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Math 140 Notes and Activity Packet (Word)ProbabilityGo over Probability Notes on Empirical Rule and Z-scores (PDF online only) before doing Probability Act 1 and Act 2Probability ActivityEmpirical Rule for Normal DistributionsDirections: Use the Empirical Rule (68-95-99.7) Rule for normal distributions to answer the following questions. 1. According to a 2014 JD Power report, the mean average monthly cell phone charges for Verizon customers is $148 with a standard deviation of $18. Assume that the cell phone bills are normally distributed.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. b) What percent of Verizon customers have a cell phone bill between $130 and $148 per month?c) What are the two cell phone charges that the middle 95% of Verizon customers are in between?d) What percent of Verizon customers have a monthly cell phone bill between $166 and $184.e) Find the cell phone bill that 99.85% of Verizon customers 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. 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. 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. Math 140 - Probability Activity#2Finding and Interpreting z-scoresA 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. 1. One woman had a height of 67 inches. Find and interpret the z-score corresponding to this height. Is she unusually tall compared to the rest of the women in the sample?2. One woman had a height of 57 inches. Find and interpret the z-score corresponding to this height. Is she unusually short compared to the rest of the women in the sample?3. One woman had a height of 73 inches. Find and interpret the z-score corresponding to this height. Is she unusually tall compared to the rest of the women in the sample?4. 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?IQ tests are normally distributed with a mean of 100 and a standard deviation of 15.5. One man scored a 140 on the IQ test. Find and interpret the z-score for this IQ. Is it unusual for someone to score a 140?6. A woman scored a 90 on the IQ test. Find and interpret the z-score for this IQ. Is it unusual for someone to score a 90?7. One man scored a 120 on the IQ test. Find and interpret the z-score for this IQ. Is it unusual for someone to score a 120?8. 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?ACT scores are normally distributed with a mean average of 21 and a standard deviation of 5.9. Juliann scored a 29 on her ACT exam. Find and interpret the z-score for this ACT exam. Is it unusual for someone to score a 29 on the ACT?10. Richard scored a 32 on his ACT exam. Find and interpret the z-score for this ACT exam. Is it unusual for someone to score a 32 on the ACT?11. Michael’s z-score for his ACT exam was -2.1 and Zach’s z-score for his ACT exam was -2.4 . Who did better on the ACT exam, Michael or Zach? Which score was more unusual? Probability Notes – Normal ProbabilitiesReview the difference between inequality symbols less than (points to left), less than or equal to, greater than (points to right), greater than or equal to.Discrete Probability – Probabilities that involve countable totals. Think categorical variables like winning and losing. For example the probability of rolling a die and getting a three. This we can count. There is 1 side that has three dots out of 6 total sides, so the probability is 1/6Continuous Probability – Probabilities about numbers in a continuous scale. Quantitative data is in involved that falls in a continuous scale. For example the probability that a person’s height is greater than 1.602719 meters. Notice there are infinitely many possibilities for this decimal. So we cannot directly count the total. The total is infinite. Also the probability that someone’s height is exactly 1.602719 meters is about zero.Continuous probabilities are difficult to calculate. We use a probability density curve.Probability Density Curve: A curve of a particular shape where the area under the curve is equal to 1 (or 100%). So to find probabilities we can either use calculus or technology to calculate the area under the curve.Normal Probability Density Curve: For continuous quantitative data that is bell shaped, we can use the normal calculator in StatCrunch. Remember this corresponds to probabilities for continuous quantitative data when it is impossible to count the total. What do we Need? Bell shaped quantitative dataMean and Standard Deviation (remember only accurate when bell shaped)The number you are finding the probability of is X Notation: P ( X ) means probability of X happening. For example, find the probability that a person is more than 1.602719 meters tall. P ( X > 1.602719 ). Remember the probability of equals is about zero in continuous data so P ( X > 1.602719 ) is about the same as P ( X ≥ 1.602719 ). Normal Probabilities with StatCrunch: Go to Stat button and click on “Calculator”. Then