the other group look the same as the line of best fit that was drawn on your paper? If not, what do you think went wrong?
9. In the next activities we will investigate how we can determine a better linear model for scattered data.
[pic][pic]
Extra graph in case no poster is provided (or for errors).
[pic][pic]
Module 11 Linear Activity 2: Lines of Best Fit
An eighth grade class was investigating how high a golf ball bounces when it is dropped from different heights. Students were given eight set heights to drop the ball from. Then they dropped a golf ball from each of those heights and measured how high the ball bounced back up. Below is the scatterplot they made of their data.
[pic]
Draw the line of best fit after you discuss with your group.
a) What criteria did your group use to decide the line of best fit?
b) Use your drawn line (do not find the equation, just use the graph) to predict the height the golf ball will bounce to if it is dropped from 60 centimeters. Is your prediction reasonable?
Module 11 Activity 4
Determine if the following would have a positive correlation, negative correlation or no correlation. Think of it this way: The more INPUT, does the OUTPUT increase or decrease?
1. The distance your drive in town (input), and the number of traffic lights you go through (output).
1. Mr. Hat drinks caffeinated coffee two hours before his bed time. The number of cups of coffee he drinks (input), the hours he sleeps (output).
3. Jimmy is visiting the Bahamas. The time he spends snorkeling (input), the tropical fish he spies (output).
4. Ayesha is purchasing a plane ticket. The days between the purchase date and the departure date (input), the money she will spend on airfare (output).
5. Ice cream sales and temperature
6. Hot chocolate sales and temperature
7. SAT scores and college achievement
8. Come up with one POSITIVE and one NEGATIVE correlation.
a. POSITIVE
b. NEGATIVE
Module 11 Activity 5
PLEASE USE COMPLETE SENTENCES
for your responses.
1. The scatterplot shows
x = internet users
per 1,000 people
y = life expectancy
(years)
for the 20 countries with the largest population for 2009.
(World Almanac Book of Facts, 2009)
Describe what the data point (2, 62.5) tells about Bangladesh.
2. In this group of 20 countries does an increase in the density of internet users (i.e., the number of internet users per 1,000 people) tend to be associated with an increase or a decrease in life expectancy?
3. The correlation coefficient is 0.62. The value of the correlation coefficient indicates a fairly strong positive linear relationship. Based on this observation, someone might suggest that an easy way to increase a country’s life expectancy would be to get more people online. Do you think this is a reasonable conclusion? Why or why not?
It is easy and fun to construct silly examples of correlations that do not result from causal connections. Here are some examples from John Allen Paulos, a mathematics professor at Temple University who is well known for his popular books on mathematical literacy.
4. Read this excerpt from A Mathematician Reads the Newspaper[1] by Paulos. (on the next page)
Ex. 1: A more elementary widespread confusion is that between correlation and causation. Studies have shown repeatedly, for example, that children with longer arms reason better than those with shorter arms, but there is no causal connection here. Children with longer arms reason better because they’re older!
Ex 2: Consider a headline that invites us to infer a causal connection: BOTTLED WATER LINKED TO HEALTHIER BABIES. Without further evidence, this invitation should be refused, since affluent parents are more likely both to drink bottled water and to have healthy children; they have the stability and wherewithal to offer good food, clothing, shelter, and amenities.
Making a practice of questioning correlations when reading about “links” between two variables is a good statistical habit.
Identify the explanatory, response, and confounding (lurking) variables in Paulos’ examples.
Example 1
Explanatory variable:
Response variable:
Confounding variable:
Example 2
Explanatory variable:
Response variable:
Confounding variable:
5. Paulos also writes a column for called Who’s Counting? In his February 1, 2001, column, Paulos discusses the idea that correlation does not imply causation. He points out that the consumption of hot chocolate is negatively correlated with crime rate. Obviously, drinking more hot chocolate does not lower the crime rate.
For this situation assume that the data describe large cities in the United States.
A What is the explanatory variable?
B What is the response variable?
C Identify a plausible confounding variable in this scenario
6. Describe a scenario with two quantitative variables that are probably highly correlated due to a third confounding variable
Module 11 Activity 6
Introduction to the Correlation Coefficient and Its Properties
1. Which of the eight graphs on the following page shows a positive association between x and y, a negative association between x and y, or no association between x and y?
Scatterplot 1: __________ Scatterplot 2:___________
Scatterplot 3: __________ Scatterplot 4: __________
Scatterplot 5: __________ Scatterplot 6:___________
Scatterplot 7: __________ Scatterplot 8: __________
|[pic] |[pic] |
|Scatterplot 1 |Scatterplot 2 |
|[pic] |[pic] |
|Scatterplot 3 |Scatterplot 4 |
|[pic] |[pic] |
|Scatterplot 5 |Scatterplot 6 |
|[pic] |[pic] |
|Scatterplot 7 |Scatterplot 8 |
2. Use Statcrunch to find the correlation coefficient ([pic]) for each of the above scatterplots. Use the data given in the tables (on the next page). Divide the work among pairs. Then write the value of r that you find next to the correlation on your paper. [Copy and paste the data onto Statcrunch if you have access to it on the internet (Control V)]. On Statcrunch go to Stat-Regression-simple linear-select X variable-select Y variable-Compute. Record the R value (correlation coefficient).
Scatterplot 1: __________ Scatterplot 2:___________
Scatterplot 3: __________ Scatterplot 4: __________
Scatterplot 5: __________ Scatterplot 6:___________
Scatterplot 7: __________ Scatterplot 8: __________
Data for Calculating the Correlation Coefficient1
|x1 |y1 |x2 |y2 |x3 |y3 |x4 |y4 |
|150 |60 |50.00 |160.0 |150 |10 |150 |50 |
|155 |72 |50.25 |160.5 |155 |75 |155 |90 |
|160 |94 |50.50 |161.0 |160 |10 |160 |36 |
|160 |80 |50.75 |161.5 |165 |50 |160 |70 |
|165 |82 |52.00 |164.0 |170 |140 |165 |5 |
|165 |90 |54.00 |168.0 |175 |70 |165 |130 |
|170 |97 |54.25 |168.5 |180 |120 |170 |39 |
|170 |110 |55.00 |170.0 |165 |120 |170 |80 |
|175 |112 | | |170 |65 |175 |40 |
|180 |119 | | |160 |120 |180 |80 |
| | | | |170 |100 | | |
| | | | |160 |50 | | |
| | | | | | | | |
|x5 |y5 |x6 |y6 |x7 |y7 |x8 |y8 |
|20.8 |44 |150 |180 |150 |250 |35 |65 |
|21.4 |25 |155 |150 |155 |225 |20 |50 |
|21.2 |71 |160 |160 |160 |200 |30 |60 |
|24.2 |25 |160 |80 |165 |175 |40 |55 |
|24.7 |37 |165 |80 |170 |150 |23 |47 |
|22.5 |37 |165 |112 |175 |125 |30 |53 |
| | |170 |38 |180 |100 |18 |43 |
| | |170 |5 | | |28 |52 |
| | |175 |50 | | | | |
| | |180 |10 | | | | |
1Scatterplot 1 is x1 versus y1, Scatterplot 2 is x2 versus y2, and so on.
3. Now look for patterns by comparing scatterplots and [pic]-values. How does the value of [pic] seem to be related to the patterns you see in the scatterplots?
4. Answer the following
A. What do you think the correlation coefficient ([pic]) measures?
B. Is there a largest possible value for [pic], or can it have larger and larger values without limit? What makes you think so?
C. Is there a smallest possible value for [pic], or can it have smaller and smaller values without limit? What makes you think so?
5. Go to Correlations/
You will now practice guessing correlations for different scatterplots.
Match the values of the correlation coefficient with the corresponding scatterplot using what you know about strength and direction of linear relationships. Click answers to check your answer. Click new plots for a new set of scatterplots. The applet keeps a running count of how many correct matches you have made. Continue until you have at least 25 correct matches.
Module 11 Scatterplots, Linear Relationships and Correlation
1. Match each description (A, B, and C), of a set of measurements, to a scatterplot. Briefly describe your reasoning. Then describe what a dot represents in each graph.
[pic] [pic][pic]
a. [pic]= average outdoor temperature on a winter day and [pic] = heating costs for a residence on the day
What does a dot represent?
b. [pic] = height of an adult (inches) and [pic]= shoe size for the adult
What does a dot represent?
c. [pic]= height of a teenager (inches) and [pic] = score on an intelligence test for the teenager
What does a dot represent?
2. Researchers gathered data about the amount of fat, sugar, and carbohydrates in 22 fast food hamburgers. They gathered the information from fast food companies’ websites. To keep their measurements consistent, all data was described in grams. Using the companies’ websites, the researchers also identified the number of calories for each hamburger. The researchers wanted to know if the amount of calories in the hamburgers depended on how much of each ingredient (fat, sugar and carbohydrates) was in the burger. That is, the researchers wanted to know whether there was a relationship between the amount of fat, sugar, and carbohydrates and the amount of calories in a hamburger. A line has been added to each graph to help you see the patterns more clearly.
a. About how many calories would you predict for a burger that has 20 grams of fat?
b. About how many calories would you predict for a hamburger that has 40 grams of carbohydrates?
c. Which prediction is likely to be more accurate? Why do you think this?
d. Which ingredient has the weakest impact on calories? Why do you think this?
e. What does the idea of strength tell you about whether an ingredient is a good predictor of calories?
f. What is the direction of the fat/calories graph? What does the direction of the line tell you about the association between the amount of fat and the calories in fast food hamburgers?
3. Suppose you gathered the following information from students at a local high school:
▪ GPA (grade point average),
▪ Average weekly hours spent working at a job,
▪ Average weekly hours spent doing homework,
▪ Average hours of sleep a night,
▪ Hourly wage,
▪ Height,
▪ Weight,
▪ Length of the left foot,
▪ Age of the oldest child in the student’s immediate family,
▪ Number of children in the student’s immediate family,
▪ Gender,
▪ Race, and
▪ Age.
From this list of variables, choose:
i Two variables that you think will show a positive linear association.
ii Two variables you think will show a negative linear association.
iii Two variables you think will not show an association in a scatterplot.
You may use the same variable for more than one comparison.
Explain why you chose the two-variable pairs. What was your reasoning for each pair?
MODULE 11 Scatterplots, Linear Relationships and Correlation (2)
1 DESCRIPTIONS “A” AND “B”, BELOW DESCRIBE A SET OF MEASUREMENTS IN A SCATTERPLOT. THE EXPLANATORY VARIABLE (X) IS REPRESENTED BY THE HORIZONTAL AXIS AND THE RESPONSE VARIABLE (Y) IS REPRESENTED BY THE VERTICAL AXIS. MATCH EACH DESCRIPTION TO A SCATTERPLOT, AND BRIEFLY EXPLAIN YOUR REASONING.
Scatterplot 1 Scatterplot 2
A [pic] = city miles per gallons and [pic] = highway miles per gallon for 10 cars
What does a dot represent?
B [pic]= sodium (milligrams/serving) and [pic] = Consumer Reports quality rating for 10 salted peanut butters
What does a dot represent?
