Mathematics 20-2



MATHEMATICS 20-2Statistical ReasoningHigh School collaborative venture withEdmonton Christian, Institutional Services, Jasper Place, Millwoods Christian, Queen Elizabeth and Victoria SchoolsEdmonton Christian: Jenn JohnsonInstitutional Services: Eric HansonJasper Place: Jessica NoselskiMillwoods Christian: Ken ScharfQueen Elizabeth: David Hernandez-RiveraVictoria: Gina MacKechnieFacilitator: John Scammell (Consulting Services)Editor: Jim Reed (Contracted)2010 - 2011TABLE OF CONTENTSSTAGE 1 DESIRED RESULTSPAGEBig IdeaEnduring UnderstandingsEssential Questions444Knowledge Skills56STAGE 2 ASSESSMENT EVIDENCETransfer TaskTalking Feet ProjectTeacher Notes for Transfer TaskTransfer TaskRubricPossible Solution791617STAGE 3 LEARNING PLANSLesson #1 Standard Deviation23Lesson #2 The Normal Curve30Lesson #3 Z-scores35Lesson #4 Confidence Interval 43Lesson #5 Confidence Intervals in Print and Media47 Mathematics 20-2 Statistical ReasoningSTAGE 1 Desired Results Big Idea: Statistics summarizes data and predict future outcomes in areas such as advertisement, sales, sports and academics. We can make logical assumptions and back them up with numerical data.Implementation note:Post the BIG IDEA in a prominentplace in your classroom and refer to it often. Enduring Understandings:Students will understand that…There are different measures of central tendency, which may be appropriate in different situations.Data can be presented in a misleading fashion.Certain data is normally distributed (bell curve).Predictions based on statistics will contain error. Essential Questions:Which measure of central tendency is better in certain situations?When is the mean the best measure of central tendency to use?When is the median the best measure of central tendency to use?When is the mode the best measure of central tendency to use?When is it appropriate to use a sample set instead of an entire population?Why would people misrepresent data using statistics?When is the size of the standard deviation important?Why are some sets of data normally distributed and others are not?Implementation note: Ask students to consider one of the essential questions every lesson or two.Has their thinking changed or evolved? Knowledge:Enduring UnderstandingSpecific OutcomesDescription ofKnowledgeStudents will understand…There are different measures of central tendency, which may be appropriate in different situations. *S1.3Students will know …how to identify measures of central tendencyStudents will understand…Data can be presented in a misleading fashion. S1.6Students will know …how standard deviation effects the curve and area under the curveStudents will understand…Certain data is normally distributed (bell curve). S1.1, 1.4, 1.5, 1.7, 1.8, 1.9Students will know …what standard deviation isthe characteristics of a normal distributionhow standard deviation effects the curve and area under the curvewhat a z-score is and how it applies to the normal distributionStudents will understand…Predictions based on statistics will contain error. S2.1, 2.2Students will know …how confidence levels, margin of error and confidence intervals may vary depending on the size of the random samplethe significance of a confidence interval, margin of error or confidence level8888I*S = Statistics Skills: Enduring UnderstandingSpecific OutcomesDescription of SkillsStudents will understand…There are different measures of central tendency, which may be appropriate in different situations. *S1.5 S2.6Students will be able to…compare the properties of two or more normally distributed data setssupport a position by analyzing statistical data presented in the mediaStudents will understand…Data can be presented in a misleading fashion. S2.5 S2.3 S2.4Students will be able to…interpret and explain confidence intervals and margin of error, using examples found in print of electronic mediamake inferences about a population from sample data using given confidence intervalsprovide examples from print or electronic media in which confidence intervals are used to support a particular positionStudents will understand…Certain data is normally distributed (bell curve). S1.7, S1.4 S1.8 S1.5, S1.9 S1.2Students will be able to…determine if a set approximates a normal curvedetermine the z-score for a given value in a normally distributed data setsolve a contextual problemcalculate the population standard deviation of a data setStudents will understand…Predictions based on statistics will contain error. S2.3 S2.4Students will be able to…make inferences about a population from sample data using given confidence intervalsprovide examples from print or electronic media in which confidence intervals are used to support a particular positionImplementation note:Teachers need to continually askthemselves, if their students are acquiring the knowledge and skills needed for the unit.