go down the menu to “Normal”.Enter the mean and standard deviation for your bell shaped data. Decide on “standard” ( for probabilities involving ≤ or ≥ ) or click “between” (for probabilities in between two x values).Put in either the x value or the probability given. If you put the x value, StatCrunch will calculate the probability (Area). If you put the probability (Area), StatCrunch will calculate the x value. Put whether you want greater than or equal to or less than or equal to. Remember probability of equal is zero.Example 1: Suppose heights of men are bell shaped with a mean average of 1.76 meters with a standard deviation of 0.071 meters. Draw a bell shaped curve with 1.76 in the center. Find the probability that the height of a man is greater than or equal to 1.602719 meters? (Mark off this area on your curve)P ( X ≥ 1.602719 ) = The curve should look like the one you drew. The answer is 0.987 or 98.7% of men.Example 2: Now find the height in meters that about 15% of men are less than. Plug in 0.15 where the probability goes and make sure to press the less than or equal to. P ( X ≤ ?? ) = 0.15You should get a height of about 1.686 meters.Example 3: Now find the probability that a man is between 1.8 meters and 1.9 meters tall. Remember to click the between button and use the same mean and standard deviation. Draw the bell shaped curve with 1.76 in middle. Where is 1.8 and 1.9. Estimate. Shade the area under the curve. Now put 1.8 and 1.9 into the between normal calculator and get the answer.P ( 1.8 ≤ X ≤ 1.9 ) = Math 140 Probability Activity#3Using Statcrunch to find probabilities and x valuesfor a normal distributionIn an effort to increase customer satisfaction, a branch of Bank of America took a random sample of how long people wait in line at their bank. They found that the wait times were normally distributed with a mean of 169 seconds and a standard deviation of 36 seconds. What percent of customers are waiting more than 3 minutes (180 seconds)? Do you think this is a problem for the bank? Why?Find the number of seconds that 20% of customers wait less than. What percent of customers wait between 1 and 2 minutes (60 seconds to 120 seconds)? Give a recommendation to the bank. Do you believe that customers that complain about waiting too long have a reason to complain? Why?According to a 2014 JD Power report, the mean average monthly cell phone charges for Verizon customers is $148 with a standard deviation of $18. Assume that the cell phone bills are normally distributed.a) What percent of Verizon customers have a cell phone bill greater than $100 per month?b) What percent of Verizon customers have a cell phone bill less than $175 per month? c) What are the two cell phone charges that the middle 60% of Verizon customers are in between? d) What percent of Verizon customers have a monthly cell phone bill between $125 and $150?e) Find the cell phone bill that 90% of Verizon customers pay less than.f) Find the 67th percentile for cell phone bills. This is the bill that 67% of Verizon customers are less than. 3. Human pregnancies are normally distributed and last a mean average of 266 days and a standard deviation of 16 days. a) What percent of pregnancies last more than 279 days?b) What percent of pregnancies last less than 251 days?c) What percent of pregnancies last between 245 days and 272 days?d) Find the 70th percentile. This is the number of days that 70% of pregnancies are less than.e) Find two pregnancy lengths that the middle 80% of people are in between. Binomial ProbabilitiesA Binomial Probability is a type of discrete probability with only two outcomes (male or female, win or lose, have disease or don’t have disease)Success Category (have disease)Failure Category (don’t have disease)For a binomial probability, individual observations should be independent of each other with a consistent probability of success. (For example, winning at cards often fails this assumption because the number of cards and the probabilities are always changing.)Calculate Binomial Probabilities with StatCrunchStat Menu => Calculator => Binomial => Standard or BetweenWhat you need?X = # of successes and what you are finding the probability ofp = the probability of success in 1 observationn = total number of observationsNote about inequality symbols. Normal Probabilities: When dealing with continuous quantitative data with decimals, we had infinite totals so the probability of less than 3 kilograms is 2.999999999… or below. Hence for normal probabilities the probability of less than 3 is about the same as less than or equal to.Binomial Probabilities: This is not the case for binomial probabilities. Winning a game less than 3 times means winning less than or equal to 2 times. So be careful about the wording with inequalities. For Binomial, StatCrunch gives the options of = , < , > , ≤, ≥ Remember greater than points right and less than points left.Wording examples= “probability that exactly 5 people have the disease”> “ probability that she wins more than 4 times “≥ “ probability that she wins 4 or more times “ or “ at least 4 times”< “ probability that he wins less than 6 times “≤ “ probability that he wins 6 times or less “ or “ at most 6 times”Let’s look at an example. Sarah likes to play slot machines in a Casino in Las Vegas. The particular slot machine she is playing has a 7% chance of winning. Suppose Sarah plays the game 35 total times. 