2 These scatterplots show body measurements for 34 physically active adults. Match each description (A, B, and C) to a scatterplot. Briefly explain your reasoning.
Scatterplot 1 Scatterplot 2 Scatterplot 3
[pic]
A [pic]= forearm girth (centimeters), [pic]= bicep girth (cm). The forearm is the part of the arm between the elbow and wrist. The bicep is the part of the arm between the shoulder and elbow.
What does a dot represent?
B [pic] = calf girth (cm), [pic] = bicep girth (cm). The calf is the part of the leg below the knee. (Girth is the measurement around a body part.)
What does a dot represent?
C [pic]= age (years), [pic] = bicep girth (cm)
What does a dot represent?
3 Match each description, A to F, to a scatterplot. Briefly explain your reasoning.
Scatterplot 1 Scatterplot 2 Scatterplot 3
Scatterplot 4 Scatterplot 5 Scatterplot 6
A [pic] = month number (January = 1) and [pic] = rainfall (inches) in Napa, California. Napa has several months of drought each summer.
What does each dot represent?
B [pic] = month number (January = 1) and [pic] = average temperature in Boston, Massachusetts. Boston has cold winters and hot summers.
What does each dot represent?
C [pic]= year (from 1970) in five-year increments and [pic]= Medicare expenditures ($). The yearly increase in Medicare costs has been getting bigger over time.
What does each dot represent?
D [pic] = average temperature (°C) each month in San Francisco, California and [pic] = average temperature (°F) each month in San Francisco, California.
What does each dot represent?
E [pic]= chest girth (cm) and [pic] = shoulder girth (cm) for a sample of men.
What does each dot represent?
F [pic] = engine displacement (in liters) and [pic] = city miles per gallon for a sample of cars. Engine displacement is roughly a measurement of the size of the engine. Larger engines tend to use more gas.
What does each dot represent?
Module 12 Introduction: Class Example
[pic]
1. Using the linear equation above, find the predicted max distance each age can read:
➢ Age 57
➢ Age 8
2. When can we not predict? ____________________________________________________
3. What is the slope? _________ Interpret the meaning of the slope in context to the problem.
4. What is the y- intercept? ______________ Interpret the meaning of the y-intercept in context to the problem. Does it have meaning here?
5. Does the older you get, the further you can read?
6. For every year you get older, how much less can we predict you will be able to read?
7. Can we make a prediction for Betty, if she is 93 year old?
Mod 12 Interpreting Slope and Y-intercept
For each of the following, you are given two variables and a linear equation relating those variables.
a) Write a sentence interpreting the y-intercept for given equation (include the units). Remember, this is a prediction equation and the y-intercept doesn’t always make sense.
b) Write a sentence interpreting the slope for the given equation (include the units).
1. y = number of owls in a town x = number of barns in the town
Owls =18 + 5(Barns)
2. y = number of cars parked on a street x = number of houses on the street
Cars = 12 + 2 (houses)
3. y = Cost in dollars for toner x = number of pages printed
Cost = 20 + .03(page)
4. y = Cost of taking a cab x =miles driven by the cab
Cost = 1.20(miles) + 10
5. y = Gallons of milk that cows on a farm produce per day x = Acres of land available on the farm
Gallons of milk = 1200(acres of grass) + 1000
6. y = the height in inches of a male student in class x = the height in inches of his father
Son’s height = 60 + .02(father’s height)
7. y = A company’s monthly cost of electricity in thousands of dollars x = kilowatts of electricity used
Electricity bill = 0.003(kilowatt) + 4
Mod 12 Linear Equation Word Problems
1. Suppose that the water level of a river is 34 feet and that it is receding at a rate of 0.5 foot per day. Write an equation for the water level, L, after d days. In how many days will the water level be 26 feet?
2. For babysitting, Nicole charges a flat fee of $3, plus $5 per hour. Write an equation for the cost, C, after h hours of babysitting. What do you think the slope and the y-intercept represent? How much money will she make if she baby-sits 5 hours?
3. In order to “curve” a set of test scores, a teacher uses the equation y = 2.5x + 10, where y is the curved test score and x is the number of problems answered correctly. Find the test score of a student who answers 32 problems correctly. Explain what the slope and the y-intercept mean in the equation.
4. A plumber charges $25 for a service call plus $50 per hour of service. Write an equation in slope-intercept form for the cost, C, after h hours of service. What is a reasonable domain for this situation?
5. Rufus collected 100 pounds of aluminum cans to recycle. He plans to collect an additional 25 pounds each week. Write and graph the equation for the total pounds, P, of aluminum cans after w weeks. What does the slope and y-intercept represent? How long will it take Rufus to collect 400 pounds of cans?
6. A canoe rental service charges a $20 transportation fee and $30 dollars an hour to rent a canoe. Write and graph an equation representing the cost, y, of renting a canoe for x hours. What is a reasonable domain for this situation?
7. An attorney charges a fixed fee on $250 for an initial meeting and $150 per hour for all hours worked after that. Write an equation in slope-intercept form. Find the charge for 26 hours of work.
8. A water tank already contains 55 gallons of water when Baxter begins to fill it. Water flows into the tank at a rate of 8 gallons per minute. Write a linear equation to model this situation. Find the volume of water in the tank 25 minutes after Baxter begins filling the tank.
9. A video rental store charges a $20 membership fee and $2.50 for each video rented. Write and graph a linear equation to model this situation. If 15 videos are rented, what is the revenue? If a new member paid the store $67.50 in the last 3 months, how many videos were rented?
10. Casey has a small business making dessert baskets. She estimates that her fixed weekly costs for rent and electricity are $200. The ingredients for one dessert basket cost $2.50. If Casey made 40 baskets this past week, what were her total weekly costs? Her total costs for the week before were $562.50. How many dessert baskets did she make the week before?
11. Tim buys a snow thrower for $1200. For tax purposes, he declares a depreciation (loss of value) of $200 per year. Let y be the declared value of the snow thrower after x years.
a) What is the slope of the line that models this depreciation?
b) What is the y-intercept of the line.
a) Write a linear equation in slope-intercept form to model the value of the snow thrower over time.
b) What is a reasonable domain for this function?
c) Find the value of the snow thrower after 4.5 years.
Module 12 Homework: Slope and y-intercept
Interpret the meaning of the slope and the y-intercept in context for each of the following in complete sentences for the following examples.
1. The cost of hiring a private tutor is $20 plus an additional $4 per hour. The equation representing this information is given by C= 20+4h.
a. Slope: (m= 4, interpret this in context)
Sentence:
b. Y-intercept: (b=20, interpret this in context)
Sentence:
2. Max bought a car for $25,000. It will depreciate by $3000 per year. The equation representing this is : V= 25000-3000t.
a. Slope = ________
Sentence:
b. Y-intercept = _______
Sentence:
3. A tree is planted with an initial height of 12 feet. It will grow about 2 feet per year. The equation representing this information is given by H=12+2t
a. Slope = ________
Sentence:
b. Y-intercept = _______
Sentence:
Module 12 Class Example: Finding the Least Squares Regression Line
We want lines of the form: y = b + mx
We can also use descriptive statistics from Statcrunch to calculate slope and y intercept.
[pic]: is standard deviation of all the y values which measures the variability of the y coordinates
[pic]: is standard deviation of all the x values which measures the variability of the x coordinates
The r value will place the negative or positive sign on the slope.
To find b, the y-intercept, we need a point to plug into y = b + mx. The point that is chosen is [pic]which represents the mean y value and mean x value.
Using the following computer generated statistics, we can find the least squares regression line. Computer technology can give us the equation directly as well.
[pic]
Module 12 Finding Equations of Regression Lines Using Statcrunch
To access the men’s health and bear data:
Open Statcrunch. Select: explore – data – under ‘browse al’ type: Men’s health and bear data – select to load the data.
[pic]
[pic]
Module 12 Fitting a Line
The Following ordered pair data describes food trash, paper trash, plastic trash and total trash in tons. Use the following information to find the equation of the regression line that best fits the data. Then graph the line on the scatterplot. How well does the line fit the data? Use the following formulas
[pic] [pic] [pic]
1)
2)
3)
Module 13 Standard Error and Residual Plots
The relationship between height and arm-span measurement (in cm) was investigated by a group of students (in 2010). Forty subjects were measured and the data was analyzed.
Computer technology gave the following results:
Least squares regression equation: Height = 7.333. + 0.949*(Arm Span)
r = 0.788 [pic]= 4.8 cm
The domain of the explanatory variable (arm span) was: 145 cm to 180 cm.
Answer the following questions. . Make sure to use appropriate units and complete sentences.
1. Explain the standard error [pic] in context.
2. One of the subjects (Trevor) had an arm span of 176.5 cm. Use the linear regression equation to predict his height.
3. Trevor’s actual height was 169.5 cm. Calculate the residual (error). Explain in context using a complete sentence.
4. The standard error is shown to be 4.8 cm. What is the difference between this error and the error found in problem 3 (from the residual)?
5. What is the response variable?
[pic]
6. Match the scatterplot with the residual plot.
7. Which of the above scatterplots could a linear regression equation be used to make predictions? Explain.
Module 13 Introduction to r squared (Coefficient of Determination): Class Example
Using the Value of [pic]to Assess the Fit of a Linear Model
How do we know if the explanatory variable we used is truly the best predictor of the response variable? Are there other variables that may also be good predictors? Note that the regression line only considers the explanatory variable we chose and not any other possible contributing factors.
Question: What proportion of the variation in the response variable does our linear regression model explain?
Answer: The value of [pic] tells us this:
[pic]
Example: The following examines the relationship between a man’s weight (in pounds) and waist size (in cm) where r = 0.889. There is variation between what the linear model is predicting at the actual data value. Sometimes there is very little difference and for other data there is a larger difference. What is causing this variation?
[pic]
What is the value of [pic]? _______________What does this tell us?
Men’s weight explains __________% of the total variation in men’s waist sizes. Consequently, ______% of the total variation is due to other causes.
Or
The linear relationship between a man’s weight and his waist size accounts for _____% of the variation in the waist size. Consequently, ______% of the total variation is due to other causes.
Or
The linear regression model explains ____% of the total variation in the (response variable). Consequently, (100-[pic])% of the total variation remains unexplained.
Other contributing factors (causes) can be:______________________
General formats of interpretations of [pic]:
(The explanatory variable) explains ________% of the total variation in (the response variable). Consequently,
(100-[pic])% of the total variation is due to other causes.
The linear relationship between (the explanatory variable) and (the response variable) accounts for _____% of the variation in (the response variable). Therefore, (100-[pic])% of the total variation is due to other causes.
The linear regression model explains ____% of the total variation in the (response variable). Consequently,
(100-[pic])% of the total variation remains unexplained.
Module 13 Finding and Interpreting r-squared
Use the given graphs and r-values to complete the following. Find the value of r-squared and write a sentence interpreting r-squared percentage in the context of data. Be sure to include the appropriate units. Now find possible other variables that may also account for the variability in y. What does this imply about making causal statements between the x variable and the y variable?
[pic] [pic]
[pic] [pic]
Module 13 Finding and Interpreting r-squared Activity 2
For each of the following, two variables and a correlation coefficient is given.