*S = StatisticsSTAGE 2 Assessment Evidence1 Desired Results Desired Results Talking Feet Project Teacher NotesThere is one transfer task to evaluate student understanding of the concepts relating to statistical reasoning. A photocopy-ready version of the transfer task is included in this section.Implementation note:Students must be given the transfer task & rubric at the beginning of the unit. They need to know how they will be assessed and what they are working toward.Note: This task will take several days to complete. The task is best used as a final cumulative activity.Question 8: Students should have the same scale for male and female foot length, as well as for male and female height. This will allow for comparison of the distribution by inspection.Students must know how to:calculate z-scores using tables or calculatoruse the confidence interval formulacalculate standard deviation and meancalculate margin of errorEach student will:demonstrate their understanding of normal distribution including standard deviation and z-scoresinterpret statistical data using confidence intervals and margin of errorsolve problems involving interpretation of standard deviation Implementation note:Teachers need to consider what performances and products will reveal evidence of understanding?What other evidence will be collected to reflectthe desired results?4572000106680 Talking Feet Project - Student Transfer TaskWhat does your foot length say about your height?Can you predict people's height by how long their feet are?If a Grade 11 student's foot is 27 cm long, how tall is that student likely to be? In this activity, you will use data collected from your class to determine whether a relationship exists between foot length and height.Example: Students from South Africa, the United Kingdom, Australia and New Zealand measured and recorded their height and foot length in centimetres and entered this information into the Census at School database. A random sample was selected from the combined data to create the scatter plot below, where each dot shows an ordered pair of height relative to foot length. Scatter plot of height and foot length for 15-year-old studentsSource: : Feet ProjectPart 1: Data CollectionCollect data on foot length in centimetres and height in centimetres from at least two grade 11 students not in your class (at least one male, one female) using the Talking Feet Survey Form provided.Fill in the Predicting Height from Foot Length Class Data table provided for all data collected by your class.Create a scatter plot similar to the one shown on the previous page.Manually draw a line of best fit on the scatter plot.From the graph:Is there a relationship that exists between foot length and height?Is your graph similar to the above example from other countries?Determine the mean and standard deviation for the following:Male foot lengthMale heightFemale shoe lengthFemale heightCreate a normal distribution curve for:Male foot lengthMale heightFemale foot lengthFemale heightLabel the following distribution curve with:Mean3 standard deviations above and below the meanUsing your normal distribution curve answer the following:What percent of students have a height less than 150 cm?a malea femaleWhat percent of students have a foot size larger than 26 cm? a malea femalec. In two or three sentences explain your findings regarding the percentages found in questions (a) and (b).Talking Feet ProjectCalculate the z-score for your foot size and height.What standard deviation for your foot length would be necessary so that you would have the same z-score for your height (prove your answer algebraically)?Of the four sets of data collected which ones, if any, falls within the 68-95-99 Rule for a normal distribution curve? Provide an explanation if necessary.Calculate the 95% confidence interval for each set of data and explain what it means.What is the margin of error for each set of data? Explain what factors could influence this.Conclusion: What does your foot length say about your height?Can you predict people's height by how long their feet are?If a Grade 11 student's foot is 27 cm long, how tall is that student likely to be?52578000Talking Feet Survey