1. What is the probability that Sarah wins more than 3 times?P (x > 3) =???We will need to go to StatCrunch and click the stat menu, then calculator, then binomial. Since this is not a “between” problem, click on the standard button. Notice n = 35, and p = 0.07 and x =3. Click the greater than but not the greater than or equal to.2. What is the probability that Sarah wins at most 2 times?We will need to go to StatCrunch and click the stat menu, then calculator, then binomial. Since this is not a “between” problem, click on the standard button. Notice n = 35, and p = 0.07 and x =2. Click the less than or equal to sign. 3. Find the probability that Sarah wins 5, 6 or 7 times (between 5 and 7 inclusively)?We will need to go to StatCrunch and click the stat menu, then calculator, then binomial. Since this is a “between” problem, click on the between button. Notice n = 35, and p = 0.07 and x =5 and x = 7. Math 140 Probability Activity#4Binomial ProbabilitiesMany probabilities are binomial in nature. When we collect a random sample of size n from a population and the data collected falls into just two outcomes, we call this binomial. The two outcomes are often called success and failure. The probability of success is p and must be the same for every trial. Also each outcome must be independent of one another. The probability of X successes in n trials can be calculated using StatcrunchDirections: For each of the following, verify that the situation is binomial in nature and find the number of trials n, the number of outcomes X, and the probability of success p. Then use Statcrunch to find the given probabilities. Steve Nash is one of the best free throw shooters in the NBA. The probability he will make a free throw is 92%. Let us suppose that Nash shoots 16 free throws in a game. What is the probability that he makes exactly 13 free throws out of the 16 tries? What is the probability that he makes less than 12 free throws in the game?What is the probability that he makes 14 or more free throws in the game?Suppose that a company manufactures I-pods. It has been found that 3% of the I-pods made will be defective. The company ships a box of 50 total I-pods.What is the probability that the box will contain exactly 4 defective I-pods?What is the probability that the box will contain 3 or less defective I-pods?What is the probability that the box contains 5 or more defective I-pods?Betting on an individual number in roulette, a person has a 1/38 or 2.63% chance of winning. Suppose Rick plays roulette a total of 24 times. What is the probability that he wins at least once (1 or more)? What is the probability that he doesn’t win at all? What do you notice about the probability of at least one and the probability of none? How are they related?What is the probability that he wins less than 3 times?To win at a dice game, the player must role two dice and get a 7 or 11 sum. There is an 8/36 or 22.2% chance of winning. Suppose Lacy roles the dice a total of 18 times.What is the probability that she wins 2 or more times? What is the probability that she wins exactly 4 times? What is the probability that she doesn’t win at all? A car company found that their Minivan transmissions have a 12% defective rate. A total of 72 Minivans were brought in for service this month. What is the probability that exactly 9 of them need to have their transmission replaced? What is the probability that more than 10 of the minivans will need their transmission replaced? What is the probability that 6 or less of the minivans will need their transmission replaced?Two Way Table Probability NotesComparing Two Categorical Data SetsCreating a Two Way Table with StatCrunchOne of the best ways to summarize relationships between categorical data is with a “Two-Way Table”. Statcrunch can make a two-way table from raw categorical data. Go to the Stat menu, then Tables, then Contingency. Pick which variable you want for the rows and which variable for the columns. Then press compute. Example 1: Open the Fall 2015 Math 140 Survey Data in StatCrunch and create a two-way table summarizing gender and hair color. Draw your two-way table below.Contingency table results:Rows: What is your gender?Columns: What is the natural color of your hair?BlackBlond(e)BrownOtherRedTotalFemale382313232198Male63224804137Total1014518036335ORContingency table results:Rows: What is the natural color of your hair?Columns: What is your gender?FemaleMaleTotalBlack3863101Blond(e)232245Brown13248180Other303Red246Total198137335Example 2: Open the Fall 2015 Math 140 Survey Data in StatCrunch and create a two-way table summarizing what campus the math 140 student goes to (Valencia or Canyon Country) and whether or not the student attended high school in Santa Clarita CA. Draw your two-way table below.Contingency table results:Rows: At which campus are you taking Math 075?Columns: Did you attend high school in Santa Clarita Valley?NoYesTotalCanyon Country Campus5357110Valencia Campus88129217Total141186327ORContingency table results:Rows: Did you attend high school in Santa Clarita Valley?Columns: At which campus are you taking Math 075?Canyon Country CampusValencia CampusTotalNo5388141Yes57129186Total110217327Creating a Stacked Bar Chart with StatCrunchAnother graph that is very useful when dealing with categorical data is a stacked bar chart. Just go to the Graph menu in Statcrunch. Click on “Bar Plot”. Highlight one of your categorical variables. Click on “Group By” and pick a different variable. Then click on Group options and click on “stack bars”. Don’t forget to check the box for “value above the bar”. This graph gives you a visual look at your two way table.Example 3: Open the Fall 2015 Math 140 Survey Data in StatCrunch and create a stacked bar chart summarizing gender and hair color. Draw your stacked bar chart below.ORExample 4: Open the Fall 2015 Math 140 Survey Data in StatCrunch and create a stacked bar chart summarizing what campus the math 140 student goes to (Valencia or Canyon Country) and whether or not the student attended high school in Santa Clarita CA. Draw your stacked bar chart below.ORCreating Pie Charts for each group with “Group By” on StatCrunchAnother graph that is very useful when dealing with categorical data is by making a pie chart for each group. Just go to the Graph menu in Statcrunch. Click on “Pie-chart” and “with data”. Highlight one of your categorical variables. Click on “Group By” and pick a different categorical variable. This graph gives you a great way to compare categorical data from various groups. It also gives you the percentages.Example 3: Open the Fall 2015 Math 140 Survey Data in StatCrunch and create pie charts to summarize gender and hair color. Draw your pie charts below.ORExample 4: Open the Fall 2015 Math 140 Survey Data in StatCrunch and create pie charts to summarize what campus the math 140 student goes to (Valencia or Canyon Country) and whether or not the student attended high school in Santa Clarita CA. Draw your pie charts below.ORProbability Activity 5Creating Stacked Bar Charts, Pie Charts, and Two Way TablesTwo-Way Table: One of the best ways to summarize relationships between categorical data is with a “Two-Way Table”. Statcrunch can make a two-way table from raw categorical data. Go to the Stat menu, then Tables, then Contingency. Pick which variable you want for the rows and which variable for the columns. Then press compute. Stacked Bar Chart: Another graph that is very useful when dealing with categorical data is a stacked bar chart. Just go to the Graph menu in Statcrunch. Click on “Bar Plot”. Highlight one of your categorical variables. Click on “Group By” and pick a different categorical variable. Then click on “Group options” and click on “stack bars”. Don’t forget to click on “value above bar”. This graph gives you a visual look at your two-way table.Multiple Pie Charts: Another graph that is very useful when dealing with categorical data is by making a pie chart for each group. Just go to the Graph menu in Statcrunch. Click on “Pie-chart” and “with data”. Highlight one of your categorical variables. Click on “Group By” and pick a different categorical variable. This graph gives you a great way to compare categorical data from various groups. It also gives you the percentages.1. Open the Math 140 Survey Data in Excel. Copy and paste the gender data and what campus the students go to. Create a two-way table, a stacked bar chart and pie charts for each group with Statcrunch. Copy and paste or draw the table, the stacked bar chart and all of the pie charts below. Write down one interesting fact you learned about gender and which campus students go to.2. Open the Math 140 Survey Data in Excel. Copy and paste the type of transportation data and the texting and driving data. Create a two-way table, a stacked bar chart and pie charts for each group with Statcrunch. Copy and paste or draw the table, the stacked bar chart and all of the pie charts below. Write down one interesting fact you learned about transportation and texting and driving.3. Open the Math 140 Survey Data in Excel. Copy and paste the type of smoking cigarettes and the political party data. Create a two-way table, a stacked bar chart and pie charts for each group with Statcrunch. Copy and paste or draw the table, the stacked