Write two sentences interpreting the value of r2. Try to be specific about other contributing factors.
1. y = number of owls in a town x = number of barns in the town r = .65
2. y = number of cars parked on a street x = number of houses on the street r = .72
3. y = Cost in dollars for toner x = number of pages printed r = .92
4. y = number of pixels on a computer screen x = the width of the screen r = .96
5. y = Gallons of milk that cows on a farm produce per day, x = Acres of land available on the farm
r = . 55
6. y = the height in inches of male students in class x = the height in inches of their fathers
r = .82
7. y = A company’s monthly cost of electricity in thousands of dollars x = kilowatts of electricity used
r = .99
Module 13 Class Example: SSE and Residual Plots
Sum of Squared Errors (SSE)
[pic]
[pic]
[pic]
Residuals
To create a residual plot, we will take the residuals and plot these errors as distances from a base line described by the explanatory or x-variable.
[pic]
[pic]
Example: A
[pic]
Example: B
[pic]
Module 12 Review Homework
Daughter and mother height LAMC students Spring 2016
Simple linear regression results:
Dependent Variable: daughter's height
Independent Variable: mother's height
daughter's height = 3.5775438 + 0.97379782* mother's height
Sample size: 25
R (correlation coefficient) = 0.94821641
R-sq = 0.89911437
Estimate of error standard deviation: 4.0885799
[pic]
1. Describe the association between a mother’s and daughter’s height.
2. Use the regression line equation (if applicable) to predict a daughter’s height if the mother’s height is 59 inches.
3. Use the regression line equation (if applicable) to predict a daughter’s height if the mother’s height is 88 inches.
4. What is the slope of the regression line? What does the slope mean in this context? (Hint: No need to compute.)
5. What is the y-intercept of the regression line? What does the y-intercept mean in this context? (No need to compute). Does it have meaning in this case?
The following statistics for this scatterplot were found to be:
6. Given the scatterplot and these statistics, how well does the regression line fit this data? How confident could one be in making predictions with the regression line? (Use the value of r to support your answer.)
7. Explain the meaning in context of [pic]with respect to this problem.
8. Are there any outliers? If so, state the ordered pairs that are outliers.
Unit 4: Module 11 – 13 Essay
Essay question: The following Scatterplot and residual plot compare a father’s height (x-axis) to their son’s height
(y-axis) in inches (on the next page).
Using a separate paper (preferably typed) write an essay analyzing the data. Include the following in your essay:
• Write an introduction sentence.
• Write a sentence describing the association between a father and son’s height.
• Describe the direction, form, strength, and potential outliers (if any) for the graph.
• Interpret slope, y-intercept and the standard error.
• Write a sentence describing what the residual plot is showing in context to this data. Does the residual plot suggest that the linear model is or is not appropriate to use in making predictions?
• For what domain of the father’s height can we make predictions for? Use appropriate units in your explanations.
• In your conclusion, write about what the data is suggesting, and why this could be important and who would be interested in this data. How confident would you be in making predictions using the regression line (use and quote the statistics provided below to support your decision)?
Correlation coefficient r = 0.977
Standard error of estimate = 0.9839
[pic]
[pic]
Module 12 and 13: Homework Review
The relationship between height and arm-span measurement (in cm) was investigated by a group of students (in 2010). Forty subjects were measured and the data was analyzed.
Computer technology gave the following results:
Least squares regression equation: Height = 7.333. + 0.949 (Arm Span)
r = 0.788 [pic]= 4.8 cm
The domain of the explanatory variable (arm span) was: 145 cm to 180 cm
Answer the following questions. . Make sure to use appropriate units and complete sentences.
1. Predict the height of a person whose arm span is 163 cm (about 64 inches which is 5’4”).
2. Predict the height of a person whose arm span is 185 cm (about 73 inches which is 6’1”).
3. Explain the slope in context.
4. Explain the y-intercept in context. Does it have meaning in this context?
5. Using the correlation coefficient, is the correlation between arm span and height weak, moderately weak, strong, or moderately strong?
6. Check your answer for problem 2. The answer should have been something like: No predictions can be made since 185 cm is beyond the scope of the data. This would be extrapolation!
[pic][pic]
7. What is the predicted fat content for a BK Broiler chicken sandwich that has 30g of protein?
8. The actual fat content ends up being 32.6g. What is the residual for the BK Broiler chicken sandwich? What does this mean (explain using complete sentences)?
9. The SE is 5 grams of fat. Interpret the meaning in context.
10. Interpret the slope and y-intercept in context to this problem (don’t just write what it came out to be). Don’t forget to also use appropriate units.
■ Slope:
■ y-intercept:
Module 14 Exponential Functions Introduction
[pic]
Exponential growth: Exponential decay:
Module 14 Exponential Regression (1)
1. The number of people living in a city after t years is given by: [pic]
a) How many people were living in the city initially?
b) Is the population growing or decreasing?
c) At what rate?
2. A group of scientists observed a population of birds in a remote area and counted the birds in 2002. After returning every year and counting the birds, they came up with the following formula for the number of birds: [pic].
a) Is the population growing or decreasing?
b) At what rate?
c) How many birds used to live in the area in 2002?
d) Predict the population in 2010.
3. The initial number of bacteria in a sample is 156,000 and it grows at a rate of 14% every day.
a) Write an exponential equation for the number of bacteria after t days.
b) Use your equation to predict the number of bacteria after 4 days.
4. The population in a small city is growing at a rate of 4.5% every year. There are 18,000 people living in the city now.
a) Write an exponential equation for the population after t years
b) Use the equation to predict the population after 5 years.
5. The bone mass in the average person’s legs decreases by 0.24 % every year. A person’s bone mass is currently 5 kg.
a) Write an exponential equation for the mass after t years
b) Use the equation to predict the mass after 10 years.
Module 14 Exponential Regression (2)
1. The number of people living in a city after t years is given by: [pic]
a) How many people were living in the city initially?
b) What is the rate of growth for the city’s population?
2. A group of scientists observed a population of birds in a remote area and counted the birds in 1998. After returning every year and counting the birds, they came up with the following formula for the number of birds: [pic].
a) Is the population growing or decreasing?
b) By what rate?
c) How many birds lived in the area in 1998?
d) Predict the population in 2010.
3. The initial number of bacteria in a sample is 120,000 and it grows at a rate of 12% every day.
a) Write an exponential equation for the number of bacteria after t days.
b) Use your equation to predict the number of bacteria after 4 days.
4. The population in a small city is growing at a rate of 3.2% every year. There are 15,000 people living in the city now.
a. Write an exponential equation for the population after t years
b. Use the equation to predict the population after 8 years.
5. The bone mass in the average person’s legs decreases by .1% every year. A person’s bone mass
is currently 3.4 kg.
a) Write an exponential equation for the mass after t years
b) Use the equation to predict the mass after 12 years.
Module 14 Exponential Functions (3)
Write an exponential equation for a regression model with the following initial amount and rate of change:
|Initial amount |Rate of change |Equation |
|5000 |20% increase | |
|5000 |20% decrease | |
|5000 |2% increase | |
|5000 |2% decrease | |
|5000 |.2% increase | |
|5000 |.2% decrease | |
|5000 |.02% decrease | |
2. Find the percent increase or decrease for each of the following:
a) [pic]
a) [pic]
b) [pic]
c) [pic]
Module 14 Making Predictions with Exponential Functions
The following exponential functions were found to be reasonably good models for data sets. Use the functions and the scope of the data (domain) to make predictions. Remember if the x value is out of the scope of the data, do not make a prediction.
1. [pic] (x value = the number of months since a local trash dump opened up,
y value = number of tons of trash dumped, Domain: [pic])
a) Predict the number of tons of trash dumped in the 6 month after opening.
b) Predict the number of tons of trash dumped after 1 year (12 months).
c) Predict the number of tons of trash dumped after 3 years (36 months).
2. [pic] (x value = percent of pollutants removed from a lake,
y value = cost of cleanup in dollars, Domain: [pic].)
a) Predict the cost to clean up 60 percent of the pollutants in the lake?
b) Predict the cost to clean up 10 percent of the pollutants in the lake?
c) Predict the cost to clean up 90 percent of the pollutants in the lake?
3. [pic] (x value = number of years since 1995, y value = amount in savings account,
Domain: [pic])
a) Predict the amount in the account in 2003 (x = 8)?
b) Predict the amount in the account in 2025 (x = 30)?
c) Predict the amount in the account in 2012 (x = 17)?
4. [pic] (x value = age of bear in months, y value = weight of bear in pounds,
Domain [pic])
a) Predict the weight of a bear that is 5 years old (60 months)?
b) Predict the weight of a bear that is 10 years old (120 months)?
c) Predict the weight of a bear that is 20 years old (240 months)?
Math 137 Module 14 Intro to Quadratic Functions
Definition:
A quadratic function is of the form: [pic] where a, b, and c are real numbers (with a≠0). The graph of a quadratic function is called a parabola.
Example 1 The graph below shows the graph of the quadratic function: [pic]
[pic]
Note that the parabola opens up and has a lowest point at (2, - 3). Since a = 1 which is positive, the graph will open up. The lowest point is called the vertex.
The x-coordinate of the vertex can be found by using the following formula: [pic] . The y-coordinate can then be found by plugging in the x value found.
Example 2 The graph below shows the graph of the quadratic function: [pic]
[pic]
Note that the parabola opens down and has a highest point at its vertex (1, 5). Since a = - 2 is negative, the graph will open down.
Let’s find the vertex using the quadratic function:
Module 14B Quadratic Relations (1)
1. A researcher has collected data on the price of gasoline from 1990 to 2010 and has found that the price in dollars after t years can be predicted using the equation: [pic]
a) According to this model what was the price of gas in1990?
b) Using this model predict the price of gas 1998.
c) Based on the equation, what year had the most expensive gas?
d) How much did gas cost in that year?
2. The number of students attending Los Medanos community college from 2000 to present can be predicted using the equation: [pic].
a) What direction does the parabola open?
b) What year had the lowest enrollment?
c) What was the enrollment in that year?
d) What was the enrollment in 2002?
3. Based on annual data collected by the Center of Disease Control, the number of cases of people infected with flu during any given month between September and December can be modeled using a quadratic equation: [pic] where x = 0 is September 15 and y is thousands of people.
a) If possible, predict the number of flu cases on November 15.
b) If possible, predict the number of flu cases on January 15.
c) What month had the highest number of flu cases? How many people were infected at that time?
4. Match the graph to the equation:
[pic] [pic] [pic] [pic] [pic]
Module 14B Quadratic Relations (2)
1. A researcher has collected data on the price of gasoline from 1995 to 2012 and has found that the price in dollars after t years can be predicted using the equation: [pic]
a) According to this model what was the price of gas in 2002?
b) Using this model predict the price of gas in 1995.
c) Based on the equation, what year had the most expensive gas?
d) How much did gas cost that year?
2. The number of students attending Glendale community college from 1990 to present can be predicted using the equation: [pic]
a) What direction does the parabola open?
b) What year had the lowest enrollment?
c) What was the enrollment that year?
d) What is the prediction for their current enrollment?