FormParticipants NameGender (M / F)GradeFoot Length (cm)Height (cm)1111111111111111111111References:US & Canada Shoe SizeM3?44?55?66?77?88?910?11?12?14F55?66?77?88?99?1010?12131415?Foot Length (cm)---22.823.123.523.824.124.524.825.125.425.72626.727.327.928.829.2Height in feet and inchesHeight (cm)Height in feet and inchesHeight (cm)Height in feet and inchesHeight (cm)4 feet 0 inches121.925 feet 0 inches152.406 feet 0 inches182.884 feet 1 inches124.465 feet 1 inches154.946 feet 1 inches185.424 feet 2 inches127.005 feet 2 inches157.486 feet 2 inches187.964 feet 3 inches129.545 feet 3 inches160.026 feet 3 inches190.504 feet 4 inches132.085 feet 4 inches162.566 feet 4 inches193.044 feet 5 inches134.625 feet 5 inches165.106 feet 5 inches195.584 feet 6 inches137.165 feet 6 inches167.646 feet 6 inches198.124 feet 7 inches139.705 feet 7 inches170.186 feet 7 inches200.664 feet 8 inches142.245 feet 8 inches172.724 feet 9 inches144.785 feet 9 inches175.264 feet 10 inches147.325 feet 10 inches177.804 feet 11 inches149.865 feet 11 inches180.345029200-109855Predicting Height from Foot Length Class DataGrade 11 Male Data: Grade 11 Male Data Continued: Foot Length (cm)Height (cm)Foot Length (cm)Height (cm) 5026025114300Predicting Height from Foot Length Class DataGrade 11 Female Data: Grade 11 Female Data Continued:Foot Length (cm)Height (cm)Foot Length (cm)Height (cm) Glossary central tendency – Measures of central tendency are numbers that indicate the center of a set of ordered numerical data. The three common measures of central tendency are the mean, median and the mode.confidence interval – The interval within which the value of a random variable is estimated to lie with a stated degree of probabilityhistogram - A graph consisting of bars used to visually represent a frequency table where at least one of the scales represents continuous dataline of best fit - A line on a scatter plot that best defines or expresses the trend shown in the plotted pointsmargin of error - The proportion added to and subtracted from the result to construct the confidence intervalmean – The mean (or "arithmetic mean") is a measure of central tendency of a set of data represented by numbers. Adding all of the values and dividing by the number of values calculate the mean. The symbol for the arithmetic mean is a letter with a segment above it.median - The median is a measure of central tendency of a set of data represented by numbers. The median is the "middle" of a set of numbers in ascending or descending order. The symbol for the median is usually the letter "M".mode – The mode is a measure of central tendency of a set of data represented by numbers. The mode is the most frequently occurring number. There is no standard symbol associated with the mode.normal distribution curve - A bell-shaped curve showing a particular distribution of probability over the values of a random variable.scatter plot - A graph consisting of individual points whose coordinates represent values of an independent and a dependent variablestandard deviation - A measure of the dispersion of a frequency distributionGlossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.z-score - A standard score that measures how many standard deviation units away from the mean a particular value lays. AssessmentMathematics 20-2Statistical ReasoningRubric LevelCriteriaExcellent5Proficient4Adequate3Limited2Insufficient1ContentProcedures and calculations are efficient and effectiveMay contain minor errors that does not affect understandingProcedures and calculations are reasonable and may contain minor errorsMay contain error(s) that hinder a complete solutionProcedures and calculations are basic May contain a major mathematical error or omissionProcedures and calculations are basic May contain several major mathematical errors or omissionsDevelops an initial start that may be partially correct or could led to a correct solutionReasoningMakes significant comparisons and connections with dataMakes reasonable comparisons and connections with dataMakes some comparisons and connections with data Makes minimal comparisons and connections with dataMakes minimal or no comparisons and connections with dataCommunicationUses significant mathematical language to explain understandingUses mathematical language to explain understandingUses common language to explain understandingCommunication is weakCommunication is weak or absentWhen work is judged to be limited or insufficient, the teacher makes decisions about appropriate intervention to help the student