bar chart and all of the pie charts below. Write down one interesting fact you learned about smoking and political party.4. Open the Math 140 Survey Data in Excel. Copy and paste the prescription glasses and contact lens data and the car accident data. Create a two-way table, a stacked bar chart and pie charts for each group with Statcrunch. Copy and paste or draw the table, the stacked bar chart and all of the pie charts below. Write down one interesting fact you learned about glasses, contacts and car accidents.5. Open the Math 140 Survey Data in Excel. Copy and paste the tattoo data and social media data into StatCrunch. Create a two-way table, a stacked bar chart and pie charts for each group with Statcrunch. Copy and paste or draw the table, the stacked bar chart and all of the pie charts below. Write down one interesting fact you learned about tattoos and social media.Probability Notes (Go over before doing Prob Act 6 & Act 7) Two-Way Table Probability AnalysisWe created a two-way table from the Math 140 students in Fall of 2015. The two-way table describes the relationship between whether or not a student still lives with their parents and the type of transportation they use when they go to school.Contingency table results:Rows: Do you live at home with your parents?Columns: What type of transportation do you take to campus? Bicycle Carpool Drive alone Dropped off by someone Public transportation WalkTotalNot live with parents0758121078Lives with parents1232051740250Total13026318610Grand Total = 328Basic Marginal ProbabilityRemember a percentage (proportion) is an amount out of the total. Multiply by 100% if you want to convert the proportion into a percentage. Example 1: Suppose we want to find the probability that a student does not live with their parents. We have a common notation when finding the probability. We would write that as P ( does not live with parents )This probability can be calculated by finding the amount out of the total. P ( does not live with parents ) = What is the amount? What is the total? Let us start with the total. In a two way table, always use the grand total (sum of all the cells) unless instructed otherwise. That means you will always use the grand total unless we have a condition like restricting our answer to only people that drive alone to school. If there is no condition, always use the grand total. What about the amount? We need the amount of people that do not live with their parents. That is actually the first row total. So our answer will be… Notice we usually round probabilities to 3 significant figures or the thousandths place.How can we convert this answer into a percentage? Multiply by 100%. So 0.238 = 0.238 x 100% = 23.8%So 23.8% of math 140 students do not live with their parents.Contingency table results:Rows: Do you live at home with your parents?Columns: What type of transportation do you take to campus? Bicycle Carpool Drive alone Dropped off by someone Public transportation WalkTotalNot live with parents0758121078Lives with parents1232051740250Total13026318610Grand Total = 328Example 2What percentage of math 140 students drive alone?Again find the amount out of the total. Since there is no condition given about the student, we will use the grand total. The amount that drive alone is the total in the drive alone column.So the percent of math 140 students that drive alone to school is 0.802 x 100% = 80.2%Joint ProbabilitiesSometimes we want to find a probability when we know more than one thing about the person. These are called Joint Probabilities. In general there are two types. “AND” = means both things have to be true about the person. “OR” means either 1 of two things are true about the person. There are no conditions (If or Given) so we will use the grand total for our total.Two way table “AND” probabilities are a single cell out of the grand total.Contingency table results:Rows: Do you live at home with your parents?Columns: What type of transportation do you take to campus? Bicycle Carpool Drive alone Dropped off by someone Public transportation WalkTotalNot live with parents0758121078Lives with parents1232051740250Total13026318610Grand Total = 328Example 3:Find the probability that a student does not live with parents AND drives alone to school?P ( not live with parents AND drives alone ) = How many students both do not live with parents and drive alone? Both things have to be true about the person. Look for the cell where drive alone column meets the not live with parents row.P ( not live with parents AND drives alone ) = Two way table “OR” probabilities are more difficult because either thing could be true about the person. Example 4How would the previous problem be different if we looked for the probability of a student either not living with parents OR driving alone?Notice this will include everyone that does not live with parents (regardless of transportation status) and