3. Based on annual data collected by the Center of Disease Control, the number of cases of people infected with flu during any given month between September and February can be modeled using a quadratic equation: [pic] where x = 0 is August and y is thousands of people.
a) If possible, predict the number of flu cases in October.
b) If possible, predict the number of flu cases in January.
c) What month had the highest number of flu cases? How many people were infected at that time?
4. Match the graph to the equation:
[pic] [pic] [pic] [pic] [pic]
Additional Probability Chapter 1
The Basic Counting Principle
The basic counting principle is the method used to calculate the number of possibilities of events occurring. The events may be independent or dependent. Two events are considered to be independent if the occurrence of one does not affect the possibility of the occurrence of the other. For example, if we were to flip two coins simultaneously the outcome of one coin’s flip would not affect the other coin’s flip. So here we say that the two events are independent. Two events are said to be dependent if the occurrence of one does affect the possibility of the occurrence of the other. For example, if one card is drawn out of a deck of cards and not replaced the outcome of drawing a second card is affected by the drawing of the first since there is now one less cards in the deck.
|Basic Counting Principle |Suppose an event can occur in m different ways and another event can occur in n different ways. There are m |
| |( n ways that both events can occur. |
Example 1) Suppose that the XYZ clothing company has a line of shirts that come in three colors: blue, red and green. They also make the shirts in short sleeves and long sleeves. How many different styles do they make?
The tree diagram below shows all the possible styles that could make:
Notice that there are 6 different possibilities. Now by applying the counting principle to the same problem: there are two possibilities for the sleeve length and 3 possibilities for the color. According to the counting principle there are 2 ( 3 = 6 total possibilities, which are apparent in our diagram.
Example 2) How many different three digit numbers can be formed using the numbers 2, 3 and 5 if each number can be used more than once?
The choices of the numbers for each digit are independent (that is, the choices are not affected by each other). Therefore, the number of choices remains three for each digit.
Digits: First Second Third
Choices: 3 3 3
So, by the counting principle, there are 3 ( 3 ( 3 = 27 different three digit numbers that can be formed by 2, 3 and 5.
Example 3) How many different three digit numbers can be formed using the numbers 2, 3 and 5 if each number can be used only once?
Notice that in this example the events are dependent. That means, the choices for each digit are affected by the previous choices made. If we use a number as the first digit, we may not use it again as the second or third digit. In this case we can choose either 2, 3 or 5 for the first digit, but once we decide on a number, only two of those will still be available for the second digit, and once that choice is made there will be only one remaining number for the last digit.
Digits: First Second Third
Choices: 3 2 1
There are 3 ( 2 ( 1 = 6 different possible three digit numbers that can be made without repeating a number.
Example 4) How many different three letter combinations can be formed using the letters in the word “CARD”, if each letter can be used only once?
Letter: First Second Third
Choices: 4 3 2
There are 4 ( 3 ( 2 = 24 possible combinations.
Example 5) How many of the combinations in example 4 start with a vowel?
Here we only have one choice for our first letter (“A”) , three choices for our second letter (“C”, “R” and “D”) and once one of these letters is chosen, there will be only two choices for the last letter.
Letter: First Second Third
Choices: 1 3 2
There are 1 ( 3 ( 2 = 6 of the combinations start with a vowel.
Factorial Notation:
The factorial notation is used to represent a chain of multiplications where each number in the chain is one less than the previous number. In factorial notation, the product [pic] can be written as 3! (three factorial). This notation is commonly used in probability and statistics textbooks for convenience. For example 5! = 5[pic]4[pic]3[pic]2[pic]1, and 7! = 7[pic]6[pic]5[pic]4[pic]3[pic]2[pic]1.
In general, n factorial or n! = n [pic] (n – 1) [pic] (n – 2) [pic][pic][pic]3 [pic] 2 [pic]1.
For purposes of consistency in our calculations, we define 0! = 1.
Example 6) Evaluate [pic]
[pic]
Example 7) Evaluate [pic]
[pic]
Probability Chapter 1 Practice Problems – The Basic Counting Principle
1. In how many ways can the letters in the word "MATH" be used to make:
a) a two–letter word with no letters repeating?
b) a two–letter word with possible repeating letters?
c) a two–letter word starting with a consonant and ending in a vowel?
2. A pizza parlor offers three different size pizzas, two different types of crust, and five different toppings.
a) How many different one topping pizzas are possible?
b) How many different one topping small pizzas are possible?
3. A California license plate starts with a non–zero digit, followed by three letters and then three numbers. How many different license plate combinations are possible?
4. A certain car model comes in blue, silver, black and red with either automatic or manual transmissions. In addition, a consumer has the option for air conditioning or not and a cassette player or a CD player. How many different choices do the consumers have?
5. In Sally's closet, she has 4 different pairs of shoes, 6 different shirts, and 5 different pairs of pants. How many different outfits can she make?
6. Evaluate the following:
a) 4!
b) [pic]
c) [pic]
d) [pic]
e) [pic]
Probability Chapter 2
Permutations and Combinations
A permutation of objects is an arrangement of these objects in a certain order. To “permute” a set of objects means to arrange them in that order. As we saw in example 4 of the previous chapter, when we calculated the number of possible three letter words that could be formed using the letters in "CARD", we counted every possible permutation of three letters.
In a permutation, the order of the objects is very important. The arrangement of objects in a line is called a linear permutation.
If we want to calculate the number of ways we can choose 4 people from a group of 6 people to stand in a line, we are calculating a linear permutation. Applying the method from Chapter 1, we can find the number of possibilities as 6 ( 5 ( 4 ( 3. Notice that
6 ( 5 ( 4 ( 3 is part of 6!. We can write an equivalent expression in terms of 6!.
[pic]
Note that the denominator is the same as (6 – 4)!.
The number of ways we can arrange 4 items chosen from 6 items in a line is written as P(6,4). In general, P(n,r) is read "n objects chosen r at a time" and is defined as follows:
|P(n,r) or nPr |The number of permutations of n objects chosen r at a time is |
| |[pic] |
Example 1) The CSUN history club which has 25 members is holding elections for a president and a vice–president. In how many possible ways can the club fill these positions?
n = 25 and r = 2
P(25,2) = [pic]
Example 2) In how many ways can a store manager arrange 2 different blue dresses, 5 different green dresses, and 4 different red dresses on a straight rack if the same color dresses have to be displayed together?
We first need to calculate in how many ways the dresses of the same color can be arranged next to each other.
The blue dresses can be arranged in P(2,2) = [pic] different ways.
The green dresses can be arranged in P(5,5) = 120 different ways.
The red dressed can be arranged in P(4,4) = 24 different ways.
In addition to these choices, the store manager can also group the colors in a different order (blue, green, red as opposed to red, blue, green). This is P(3,3) = 6 different ways.
So by the basic counting principle, she has 2 ( 120 ( 24 ( 6 = 34,560 different ways of displaying the dresses.
In example 1 of this chapter, we considered the number of possibilities for choosing a president and vice–president from a group of 25. We used a permutation because order mattered. Suppose Mary and John were elected. There are two situations: either Mary is President and John is Vice–President or John is President and Mary is Vice–President. Now suppose instead that the same club wants to choose two representatives for the homecoming committee. In this case, the people are being chosen without regard to any order. So if Mary and John are chosen, there is no need to consider two situations. Such a selection is called a combination. So in combinations the order does not matter while in permutations the order is important.
We use the notation C(n,r) when choosing r objects from a group of n objects when order does not matter. This is generally read "n choose r".
|C(n,r) or nCr |The number of combinations of n objects chosen r at a time is |
| |[pic] |
Example 3) In how many ways can two members be chosen to represent the CSUN history club (from example 1) on the homecoming committee?
n = 25 and r = 2
C(25,2) = [pic]
Compare this answer with the answer in example 1.
Practice Problems – Probability Chapter 2
1. In how many ways can 6 different books be arranged on a shelf?
2. From a committee of 9 people, in how many ways can we choose a chairperson, a secretary, and a treasurer?
3. The CSUN Art Gallery has 12 paintings submitted for display. In how many ways can they choose 4 of the paintings to be displayed?
4. In how many ways can a committee of 3 people be chosen from a group of 8 people?
5. A combination lock has 15 positions labeled. It can be unlocked by moving the dial first to the right, then to the left, and then back to the right. How many 3 number combinations are possible if no number is used more than once?
6. An ice cream parlor has 31 flavors of ice cream.
a) How many different triple scoops are possible if no flavor is used more than once?
b) How many different triple scoops on a cone are possible if no flavor is used more than once? (Note – a Vanilla/Chocolate/Strawberry cone is different than a Strawberry/Chocolate/Vanilla cone.)
7. Ten students are to be divided into two groups of 4 and 6. In how many ways can this be done?
8. A catering company offers five different appetizers, 3 different main courses, and 2 different desserts. How many different menus are possible if you are allowed 2 appetizers, 1 main course, and 1 dessert for the meal?
9. A video clerk needs to arrange the new arrival shelf. There are 4 new comedies, 3 new horror movies, and 2 new action films. In how many ways can he arrange the movies on a shelf if he needs to keep them grouped by type?
Probability Chapter 3
Probability is loosely defined as likelihood of the success or failure of obtaining a desired outcome. Knowledge of how to compute probabilities is necessary in many different professional areas, including physical, biological and social sciences, and journalism, among others, and is useful in daily life. Probability theory is often used to make predictions. For example, we can use probability theory to make predictions about the average blood pressure a human being will have at a certain age, the odds of rolling a seven with two fair dice, or the chances of winning the lottery. Such random events cannot be predicted with certainty, but the relative frequency with which they occur in a long series of trials is often remarkably stable. This fact enables us to compute probabilities about such events.
The probability of a favorable outcome is defined as the number of favorable outcomes divided by the number of possible outcomes, where a favorable outcome is the result we are seeking and a possible outcome is any possible result.
|Probability of event E |[pic] |
|P(E) | |
Example 1) What is the probability that a fair coin comes up heads?
In this problem, the favorable outcome is heads on a fair coin. The number of favorable outcomes is 1 (there is only 1 way to get heads on a fair coin). The number of possible outcomes (either heads or tails) is 2. So the probability of obtaining heads is [pic].
Example 2) Find the probability of rolling a two on a fair die.
Here, the favorable outcome is obtaining a two on a fair die. The number of favorable outcomes is 1 (there is only 1 way to get a two on a fair die). The number of possible outcomes is 6. So the probability of obtaining a two on a fair die is [pic].
Example 3) Two fair dice are thrown, find the probability that a total of 7 is rolled.
Let's use the following table to help us determine the different possibilities. The horizontal rows represent the faces of one die and the vertical columns represent the faces of the other die. The body of the table, then, gives all possible outcomes of rolling the two dice.
| |First die |
|Second die | | | | | | | |
| | |2 |3 |4 |5 |6 |7 |
| | |3 |4 |5 |6 |7 |8 |
| | |4 |5 |6 |7 |8 |9 |
| | |5 |6 |7 |8 |9 |10 |
| | |6 |7 |8 |9 |10 |11 |
| | |7 |8 |9 |10 |11 |12 |
Notice that there are six different ways of rolling a 7 and there are 36 possible outcomes given in the body of the table. So the probability of rolling a seven is [pic].