improve.Possible Solution to Talking Feet ProjectTalking Feet Survey FormParticipants NameGender (M / F)GradeFoot Length (cm)Height (cm)PamF1127.5180JillF1123.0163BobM1129.5185JimM1126.5166AmaleF1125.0172KevinM1127.0178KellyF1121.0162AmaraF1123.0148RalphM1126.0167TarekM1126.0179SeanM1119.01475029200-10985500Predicting Height from Foot Length Class DataGrade 11 Male Data: Grade 11 Female Data: Foot Length (cm)Height (cm)Foot Length (cm)Height (cm)29.518527.518026.516623.016327.017825.017226.016721.016226.017923.014819.014723.014524.015525.017128.017223.015723.014123.015725.015025.516627.015923.016325.517226.0165Height (cm)Height vs. Foot Length1301351401451501551601651701751801851901517192123252729Foot Length (cm)Students will manually draw in line of best fit5. Answers will vary depending on student graphQuestions 6, 7, 8Male foot lengthMale heightFemale foot lengthFemale height162.49.4Using your normal distribution curve answer the following;What percent of students have a height less than150cm? a male a female What percent of students have a foot size larger than 26 cm? a male a femalec. In two or three sentences explain your findings regarding the per cents found in questions (a) and (b). Answers will vary.Calculate the z-score for your foot size and height. (Female sample)Height: 160 cmShoe Size: 24.5 cm11. What standard deviation for your foot length would be necessary so that you would have the same z-score for your height (prove your answer algebraically)?Adjusting Standard Deviation12. Of the four set of data collected which ones, if any, falls within the 68-95-99 Rule for a normal distribution curve? Provide an explanation if necessary.68-95-99 Rule:Male HeightFemale HeightMale Foot LengthFemale Foot Length** None of the 4 distributions fit the 68-95-99 Rule13. Calculate the 95% confidence interval for each set of data and explain what it means.95% Confidence Interval14. What is the margin of error for each set of data? Explain what factors could influence this.Margin of Errorupper confidence level - mean = errorlower confidence limit - mean = errorMale Height:Female Height:Margin of Error: Margin of Error: Male Foot Length:Female Foot Length:Margin of Error: Margin of Error: STAGE 3 Learning PlansLesson 1Standard DeviationSTAGE 1BIG IDEA: Statistics summarizes data and predict future outcomes in areas such as advertisement, sales, sports and academics. We can make logical assumptions and back them up with numerical data.ENDURING UNDERSTANDINGS: Students will understand …There are different measures of central tendency, which may be appropriate in different situations.Data can be presented in a misleading fashion.ESSENTIAL QUESTIONS: Which measure of central tendency is better in certain situations?When is the mean the best measure of central tendency to use?When is the median the best measure of central tendency to use?When is the mode the best measure of central tendency to use?When is it better to use mean versus median? When is it appropriate to use a sample set instead of an entire population?Why would people misrepresent data using statistics?When is the size of the standard deviation important?KNOWLEDGE: Students will know …how to identify measures of central tendencywhat standard deviation isthe characteristics of a normal distributionSKILLS: Students will be able to …calculate the population standard deviation of a data set.Implementation note:Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete.Lesson SummaryStudents will understand what standard deviation means.Students will be able to calculate the standard deviation of a given set of data. Lesson PlanHookTeam ATeam BBoth of these teams have the same mean height. How are the two teams different? Which team would you rather have? Is there a way to mathematically describe the differences between these two basketball teams?LessonWe use a statistical measure called the standard deviation to help describe the spread of data.We want to calculate the standard deviation for the amount of gold coins each pirate on a ship has. Source: are 100 pirates on the ship. In statistical terms, this means we have a population of 100. If we know the amount of gold coins each of the 100 pirates have, we use the?standard deviation equation for an entire population:where,σ = the standard deviationx = each value in the population= the mean of the valuesN = the number of values (the population)What if we don't know the amount of gold coins each of the 100 pirates have? For example, we only had enough time to ask 5 pirates how