everyone that drives alone (regardless of parent status).When you see “OR” , you will need add. Circle all of the people in either the drive alone column or the not live with parents row. Add all of these to get the amount of students that do not live with parents OR drive alone. Bicycle Carpool Drive alone Dropped off by someone Public transportation WalkTotalNot live with parents0758121078Lives with parents1232051740250Total13026318610Grand Total = 328P ( not live with parents OR drives alone ) = NOTE: Notice that we do not want to add things twice. The 58 is in both the drive alone column and the not live with parents row, but should only be added once.NOTE: You will get the wrong answer if you add the column total (263) and the row total (78) because you will have added the 58 twice.Example 5:Find the probability that a student BOTH lives with their parents AND carpools. (Remember a single cell out of the grand total.) Example 6:What if it was an OR problem? How would it be different? Find the probability that a student EITHER lives with their parents OR carpools. (Now we have to add all students that fall in either category and do not add the same cell twice.) Conditional ProbabilitiesSometimes we know some prior information about a situation or person. For example, what is the probability that the L.A. Clippers basketball team will win a game against the Houston Rockets?Suppose you found out that the Clippers 3 best players Chris Paul, Blake Griffin and Deondre Jordan all have the flu and will miss the game. Now what is the probability that the Clippers win?This prior information, can drastically change the probability of an event. It is called a conditional probability and usually denoted with an “IF” or a “GIVEN”. Bicycle Carpool Drive alone Dropped off by someone Public transportation WalkTotalNot live with parents0758121078Lives with parents1232051740250Total13026318610Grand Total = 328Example 7:If a student is dropped off by someone, what is the probability that that person lives with their parents?In probability theory we often write this as P ( lives with parents | dropped off ) The straight bar means “given” this is true. The secret with conditional probabilities is to circle the row or column that has the IF or Given and then only use numbers in that row or column.In this problem the IF was being dropped off by someone. So that restricts us to the dropped off by someone column. Circle the dropped off by someone column and use the amount and total from that column.P ( lives with parents | dropped off ) = Bicycle Carpool Drive alone Dropped off by someone Public transportation WalkTotalNot live with parents0758121078Lives with parents1232051740250Total13026318610Grand Total = 328Example 8:Let’s try another conditional probability. Find the probability that a person walks to school if we are given that the person does not live with their parents.P ( walks | not live with parents ) Remember circle the row or column with the given. Since the given is that the person does not live with parents, we should circle that row and only use the numbers in that row.P ( walks | not live with parents ) = Conditional Probabilities and IndependenceConditional probabilities are the key to determining if categorical variables are Independent (not related) or dependent (related).Categories are independent of each other if the condition does not matter.Look at the following two events: L.A. Clippers basketball team winning and a new British prime minister being elected. If the events are independent then one event happening does not change the probability of the other event happening.So if L.A. Clippers winning and a new prime minister being elected are Independent,P ( L.A. Clippers win ) P(L.A. Clippers win | prime minister elected) P(L.A. Clippers win | prime minister not elected)Notice the condition of prime minister did not make any difference to the probability of the L.A. Clippers winning. Since the probabilities are pretty close, this indicates that the events are Independent (NOT related)Now let’s look at the example of L.A. Clippers winning and their three best players having the flu and not playing. In that caseP ( L.A. Clippers win ) P(L.A. Clippers win | three best players have flu) P(L.A. Clippers win | three best players do not have the flu)Since the probabilities are very different, this indicates that the events are Dependent (related). Bicycle Carpool Drive alone Dropped off by someone Public transportation WalkTotalNot live with parents0758121078Lives with parents1232051740250Total13026318610Grand Total = 328Example 9:Is being dropped off by someone related to (dependent) whether or not that person lives with their parents?Look at the following probabilities. If they are close, then the events are probably independent (not related). If they are significantly different, then the events are probably dependent (related).P ( dropped off ) = P ( dropped off | Lives with parents ) = P ( dropped off | NOT Live with parents ) = Since the probabilities are pretty different, this gives us some evidence that being dropped off and living with parents or not are related (dependent).How do we know if it is significantly different?