Example 4) In a family of 2 children, what is the probability that both children are girls?
We will denote B for boy and G for girl. There are 4 possible outcomes: BB, BG, GB, GG. Our favorable outcome is GG. There is only 1 way for this to happen. So the probability of two girls is [pic].
Now suppose we are interested in calculating the probability of two different events that occur simultaneously. If the two events are independent, then the probability of both events occurring is the product of the probability of each individual event’s occurrence.
|Probability of Two | |
|Independent Events |If two events, A and B, are independent, then the probability of both events occurring is found by the |
| |following: |
| |P(A and B) = P(A) ( P(B) |
Example 5) A fair die is tossed three times. Find the probability that it comes up 6 every time.
Since the outcomes of each toss of the die are independent events, we can find the probably of three consecutive 6’s by multiplying the individual probabilities: [pic] .
Example 6) A bag contains 18 blue marbles and 6 red marbles. A marble is selected at random, and placed back in the bag, then a second marble is selected. Find the probability of the first marble being red and the second marble being blue.
In this example, the marble is replaced back in the bag, therefore the two events are independent. The probability of a red marble being drawn is [pic] and the probability of a blue marble being drawn is [pic]. So the probability of a red marble and a blue marble is found by: P(Red and Blue) = [pic] .
Practice Problems – Probability Chapter 3
1. Find the probability of drawing a red card from a regular 52-card deck.
2. A wheel has 9 sections numbered 1 through 9. Suppose that you spin the wheel three times. Find the following probabilities:
a) Spinning an even number on the first spin.
b) Spinning a number divisible by 4 on the first spin.
c) Spinning three odd numbers in a row.
d) Spinning a multiple of 3 on the first spin and a multiples of 4 on the second and third spin.
3. Given a deck of cards with 52 cards, a card is drawn. Find the following probabilities:
a) A face card (Jack, Queen or King) being drawn.
b) An Ace being drawn.
c) A 5 of hearts being drawn twice after it has been placed back in the deck and the deck is shuffled before the second drawing.
4. A coin is tossed 4 times. What is the probability of it coming up tails every time?
5. If two dice are thrown,
a) What is the probability of rolling double 6's?
b) What is the probability that the roll totals 9?
Module 15 Contingency Tables Class Example
Students were asked in the spring semester 2016 whether they typically eat breakfast. The results are summarized below.
Contingency table results:
Rows: gender
Columns: eat breakfast
| |No |Yes |Total |
|Female |45 |55 |100 |
|Male |28 |32 |60 |
|Total |73 |87 |160 |
1. What percent of the students are female?
2. What percent of the students are male?
3. What percent of students eat breakfast?
4. What percent of students do not eat breakfast?
5. What percent of female students eat breakfast?
6. What percent of male students eat breakfast?
7. Is there a significant difference between genders eating breakfast?
Module 15 Two Way Tables (1)
Creating a two way table
Directions: Here is some data taken from the medical records department at a local hospital. The data includes age, gender, blood type (A, B, AB, O), Rhesus factor (Rh + or Rh -) and part of the hospital the patient was in (Medical/Surgical, Intensive Care Unit , Same Day Surgery, Emergency Room).
1. Create a two way table that we could use to compare gender to bloodtype.
2. Create a two way table that we could use to compare the part of the hospital the patient was in to the patient’s age (18-35, 36-49, 50-64, 65 or above).
3. Create a two way table that we could use to compare blood type to Rh factor.
|Patient ID# |Age |Gender |Blood Type |Rh Factor |Floor |
|1 |23 |M |A |- |SDS |
|2 |68 |M |O |+ |ER |
|3 |51 |F |AB |+ |Med/Surg |
|4 |74 |M |O |- |ICU |
|5 |49 |F |O |+ |SDS |
|6 |62 |F |O |+ |Med/Surg |
|7 |35 |M |A |+ |SDS |
|8 |46 |F |O |+ |Med/Surg |
|9 |72 |F |O |+ |ER |
|10 |61 |M |B |+ |SDS |
|11 |43 |F |A |- |Med/Surg |
|12 |81 |M |O |+ |ICU |
|13 |65 |M |A |+ |Med/Surg |
|14 |59 |F |O |- |SDS |
|15 |44 |F |B |+ |ICU |
|16 |26 |M |O |+ |ER |
|17 |58 |F |AB |- |ER |
|18 |45 |M |O |+ |SDS |
|19 |55 |M |O |+ |Med/Surg |
|20 |71 |M |A |+ |ER |
Module 15 Two-way Tables (1) continued
Using two way tables to find percentages and probabilities
Directions: Use your two way tables created in Activity 1, to answer the following questions.
1. What percent of the patients were female?
2. What percent of the patients were male?
3. What percent of the patients had an age between 18-35 years old?
4. What percent of the patients were 65 years old or older?
5. What percent of patients had type O blood?
6. What percent of patients had type AB blood?
7. What percent of patients were having a Same Day Surgery?
8. What percent of patients were in Intensive Care?
9. What percent of patients had Rh+ blood?
10. What percent had Rh- blood?
11. Which do you think is more common, Rh+ or Rh-?
12. If a random person was chosen out of the group, what is the probability that they had a blood type of A?
13. If a random person was chosen out of the group, what is the probability that they had a blood type of B?
14. If a random person was chosen out of the group, what is the probability that the person was 36-49 years old?
15. If a random person was chosen out of the group, what is the probability that the person was 50-64 years old?
16. If a random person was chosen out of the group, what is the probability that the person went to the Emergency Room?
17. If a random person was chosen out of the group, what is the probability that the person went to the Medical/Surgical floor?
Module 15 Class Example: Probabilities and Two Way Tables
[pic]
1. What is the probability of choosing a person that is female and Republican?
2. What is the probability of choosing a person that is female or Republican?
3. If a person is male, what is the probability that he is Republican?
4. What is the probability that a person is Republican given that the person is female?
5. What is the probability that a person is Democrat?
6. What is the probability that a male is Democrat?
7. What is the probability that a female is Democrat?
8. Is there a significant difference among genders in selecting Democrat?
Module 15 Two-way Tables (2)
1. It is said the 11% of boys are left handed and 14 % of girls are left handed. Assume that 46 % of children are female. Use a convenient total for the number of children and fill out the following table using this information. Then use the table to answer the following questions.
| |Left Handed |Right Handed |Totals |
|Girl | | | |
|Boy | | | |
|Totals | | | |
a) What is the probability of a right handed child being a boy?
b) What is the probability of a child being right handed and a boy?
c) What is the probability of a boy being right handed?
2. The following table describes the gender and majors of randomly selected students at a local college. Use the table to answer the following questions.
| |Business |English |History |Music |Biology |Math |
|Female |95 |58 |75 |62 |56 |9 |
|Male |102 |64 |59 |53 |78 |14 |
a) Find all the row and column totals for the table.
b) What percent of the students are female History majors?
c) If we randomly select a student, what is the probability that the person is female and a Music major?
d) If we randomly select a student, what is the probability that the person is male and a History major?
f) What is the probability of a Math major being male?
e) What is the probability of a female student being a History major?
g) What is the probability of a student being male and an English major?
Module 15 Creating Tables from Percentages
Creating Tables from Percentage information
Directions: If we only know percent information, sometimes it is helpful to assume that we have 10,000 or 100,000 individuals, and then make a table from the percents. This table could then be used to find more complicated probabilities.
1. It is said the 9.5% of boys are left handed and 12.5% of girls are left handed. Assume that 48% of children are female. Fill out the following table using this information and a convenient number for the total. Then use the table to answer the following question. If a child is right handed, what is the probability that the child is a boy?
| |Left Handed |Right Handed |Totals |
|Girl | | | |
|Boy | | | |
|Totals | | | |
P(Boy | Right Handed) = ???
2. Approximately 24% of the U.S. population smoke cigarettes. Approximately 10% of people who smoke cigarettes will develop lung cancer. If a person has never smoked cigarettes, they have approximately a 0.3% chance of getting lung cancer. Fill out the following table. Then use the table to answer the following question. If a person has lung cancer, what is the probability that they smoke cigarettes?
| |Smokes Cigarettes |Does not Smoke Cigarettes |Totals |
|Gets Lung Cancer | | | |
|Does not get Lung Cancer | | | |
|Totals | | | |
P(Smokes Cigarettes | Has Lung Cancer) = ???
Module 15 Conditional Probability
Conditional Probabilities
Directions: The following table describes the gender and majors of 692 randomly selected students at a local college. Use the table to answer the following questions.
| |Business |English |History |Music |Biology |Math |
|Female |89 |71 |62 |48 |56 |9 |
|Male |112 |58 |59 |53 |62 |13 |
1. Find all the row and column totals for the table.
2. If a person is a business major, what is the probability that the person is female?
3. If a person is male, what is the probability that the person is a biology major?
4. If we randomly select a Music major, what is the probability that the person is male?
5. What is the probability of choosing a person that is an English major, if we are given that the
person is female?
6. What is the probability of choosing a male student if we are given that the person is a Math
major.
7. If a student is a Math or Biology major, what is the probability that the person is female?
8. Of all the history majors, what percent are male?
9. Of all the female students, what percent are English majors?
10. Of all the male students, what percent are Business or History majors?
Module 15 Two-way Tables (3)
1. This data comes from a study of the factors that impact birth weight. Here the variable Visit Doctor indicates whether a woman visited a physician during the first trimester of her pregnancy. The variable Low Weight indicates whether a baby was born weighing less than 2500 grams.
| | | Low Weight | |
|[pic] | |No |Yes |Row Totals |
| | Yes |66 |23 | |
| | No |64 |36 | |
| |Column Totals | | | |
A. Complete the table.
B. Does visiting a doctor during the early stages of pregnancy seem to be associated with a lower incidence of low weight births? Identify the explanatory and response variables, and be sure to use math to support your conclusion and show your work.
C. If a woman does not visit the doctor during the first trimester of her pregnancy, how increased is the likelihood that she will have a low weight baby?
D. What is the probability of a woman visiting a doctor and having low birth weight?
E. What is the probability of a woman that has had a low birth weight having visited a doctor?
2. This table is based on records of accidents compiled by a State Highway Safety and Motor Vehicles Office (the marginal distributions and the lower right-hand corner have been filled in for you). Are people less likely to have a fatal accident if they are wearing a seatbelt? Be sure to clearly identify the explanatory and response variables, use math to support your conclusion, and show your work.
| | |Injury | |
|[pic] | |Nonfatal Injury |Fatal Injury |Row Total |
| |Seat belt |412,368 |510 |412,878 |
| |No seat belt |162,527 |1,601 |164,128 |
| |Column Total |574,895 |2,111 |577,006 |
3. A study in Sweden looked at the impact of playing soccer on the incidence of arthritis of the hip or knee. They gathered information on former elite soccer players, people who played soccer but not at the elite level, and those who never played soccer. Fill in the marginal distributions and the lower right-hand corner. Does this study suggest that playing soccer makes someone more likely to have arthritis of the hip or knee? Be sure to clearly identify the explanatory and response variables, use math to support your conclusion, and show your work.
| | |Soccer Player | |
|[pic] | |Elite |Non-elite |Did not play |Column Totals |
| |Arthritis |10 |9 |24 | |
| |No arthritis |61 |206 |548 | |
| |Row Totals | | | | |
4. Four hundred seventy-eight students in grades 4 through 6 in selected schools in Michigan, were asked the following question.