many gold coins they have. In statistical terms this means we have a sample size of 5 and in this case we use the?standard deviation equation for a sample of a population:where,s = the standard deviationx = each value in the sample= the mean of the valuesN = the number of values (the sample size)The rest of this example will be done in the case where we have a sample size of 5 pirates; therefore we will be using the standard deviation equation for a sample of a population.Here are the amounts of gold coins the 5 pirates have:4, 2, 5, 8, 6.Now, let's calculate the standard deviation:1. Calculate the mean:= == 52. Calculate for each value in the sample:= 4 – 5 = -1= 2 – 5 = -3= 5 – 5 = 0= 8 – 5 = 3= 6 – 5 = 1x1-x = 4 – 5 = -13. Calculate : = (-1)2 + (-3) 2 + 02 + 32 + 12 = 204. Calculate the standard deviation: = 2.24The standard deviation for the amounts of gold coins the pirates have is 2.24 gold coins.Note to teachers: Show how to calculate standard deviation using the graphing calculator if desired.Practice problems:Consider the following sets of data.A = {9, 10, 11, 7, 13} B = {10, 10, 10, 10, 10} Calculate the mean of each data set.Calculate the standard deviation of each data set. Which set has the largest standard deviation? Is it possible to answer question c. without calculations of the standard deviation? 2. The frequency table of the monthly salaries of 20 people is shown below. salary ($)frequency35005400084200543002a. Calculate the mean of the salaries of the 20 people. b. Calculate the standard deviation of the salaries of the 20 people.Going BeyondA given data set has a mean μ and a standard deviation σ. What are the new values of the mean and the standard deviation if the same constant k is added to each data value in the given set? Explain. What are the new values of the mean and the standard deviation if each data value of the set is multiplied by the same constant k? Explain.ResourcesPrinciple of Mathematics - NelsonSection 5.3 (pages 254 - 265) Supporting *Online practice questions: * – The mean (or "arithmetic mean") is a measure of central tendency of a set of data represented by numbers. Adding all the values and dividing by the number of values calculates the mean. The symbol for the arithmetic mean is a letter with a segment above it.median - The median is a measure of central tendency of a set of data represented by numbers. The median is the "middle" of a set of numbers in ascending or descending order. The symbol for the median is usually the letter "M".mode – The mode is a measure of central tendency of a set of data represented by numbers. The mode is the most frequently occurring number. There is no standard symbol associated with the mode.standard deviation - A measure of the dispersion of a frequency distributionGlossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.Lesson 2The Normal CurveSTAGE 1BIG IDEA: Statistics summarizes data and predict future outcomes in areas such as advertisement, sales, sports and academics. We can make logical assumptions and back them up with numerical data.ENDURING UNDERSTANDINGS: Students will understand …Certain data is normally distributed (bell curve).ESSENTIAL QUESTIONS: When is it appropriate to use a sample set instead of an entire population?When is the size of the standard deviation important?Why are some sets of data normally distributed and others are not?KNOWLEDGE: Students will know …the characteristics of a normal distributionhow standard deviation effects the curve and area under the curveSKILLS: Students will be able to …determine if a set approximates a normal curveLesson SummaryStudents will recognise the properties of a normal curve.Students will be able to determine whether data fits a normal distribution. Lesson PlanHook: The Bell Curve Student Source: properties of a normal distribution (bell curve):The total area under the curve is equal to 1.The normal curve extends infinitely to the left and right (i.e. does not actually reach the horizontal axis).The normal curve is symmetrical about the mean (i.e. 50% of area under the curve is to the left of the mean and 50% is to the right).The area under the curve represents all the data.The mean, median and mode are the same value.Task 1: Have students order themselves from shortest to tallest. The middle person will represent the mean height. How many students represent 68% of the class? Example: in a class of 26, 17 students would represent 68%. Those students would represent one standard deviation above and below the mean. Two standard