(To determine that we will need a test statistic and P-value)!!!Probability Activity 6Two-Way Table ProbabilitiesThe following table was created from the Math 140 Survey data in Fall 2015. Use the table to answer the following probability questions. Write your answers as a fraction, decimal and percentage. (All three for each problem.)FacebookInstagramOtherSnapchatTwitterTotalsNo Tattoo5683215921240Has a Tattoo1941611885Totals75124277029Grand Total = 3251. What percent of the students have at least one tattoo?2. What is the probability that a randomly chosen student prefers Facebook?3. What is the probability that a randomly chosen math student does not have any tattoos?4. What percent of the students prefers Snapchat?5. What is the probability that a randomly chosen student has a tattoo and prefers Twitter?6. What percent of the students do not have a tattoo and prefers Instagram?7. What is the probability that a randomly chosen student either does not have a tattoo or prefers Snapchat?8. What percent of the students either have a tattoo or prefer Facebook?9. If a student has a tattoo, what is the probability that they prefer Snapchat.10. What percent of the Facebook users have a tattoo?11. If a student does not have a tattoo, what is the probability the student prefers Twitter?12. What percent of the Instagram users do not have a tattoo? Probability Activity 7Independence and Conditional ProbabilityAn important concept to consider when analyzing categorical data is “Independence”. When two categorical variables are not related, they are said to be independent. If two categorical variables are related, they are said to be dependent. The topic of independence is tied to probability theory, especially conditional probabilities. Two categorical variables are independent if one event does not affect the probability of the other event occurring.1. Let’s look at the following two way table from an art academy describing gender and whether or not someone received a scholarship.Received ScholarshipNo ScholarshipTotalsMale3380113Female3995134Totals72175Grand Total = 247 a) If a student is female, what is the probability they will receive a scholarship?b) If a student is male, what is the probability they will receive a scholarship?c) If these two probabilities in part (a) and (b) are pretty close, that may give some evidence that gender and getting a scholarship are independent (not related). However if the two probabilities are significantly different, then that would give some evidence that gender and getting a scholarship are dependent (related). Do you think gender and getting a scholarship are independent or dependent? Explain why.2. Let’s look again at the Math 140 Survey data. We made a two way table from the tattoo and political party data.DemocraticIndependentOtherRepublicanTotalNo Tattoo82476746242Has Tattoo2818231584Total110659061Grand Total = 326We want to know if political party is related to having a tattoo or not. a) Pick two conditional probabilities that are checking the same variable but have different conditions. Does this give evidence toward independence (not related) or dependence (related)?b) Pick two other conditional probabilities that are checking the same variable but have different conditions. These should be different than part (a). Does this give evidence toward independence or dependence (related)?3. Let’s look at data of race and hiring practices for a company. We looked at 516 job applications and created the following two way table. Determine whether or not there is racial discrimination going on. Race and whether or not someone is hired should be independent of each other. If they are dependent, then maybe there is some discrimination. HispanicAsianAfrican AmericanCaucasianOtherTotalsHired422731448152Not Hired70698411922364Totals1129611516330516a) Pick two conditional probabilities that are checking the same variable but have different conditions. Does this give evidence toward independence or dependence (related)?b) Pick two other conditional probabilities that are checking the same variable but have different conditions. These should be different than part (a). Does this give evidence toward independence or dependence (related)?4. Is it possible to get different answers about independence or dependence from different conditional probabilities? What does that suggest about the difficulty of determining independence from a couple of conditional probabilities? (Lucky for us, there is a better way of determining if categorical variables are independent or not.) ................
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