Which of the following would make you popular among your friends? Rank in order.
Good grades
Having lots of money
Being good at sports
Being handsome or pretty
The table below lists the number of students (by gender) who gave the indicated factor a ranking of 1 (most important factor in making them popular amongst their friends).
| | |Most Important Popularity Factor |
|Gender | |Grades |
|Aspirin |104 |10,933 |
|Placebo |189 |10,845 |
Does this data suggest that taking aspirin reduces the risk of heart attack?
Module 16 Independent and Disjoint Variables
The following two way table gives the genders and majors of 114 randomly selected science students. (Note: None of these students were double majors.)
| |Biology |Chemistry |Physics |
|Female |31 |14 |5 |
|Male |33 |19 |12 |
1. a) Are Biology majors and Physics majors disjoint? Why?
b) Find the probability of a person being a Biology major or a Physics major?
c) Is the answer in part (b) the same as the probability of someone being a Biology major plus the probability of someone being a Physics major?
2. a) Are Chemistry’s majors and males disjoint? Why?
b) Find the probability of a person being a Chemistry major or male?
c) Is the answer in part (b) the same as the probability of a Chemistry major plus the probability of someone being male?
3.
a) Are the Physics majors and females independent of one another or dependent? Why? What probabilities could we use to show that our answer is correct?
a) If we are given a person is female, what is the probability they are a Physics major? Is this the same as the probability that anyone is a physics major?
b) If we are given that a person is a Physics major, what is the probability that they are female? Is that the same as the probability of anyone being female?
c) Find the probability that a student is both female and a Physics major? Is this the same as the probability of someone being female times the probability of someone being a Physics major?
Module16 Rules of Probability Worksheet
1) A striking trend in higher education is that more women than men reach each level of attainment. Here are the counts (in thousands) of earned degrees in the United States in the 2010-2011 academic year (classified by level and gender).
| |Bachelor’s |Master’s |Professional |Doctorate |Total |
|Female |986 |411 |52 |32 |1481 |
|Male |693 |260 |45 |27 |1025 |
|Total |1679 |671 |97 |59 |2506 |
a) If you choose a degree recipient at random, what is the probability that the person you choose is a woman?
b) What is the conditional probability that you choose a woman, given that the person chosen received a doctorate?
c) Are the events “choose a woman” and “choose a doctoral degree recipient” independent? How do you know?
d) What is the probability that a randomly chosen degree recipient is a woman and the degree is a doctorate?
2) Use the College-Degree data from the previous problem to answer the following questions.
a) What is the probability that a randomly chosen degree recipient is a man?
b) What is the conditional probability that the person chosen received a bachelor’s degree, given that he is a man?
c) What is the probability that a randomly chosen degree recipient is a man and the degree conferred is a doctorate?
3) The probability that the team will win the play-off game is .7. What is the probability that the team will not win the play-off game?
4) There are 3 yellow cards, 5 blue cards, and 6 green cards. What is the probability that you pick a yellow or a blue card?
5) A piggy bank consists of 8 quarters, 15 dimes, 20 nickels, and 35 pennies. If you shake one coin out, find the following probabilities.
a) P(nickel) =
b) P(dime or a nickel) =
c) P(quarter) =
d) P(not a penny) =
6) There are purple, orange, yellow, and green gum drops in a bowl. If the probability of picking a purple is 26%, a yellow is 48%, and a green is 12%. What is the probability of picking an orange?
a) Using the probability from #6, is it more likely to pick an orange or purple or a green or yellow? Justify your answer.
b) Give an example of a probability that would be equal to 0.
c) Give an example of a probability that would be equal to 1.
d) Can you have a probability that is [pic]? Explain.
Module 16 Joint Probabilities
Directions: The following table describes the gender and majors of 692 randomly selected students at a local college. Use the table to answer the following questions.
| |Business |English |History |Music |Biology |Math |
|Female |89 |71 |62 |48 |56 |9 |
|Male |112 |58 |59 |53 |62 |13 |
1. Find all the row and column totals for the table.
2. What percent of the students are female Music majors?
3. What percent of the students are male English majors?
4. If we randomly select a student, what is the probability that the person is female and a Math major?
5. If we randomly select a student, what is the probability that the person is male and a Biology major?
6. If we randomly select a student, what is the probability that the person is female or a Business major?
7. If we randomly select a student, what is the probability that the person is male or a History major?
8. If we randomly select a student, what is the probability that the person is female or an English major?
9. What percent of the students are either female or a music major?
10. What percent of the students are either male or a math major?
11. What is the probability of a English major being male?
12. What is the probability of a male student being an English major?
13. What is the probability of a student being male and an English major?
Module 16 Probability Distribution Introduction
1. A coin is tossed two times. Write the sample space.
2. Fill out the probability chart below for getting heads.
|X |0 |1 |2 |
|P(x) | | | |
|[pic] | | | |
|[pic] | | | |
3. What is the probability of getting exactly 1 head?
4. What is the probability of getting at least 1 head?
5. If you flip a coin 200 times, how many times would you expect exactly two heads?
6. If you flip a coin 200 times, how many times would you expect no heads?
7. Find the expected value [pic]for the number of heads. The expected value is the long run mean (average) of heads expected when tossing a coin twice.
8. Find the standard deviation for the number of heads:[pic]
Classwork:
1. A family wishes to have three children.
Write the sample space.
2. Fill out the probability chart below for getting a girl when having three children.
|X | | | | |
|P(X) | | | | |
|[pic] | | | | |
|[pic] | | | | |
3. What is the probability of getting exactly 2 girls when having three children?
4. What is the probability of getting at least 1 girl when having three children?
5. What is the probability of getting no girls when having three children?
6. Find the expected value [pic]for the number of girls when having three children.
7. Find the standard deviation for the number of girls when having three children.
Section 16 Probability Distributions and Expected Values
A biology teacher assigns a large project on plants, but every semester, many of her students turn in the project late. In looking at her records she finds the following totals: Out of 400 total students that turned in a project, 342 turned it in on time, 21 turned it in one day late, 15 turned it in two days late, 11 turned it in three days late, 7 turned it in four days late, and 4 turned it in five days late.
1. Let X represent the random variable describing the number of days the project was late and P(x) represent the probability of a student turning in their project X days late. Fill out the following probability table with all the missing probabilities.
|X |0 |1 |2 |3 |4 |5 |
|P(X) | | | | | | |
| | | | | | | |
2. From your probability distribution in #1, answer the following questions.
a) If a student does turn in their project late, how many days late is it most likely to be? Why?
b) What is the probability that a student turns in their project on time or one day late?
c) What is the probability that a student does not turn in their project on time? The teacher’s current biology class has 36 students. Estimate how many of the 36 students will not turn in their paper on time?
d) Find the expected value for the probability distribution. Write a sentence explaining the meaning of the expected value in this context.
3. A casino in Las Vegas offers the following gambling game. To play you must pay $5. You role a die one time. If you role a 1, 2 or 3, you lose your $5. If you role a 4 or a 5, you win $3. (You get your $5 back plus an additional $3). If you role a 6, you win $7. (You get your $5 back plus an additional $7.)
Let X represent the random variable describing the amount of money won or lost and P(x) represent the probability of winning or losing that money. Fill out the following probability table with all the missing probabilities.
|Money Lost or Won |-$5 |+$3 |+$7 |
|P(x) | | | |
| | | | |
4. From your probability distribution in #3, answer the following questions.
a) If a person plays the dice game, what is the probability that they will win some money?
b) Find the expected value for the probability distribution. Write a sentence explaining the meaning of the expected value in this context.
c) Is this a fair game? Explain how you know?
Section 16 Probability Distributions and Expected Values Activity 2
The following are probability distributions for variable x. Answer the given questions for each:
|x |2 |3 |9 |11 |12 |
|P(x) |0.03 | |0.70 |0.10 |0.04 |
| | | | | | |
| | | | | | |
a. Fill in the blank space (under the 1).
b. Write a sentence explaining the meaning of 0.10 in context to the problem.
c. Find P(x > 2). Write your answer in a sentence interpreting the meaning in context.
d. Find P(x ≤ 1). Write your answer in a sentence interpreting the meaning in context.
e. Find P(x = 0 or x = 3). Write your answer in a sentence interpreting the meaning in context.
f. What is the probability that a household had at least 3 cars?
g. What is the probability that a household had 2 cars?
h. Find the mean (expected value) for the variable x and write the answer in a sentence interpreting the meaning in context.
i. Find the standard deviation for the variable x and write the answer in a sentence interpreting the meaning in context.
2. In the following probability distribution, the random variable x represents the number of activities a parent of a middle school student is involved in.
|x (number of activities) |0 |1 |2 |3 |4 |
|P(X) |0.073 |0.117 |0.258 | |0.230 |
| | | | | | |
| | | | | | |
a. Fill in the blank space (under the 3).
b. Write a sentence explaining the meaning of 0.258 in context to the problem.
c. Find P(x ≥ 1). Write your answer in a sentence interpreting the meaning in context.
d. Find P(x < 2). Write your answer in a sentence interpreting the meaning in context.
e. Find P(x = 3 or x = 4). Write your answer in a sentence interpreting the meaning in context.
f. What is the probability that a parent was involved in no school activities?
g. What is the probability that a parent was involved in at most 2 activities?
h. Find the mean (expected value) for the variable x and write the answer in a sentence interpreting the meaning in context.
i. Find the standard deviation for the variable x and write the answer in a sentence interpreting the meaning in context.
Module 17 Class Example
1. The mean weight for adult males is 164 ± 31 pounds. Therefore the mean weight is 164 pounds [pic]with a standard deviation of 31 pounds[pic].
a. Draw a diagram of the normal distribution showing the 68-95-99.7 tick marks for the weight of males. Then verify your cutoffs with Statcrunch.
[pic]
b. A typical male weight is between ______ and _______ pounds.
c. An unusual male weight is below ________pounds and above _______pounds.
d. An extremely unusual male weight is above ________ pounds and below ________ pounds.
Using Statcrunch find the following.
e. P( x > 170)
f. P( x > 158)
g. P( x < 150)
h. P( 160 < x < 180)
For parts (e) through (i) above write a sentence in context for each using percentages.
i. P( x > 170)
j. P( x > 158)
k. P( x < 150)
l. P( 160 < x < 180)
Module 17 Empirical Rule for Normal Distributions
Directions: Use the Empirical Rule (68-95-99.7) Rule for normal distributions to answer the following questions.
1. Consumer Reports Magazine wrote an article stating that monthly charges for cell phone plans in the U.S. are normally distributed with a mean of $62 and a standard deviation of $18.
a) Draw a picture of the normal curve with the cell phone charges for 1,2 and 3 standard deviations above and below the mean.
[pic]
b) What percent of people in the U.S. have a cell phone bill between $62 and $80 per month?
c) What are the two cell phone charges that the middle 95% of people are in between?
d) What percent of people in the U.S. have a monthly cell phone bill between $26 and $44.
e) Find the cell phone bill that 99.85% of people are less than.