deviations above and below the mean include 95% of the data. How many students represent 95% of the class? Three standard deviations above and below the mean include 98% of the data. How many students represent 98% of the class? **Be creative! Wrap the students in "caution tape" to represent the different standard deviations. Probing Question: are the people included in the 68% also included in the 95%?68-95-99 Rule: From the mean to one standard deviation above and below, 68.26% (~68%) of the data will fall under the curve. From the mean to two standard deviations above and below, 95.44% (~95%) of the data will fall under the curve.From the mean to three standard deviations above and below, 99.74% (~99%) of the data will fall under the curve. Task 2:Energizer wants to examine the data of the life of an AA Battery. The company tested 44 batteries to determine the mean life of the batteries as well as the standard deviation. The lifetime, in hours, of the batteries tested is shown below.899952830.5872874875880881882.38858991049932903922953.7952.5962.9975987997.2949901.7903904905908910915919.8920922898768.2849845897.9837.1840849.9851851.4854.8862.1Calculate the mean and standard deviation of the plete the table:Interval<750750-800800-850850-900900-950950-10001000-1050>1050# of batteries% of batteries in the intervalUsing the table, create a histogram to display the data.If we were to place a curve on the histogram, the following shape would be obtained:Label the interval & data values and the standard deviation away from the mean.What happens to our curve if we change the standard deviation? The following reference may help: .*As our standard deviation increases, what happens to the curve?*As our standard deviation decreases, what happens to the curve? Going BeyondHave students come up with their own survey question and plot the data in a histogram? Does it form a normal curve? Why or why not?ResourcesPrinciples of Mathematics - NelsonSection 5.4 (pages 266-268) Supporting Glossarycentral tendency – Measures of central tendency are numbers that indicate the center of a set of ordered numerical data. The three common measures of central tendency are the mean, median and the mode.histogram - A graph consisting of bars used to visually represent a frequency table where at least one of the scales represents continuous dataGlossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.normal distribution curve - A bell-shaped curve showing a particular distribution of probability over the values of a random variableLesson 3Z-scoresSTAGE 1BIG IDEA: Statistics summarizes data and predict future outcomes in areas such as advertisement, sales, sports and academics. We can make logical assumptions and back them up with numerical data.ENDURING UNDERSTANDINGS: Students will understand …Certain data is normally distributed (bell curve).ESSENTIAL QUESTIONS: When is the size of the standard deviation important?KNOWLEDGE: Students will know …the characteristics of a normal distributionhow standard deviation effects the curve and area under the curvewhat a z-score is and how it applies to the normal distributionSKILLS: Students will be able to …compare the properties of two or more normally distributed data setsdetermine the z-score for a given value in a normally distributed data setsolve a contextual problemLesson SummaryStudents will calculate the mean and standard deviation of two given sets of data.Students will display their findings in a bell curve.Students will calculate z-scores.Students will solve a contextual problem. Lesson PlanHookAre girls smarter then boys? Prove it. State your opinion and supporting evidence in a few sentences.Lesson GoalStudents will be able to determine and explain the z-score for a given value in a normally distributed data set. Students will be able to compare two sets of normally distributed data by converting to z-scores. Activate Prior Knowledgebell curve propertiesstandard deviation calculationmean of a sample set calculationbasic understanding of a normal distributionTask:The following marks are from a 20-2 math class. Calculate the mean and the standard deviation. BoysGirls7270605555535464759268806062656050817153Mean for boys = 65Standard Deviation = 11.8Mean for girls = 65Standard Deviation = 9.8Show your results on a bell curve. Solution:Girls DataBoys Data87630508000Leading Questions:What is the standard deviation of both graphs?How do they compare?How does it affect the shape of the graph?Are there any extreme values (high vs. low)?Who is