2. The ACT exam is used by colleges across the country to make a decision about whether a student will be admitted to their college. ACT scores are normally distributed with a mean average of 21 and a standard deviation of 5.
a) Draw a picture of the normal curve with the ACT scores for 1,2 and 3 standard deviations above and below the mean.
[pic]
b) What percent of students score higher than a 31 on the ACT?
c) What are the two ACT scores that the middle 68% of people are in between?
d) What percent of people score between a 16 and 21 on the ACT?
e) Find the ACT score that 84% of people score less than?
3. Human pregnancies are normally distributed and last a mean average of 266 days and a standard deviation of 16 days.
a) Draw a picture of the normal curve with the pregnancy lengths for 1,2 and 3 standard deviations above and below the mean.
[pic]
b) What percent of pregnancies last between 218 days and 234 days?
c) Find two pregnancy lengths that the middle 68% of people are in between. This is the range of days that pregnancies typically take.
d) A marine came home from a long tour of duty and was amazed to hear that his wife was pregnant and expecting a baby. This was especially amazing since it had been 314 days since he had last seen his wife. His wife claims that the baby is just very late in coming. What is the probability that a pregnancy would last 314 days or more? What does this tell you about the wife’s claim?
Module 17 Finding and Interpreting z-scores
1. A sample of the heights of women had a normal distribution with a mean height of 63.9 inches and a standard deviation of 2.8 inches. Draw a picture of the normal curve with the pregnancy lengths for 1,2 and 3 standard deviations above and below the mean.
[pic]
a) One woman had a height of 67 inches. Find and interpret the z-score corresponding to this height. Find the probability of a woman being taller than 67 inches. Is she unusually tall compared to the rest of the women in the sample?
b) One woman had a height of 57 inches. Find and interpret the z-score corresponding to this height. Find the probability of a woman being shorter than 57 inches. Is she unusually short compared to the rest of the women in the sample?
c) One woman had a height of 73 inches. Find and interpret the z-score corresponding to this height. Find the probability of a woman being taller than 73 inches. Is she unusually tall compared to the rest of the women in the sample?
d) One woman had a height of 59 inches and another woman had a height of 66 inches. Find the z-scores for both women. Which was more unusual?
2. IQ tests are normally distributed with a mean of 100 and a standard deviation of 15. Draw a picture of the normal curve with the pregnancy lengths for 1,2 and 3 standard deviations above and below the mean.
[pic]
a) A girl scored a 140 on the IQ test. Find and interpret the z-score for this IQ. Find the probability of someone scoring higher than 140. Is it unusual for someone to score a 140?
b) A boy scored a 90 on the IQ test. Find the probability of someone scoring lower than 90 Find and interpret the z-score for this IQ. Is it unusual for someone to score a 90?
c) One man scored a 120 on the IQ test. Find the probability of someone scoring higher than 120. Find and interpret the z-score for this IQ. Is it unusual for someone to score a 120?
d) Mike scored a 135 on the IQ test and Jake scored a 71 on the IQ test. Find the Z-scores for each. Which score was more unusual?
Module 17 Normal Distribution Activity (1)
Use Statcrunch for normal distributions to answer the following probabilities. Write all answers with a sentence in context. For Statcrunch go to: Stat-Calculators-Normal. Enter the mean and standard deviation. Then choose the correct inequality and compute.
1. The mean height for males in the U.S. is 69 inches [pic]with a standard deviation of 3 inches
[pic].
a. Draw a diagram of the normal distribution showing the 68-95-99.7 tick marks for the height of males. Then verify your cutoffs with Statcrunch.
[pic]
b. A typical male height is between ______ and _______ inches.
c. An unusual male height is below ________ inches and above _______ inches.
d. An extremely unusual male height is above ________ inches and below ________ inches.
Find the following using Statcrunch (and write your answer in a complete sentence):
e. P( x > 70)
f. P( x > 68)
g. P( x < 71)
h. P( x < 67)
i. P( 68 < x < 70)
2. Bald eagles mean wingspan is 6.7 feet [pic] with a standard deviation of 0.8 ft. [pic].
a. Draw a diagram of the normal distribution showing the 68-95-99.7 tick marks for the wingspan of bald eagles. Then verify your cutoffs with Statcrunch.
[pic]
b. A typical wingspan is between ______ and _______ feet.
c. An unusual wingspan is below ________ feet and above _______ feet.
d. An extremely unusual wingspan is above ________ feet and below ________ feet.
Find the following using Statcrunch (and write your answer in a complete sentence):
e. P( x < 6.9)
f. P( x < 6.0)
g. P( x >7.0)
h. P( x > 6.5)
i. P( 6.4 < x < 6.8)
3. The birth weights of full-term babies are normally distributed with a mean weight of μ =3400 grams and a standard deviation of σ=505 grams.
a. Draw a diagram of the normal distribution showing the 68-95-99.7 tick marks for the birth weights of babies. Then verify your cutoffs with Statcrunch.
[pic]
b. A typical birth weight is between ______ and _______ grams.
c. An unusual birth weight is below ________ grams and above _______ grams.
d. An extremely unusual birth weight is above ________ grams and below ________ grams.
Find the following using Statcrunch (and write your answer in a complete sentence):
e. P( x > 3500)
f. P( x > 3300)
g. P( x < 3600)
h. P( x < 3100)
i. P( 3350 < x < 3550)
4. The lengths of human pregnancies are normally distributed with a mean length of μ = 266 days and a standard deviation of σ = 16 days.
a. Draw a diagram of the normal distribution showing the 68-95-99.7 tick marks for the lengths of human pregnancies. Then verify your cutoffs with Statcrunch.
[pic]
b. A typical pregnancy lasts between ______ and _______ days.
c. An unusual pregnancy lasts below ________ days and above _______ days.
d. An extremely unusual pregnancy lasts above ________ days and below ________ days.
Find the following using Statcrunch (and write your answer in a complete sentence):
e. P( x < 287)
f. P( x < 255)
g. P( x >270)
h. P( x > 260)
i. P( 250 < x < 275)
Module 17 Normal Distribution Activity 2
1. Bald eagles average about 12 pounds in weight with a standard deviation of 2 pounds. Suppose that these weights follow the Normal model. Draw a picture of the normal curve with the pregnancy lengths for 1,2 and 3 standard deviations above and below the mean.
[pic]
a. What is the probability that a randomly selected eagle is within 1 standard deviation of the mean weight of 12 pounds?
b. What is the spread of weights for 1 SD?
c. What is the probability that a randomly selected eagle is within 2 standard deviation of the mean weight of 12 pounds?
d. What is the spread of weights for 2 SD?
e. What is the probability that a randomly selected eagle is within 3 standard deviation of the mean weight of 12 pounds?
f. What is the spread of weights for 3 SD?
2. Bald eagles mean wingspan is 6.7 feet with a standard deviation of 0.8 feet. Assume the wingspans follow the Normal model.
a. Draw the normal distribution curve with the 68-95-99.7 cutoffs.
[pic]
b. What is the typical wingspan of a bald eagle? Between __________ and ___________ feet.
c. Approximately, what percent of all bald eagles have a typical wingspan?
d. An unusual wingspan for a bald eagle would be greater than _________feet and less than _______feet.
e. Approximately, what percent of all bald eagles would have an unusual wingspan?
f. An extremely unusual wingspan for a bald eagle would be greater than _________feet and less than _______feet.
g. Approximately, what percent of all bald eagles would have an extremely unusual wingspan? ________
h. The largest bald eagles are found in Alaska. Some eagles in Alaska have been found to have a wingspan of 8 feet. Convert this measurement to a z score and determine if this is considered unusual. Use [pic].
i. The smallest bald eagles are found in South Carolina and have a wingspan of 6.2 feet. Convert this measurement to a z score and determine if this is considered unusual. Use [pic].
j. How does your normal curve with the 68-95-99.7 tick marks support your answers from part h and I above?
Module17 Normal Distribution activity 3
Use Statcrunch for normal distributions to answer the following:
1. For a population the mean and the standard deviation are given to be: [pic]. Find:
a) P( x > 16)
b) P(x > 12)
c) P( x < 17)
d) P(x < 13)
e) P( 15 < x < 16)
f) P( 12 < x < 16)
2. For a population the mean and the standard deviation are given to be: [pic]. Find:
a) P( x > 22)
b) P(x > 32)
c) P( x < 10)
d) P(x > 8)
e) P( 26 < x < 28)
f) P( 29 < x < 30)
3. For a population the mean and the standard deviation are given to be: [pic]. Find:
a) P( x < 10 )
b) P( x > 11 )
c) P( x > 14)
d) P(x > 6)
e) P( 6 < x < 10)
f) P( 10 < x < 14)
Journal 3
Reflect on your experiences in this course and what you have learned this semester. Then write a paragraph that addresses each of the following topics:
1. Think about your attitude towards word problems at the beginning of the semester. Are you more (or less) comfortable with word problems and understanding and analyzing information? Please explain.
2. Regarding math, describe the ways in which you have developed more of a growth mindset.
3. Regarding math, describe the ways in which you still have a fixed mindset.
4. This course introduced you to teaching methods that are not often used in a traditional math course (discovery activities, group discussions, group presentations, journals, projects, poster presentations, and immediate real-life applications of algebraic topics). Which did you like the most, and which did you like the least? Why?
5. This class was not taught in the traditional manner rather the majority of the class time was spent working in groups. Describe your experience in being in a ‘flipped’ class environment.
Please use a separate piece of paper for your responses and type or neatly handwrite this assignment using complete sentences. Before you submit it, please verify that you have addressed each question.
Z-Table (area to the left of Z)
| |0 |0.01 |
| | | |
| | | |
| | | |
| | | |
| |Points possible |Points received |
|Dotplot |2 | |
|Boxplot |2 | |
|Histogram |2 | |
|Summary statistics |2 | |
|Visual |2 | |
|Total (10) | | |
| | | |
|Introduction of data set |2 | |
|Shape |2 | |
|Center/reason for choice |3 |quote only the best measure |
|Spread/reason for choice |3 |of center and spread & put on a slide |
|Outliers |2 | |
|Percentages |2 | |
|Conclusion (including a bias) |2 | |
|Total (16) | | |
| | | |
|Grammar/Punctuation |2 | |
|Math accuracy |2 | |
|Group grade factor |10 | |
|Total (14) | | |
| | | |
|Total (40) | | |
Math 137 Group Grading for Project 1
One fourth of your grade will be based on this group evaluation. Please fill out the form below and turn in after your presentation. Fold your paper in half when turning in. This evaluation will not be shared with others in order for you to grade honestly.
Use the following rubric to grade yourself and each member of your group regarding participation and contribution to the project.
8-10 points: Meaningful contribution and active participation throughout the project.
6-7points: Contributed at least one idea and average participation and contribution to the project.
0-5 points: Very little or no effort in contributing to the project.
A.
Your Name_______________________________ Grade scale 1 to 10: ___________
Group member_____________________________ Grade scale 1 to 10: ___________
Group member____________________________ Grade scale 1 to 10: ___________
Group member____________________________ Grade scale 1 to 10: ___________
B.
Your contribution:
Provide below a description of all the tasks you were responsible for in completing the project.
Math 137 Project 2 Linear and Non-Linear Analysis: Project Decision
You will decide as a group what data you would like to investigate. Turn in one paper per group once you make a decision. You must get your data set approved before continuing.
Names of group members:
Search the internet for data (minimum of 20 ordered pairs):
Search for two quantitative variables that you think might be related.
1. Website where you found the data: __________________________________
2. Fill in the blanks with respect to your data.
Given the data, I want to see if there is a relationship between _________________ and
___________________.
2. What is the explanatory variable? ___________________________________
3. What is the response variable? _____________________________________
Math 137 Project2 Instructions Linear and Non-linear Analysis
Create a PowerPoint of your topic that has been approved by the instructor. Your presentation should be colorful and include your graphs, statistics and important or interesting highlights. You will be presenting to the class.
Your data set should have a least 20 ordered pairs. (If not, obtain permission from instructor.)
Criteria for regression modeling: How good is the fit of the regression model?
When assessing how well the linear or non-linear regression model fits the data, we examine the following criteria:
• The linear regression model must have two quantitative variables (explanatory and response).
• The form of the scatterplot is linear or curvilinear. Take note that the scatterplot follows the trend of the model you are attempting to use.
• Identify outliers and discuss whether they may be overly influential outliers. In the case of overly influential outlier, you may need to remove these points and re-examine your model.
1. Introduction: You are to introduce your topic with research information including:
a. A title slide with your project title, class, semester, year and instructor.
b. A slide stating the variables you are analyzing.
c. Provide a scatterplot of your data (without any regression curves drawn on the graphs)
d. What models you will be using to model your data.
e. Include the website from which your data was collected.
2. Linear Regression Analysis: Find the linear model of the curve that models your data set.
a. Create a scatterplot, with the linear regression line. Describe the association, if any.
b. Create a residual plot of your data about the regression line and describe it.
c. What is the slope of the prediction equation? Interpret this slope in the context of the data including units.
d. What is the y-intercept of the prediction equation? Interpret this y-intercept in the context of the data including units.
e. What is the correlation coefficient and what does this tell us?
f. What is the coefficient of determination, r2? (Do not interpret meaning here.)
g. What is the standard error of regression, Se? (Do not interpret meaning here.)
h. Does there appear to be any influential outliers? If there are, identify them in ordered pair format and decide if you should remove from the data set.
3. Non-linear Analysis: Now find the quadratic model of the curve that models your data set.
a. Create a scatterplot, with the regression curve.
b. Create a residual plot of your data about the regression curve and describe it.
c. What is the coefficient of determination, r2? (Do not interpret meaning here.)
d. What is the standard error of regression, Se? (Do not interpret meaning here.)
4. Overall Analysis: Provide a table or chart that compares the residual plots, r2, and Se of both models (use the table below). When comparing your residual plots, you will need to incorporate a slide with both of your labeled residual plots to provide a visual for the comparison.
Use your table of these criteria to determine which of your models you think would be the best to use and why.
Include the following in your presentation. Then decide which of the two (if any) is the best model for your data set.
|Model |Residual Plot |[pic] |[pic] |
| |No pattern or pattern? If there is a pattern, describe| | |
| |the shape. | | |
|Linear | | | |
| | | | |
| | | | |
|Quadratic | | | |
| | | | |
| | | | |
5. Based on your best fit model only, interpret both the r2 and Se in the context of your data.
6. Conclusion: Make an appropriate conclusion. Is your best fit model useful to make predictions within the scope of the data given the values of [pic]and[pic]? Can you conclude that there is a correlation between your two quantitative variables? What statistical evidence do you have to support your conclusion?
7. Presentation: You will also be graded on your presentation as well as your content. Your eye contact, enthusiasm, and overall presentation will be factored into your grade. Each person in the group should be present and participate in the presentation for credit. You will lose points if you are a disrespectful listener as well. If you are not listening intently for others’ presentations, you will lose points.
8. Make sure you turn in your self-grade report which is 25% of your grade. It should be a private evaluation of yourself and your group members. Fold the paper in half when you turn it in. If you are not present for the presentation, your grade is a zero.
Save your data on excel and clean up. Remember not to use symbols such as commas or $.
Statcrunch:
1. To obtain the scatterplot without regression line/curve:
Graph-scatterplot
2. Linear model:
Stat-regression-simple linear
For the residual plot
Stat-regression-simple linear-scroll down to graphs and choose ‘residual index plot’
3. Quadratic model:
Stat-regression-polynomial
For residual plot: Stat-regression-polynomial-graphs-residual index plot- compute (it will be on the second page)
Project 2: Linear and Non-linear Analysis Rubric
Names of Group Members:
[pic]
Math 137 Project 2: Group Grading
One fourth of your grade will be based on this group evaluation. Please fill out the form below and turn in after your presentation. Fold your paper in half when turning in. This evaluation will not be shared with others in order for you to grade honestly.
Use the following rubric to grade yourself and each member of your group regarding participation and contribution to the project.
8-10 points: Meaningful contribution and active participation throughout the project.
6-7points: Contributed at least one idea and average participation and contribution to the project.
0-5 points: Very little or no effort in contributing to the project.
A.
Your Name_______________________________ Grade scale 1 to 10: ___________
Group member_____________________________ Grade scale 1 to 10: ___________
Group member____________________________ Grade scale 1 to 10: ___________
Group member____________________________ Grade scale 1 to 10: ___________
B.
Your contribution:
Provide below a description of all the tasks you were responsible for in completing the project.
Math 137 Module 15 True Colors Project #3
After watching the True Colors power point presentation fill out the next page to discover your true color.
[pic]
[pic]
[pic]
A. We will make a two way table for our class color results. Copy it on your paper.
B. You will be divided into color groups.
C. Group Poster Activity:
1. Come up with a five probability questions that relate ONLY to your group’s personality color. Also, make an answer sheet to your questions which will be used later. (Make sure to incorporate marginal, conditional, and joint probabilities.) Include the fraction, decimal, and percent notation on your answer sheet for each question.
2. Put your 2-way table and numbered questions on a poster. Decorate the poster as you see fit regarding your personality color qualities and traits. DO NOT PUT THE ANSWERS TO YOUR QUESTIONS ON THE POSTER.
3. When the posters are done, your group will be instructed to rotate from poster to poster and answer the questions of each group. (In fraction, decimal, and percent notation.) Each student will write the answers on their own paper to be turned in.
4. When everyone is back at their poster, each group will put the answers to their questions on the poster for groups to check their answers.
5. Class discussion will follow for clarifications and corrections.
Math 137’s “True Colors” Classwork/Homework
1. Draw the two way table for the class:
2. What is the probability that someone in our class is GOLD? _____________
3. What is the probability that someone in our class is GREEN? _____________
4. What is the probability that someone in our class is BLUE? _____________
5. What is the probability that someone in our class is ORANGE? _____________
6. Given that someone is blue, what is the probability that they are female? _______
7. What is the probability that someone is gold and female?________________
8. What is the probability that someone is an orange or blue? _____________
9. What is the probability that someone is a green or male? _____________
10. What is the probability that if they are orange, that they are male? __________
11. Given a male is chosen, what is the probability he is blue? _______________
12. What is the probability that someone is a male and green? _______________
13. What is the probability someone is female, given that they are gold? _________
14. If a blue person is chosen, what is the probability they are male? __________
-----------------------
[1]Paulos, J.A. (1995). A mathematician reads the newspaper (p. 137). New York: Basic Books.
-----------------------
Edited by Bamdad Samii & Cl
[pic]
a) What percentage of employees made a salary of less than $35,000?
b) What percentage of employees made a salary of more than $80,000?
c) 60% of employees made a salary of less than _______________
d) How many employees made a salary of less than $35,000?
0
5
10
15
20
25
1. Which of the four graphs is not a histogram? How can you tell?
2. For the histograms only, name the variable in each graph.
3. What is the y-axis measuring in each of the histograms?
Steps to create a boxplot:
• Step1: Draw a box from Q1 to Q3
• Step2: Draw a line through the box at the median
• Step3: Extend a line (tail) from Q1 to the smallest value that is NOT an outlier and another line (tail) from Q3 to the largest value that is NOT an outlier.
• Step4: Indicate outliers with asterisks (*)
Lo
Q1
Q3
Hi
Q2
0
50
100
Female Life Expectancy
Descriptive Statistics
|Variable |Mean |Standard |
| | |Deviation |
|Female |80.096 |1.399 |
|Variable |Q1 |Median |Q3 |IQR |
|Female |79.1 |80.1 |81.2 |2.100 |
|Variable |Min |Max |Range |
|Female |77.4 |83.7 |6.300 |
Male Life Expectancy
Descriptive Statistics
|Variable |Mean |Standard |
| | |Deviation |
|Male |74.390 |1.821 |
|Variable |Q1 |Median |Q3 |IQR |
|Male |73.6 |74.8 |76.0 |2.400 |
|Variable |Min |Max |Range |
|Male |68.6 |77.2 |8.600 |
[pic]
[pic]
[pic]
The line we will use to describe this scatterplot is:
D = -3A + 576
where A is age of the driver, and D is the distance (in feet) they can read a highway sign.
What do each of the variables mean?
[pic]
[pic]
[pic]
r = 0.729
Variable Mean StDev
PAPER 9.428 4.168
TOTAL 27.44 12.46
r = 0.586
Variable Mean StDev
METAL 2.218 1.091
PLAS 1.911 1.065
r = 0.583
Variable Mean StDev
FOOD 4.816 3.297
TOTAL 27.44 12.46
1. The x variable is describing the number of tons of paper trash and the y variable is the number of tons of total trash. (r = 0.729)
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hÑb¬hŸdÅhÑb¬hŸdÅ;?OJQJhÑb2. The x variable is describing the number of tons of plastic trash and the y variable is the number of tons of metal trash. (r = 0.586)
4. The x variable is describing the horsepower of an automobile and the y variable is describing the miles per gallon. (r = -0.869)
3. The x variable is describing the number of tons of food trash and the y variable is the number of tons of total trash. (r = 0.583)
The estimate made from a model is the predicted value (denoted as [pic] ).
Residual = observed – predicted
= y - [pic]
Which line is better? What’s a way we can determine which is better?
When we compare the sum of the areas of the yellow squares, the line on the left has an SSE of 57.8 (using computer technology). The line on the right has a smaller SSE of 43.9.
So the line on the right fits the points better, but is it the best fit?
Computer technolgy finds this best fit line where the SSE is the mimimum.
Recall that the error or residual is the distance from the data point and the line of regression which is given by:
y – ŷ
Take these distances and plot them as vertical distances based on the x-value.
Here we are showing the graph of the points with an attached line which shows the distances.
If there is NO PATTERN in the residual plot then the linear model is a good fit.
If there IS A PATTERN in the residual plot than the linear model is not the best fit and perhaps another equation would be a better model for the data.
r=0.948
[pic]
The regression line for the Burger King data fits
the data well:
[pic]
base > 1, growth
[pic]
(C>0, C is pos.)
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