smarter? How can you tell?Notes:A z-score is a standard score that measures how many standard deviation units away from the mean a particular value lays.Converting values into z-scores allows for an easy comparison of two data sets.Convert the boys graph into z-scores using the following z-score formulaz-score = z = Solution:z = z = z = z = z = z = Notes:So, your bell curve for the boy’s turns into:Using the mean and standard deviation from the boy’s graph, we can convert the girl’s values in order to compare the two sets of data.i.e. If the girls mark was 74.8% where would this value lie on the boys curve? Girls Data Boys Dataz = z = z = z = z = z = When the two sets of z-scores are compared on one bell curve the following results: Ray’s final exam marks are shown below, together with the class mean and standard deviation for each subject. By calculating z-scores, determine in which subject Ray performed best relative to the rest of his class. Display your findings in a bell curve.SubjectRay’s MarkMean MarkStandard DeviationMath746812English797314Social686611Solution:Convert to z scoresz(Math) = z(English) = z(Social) = 2098281600Using the data from task 2, what percent of students in Ray’s class scored:lower than him in Math (use tables)lower than him in English (technology)higher than him in Social (either)Solution:69%67%43% Going BeyondManipulate the z-score formula to find the value (x), the mean (?) or the standard deviation (σ). AssessmentPractice Questions:An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed, what percent of Acme light bulbs will last at most 365 days? Suppose scores on an IQ test are normally distributed. If the test has a mean of 100 and a standard deviation of 10, what percent of students will score between 90 and 110? Find the area under the standard normal curve for the following, using the z-table. Sketch each one.between z = 0 and z = 0.78between z = -0.56 and z = 0between z = -0.43 and z = 0.78between z = 0.44 and z = 1.50to the right of z = -1.33It was found that the mean length of 100 parts produced by a lathe was 20.05 mm with a standard deviation of 0.02 mm. What percent of parts selected will have a lengthbetween 20.03 mm and 20.08 mmbetween 20.06 mm and 20.07 mmless than 20.01 mmgreater than 20.09 mmA company pays its employees an average wage of $3.25 an hour with a standard deviation of $0.60. If the wages are approximately normally distributed, determinethe proportion of the workers getting wages between $2.75 and $3.69 an hourthe minimum wage of the highest 5%The average life of a certain type of motor is ten years, with a standard deviation of two years. If the manufacturer is willing to replace only 3% of the motors that fail, how long a guarantee should he offer? Assume that the lives of the motors follow a normal distribution.ResourcesPrinciples of Mathematics - NelsonSection 5.5 (pages 283 - 295) Supporting Practice Questions: GlossaryGlossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.z-score - A standard score that measures how many standard deviation units away from the mean a particular value laysLesson 4Confidence IntervalSTAGE 1BIG IDEA: Statistics summarizes data and predict future outcomes in areas such as advertisement, sales, sports and academics. We can make logical assumptions and back them up with numerical data.ENDURING UNDERSTANDINGS: Students will understand …Data can be presented in a misleading fashion.Predictions based on statistics will contain error.ESSENTIAL QUESTIONS: Why would people misrepresent data using statistics?KNOWLEDGE: Students will know …how confidence levels, margin of error and confidence intervals may vary depending on the size of the random samplethe significance of a confidence interval, margin of error or confidence levelSKILLS: Students will be able to …make inferences about a population from sample data using given confidence intervalsLesson SummaryStudents will review z-scores, median, standard deviation and 68-95-99 Rule through warm up. An example is provided for students, which demonstrates how a confidence intervals is derived and used (using 95%C.I. =)Students will work on tasks that require them to use mean, standard deviation, z-scores to determine confidence interval. Lesson PlanLesson GoalStudents will determine the 95% confidence intervals for data that is given/collected.Students will determine confidence intervals for data that is given/collected using the formula: 95%C.I. = ? ± 1.96σ.Activate Prior Knowledgecalculate standard deviation, mean and z-scores68-95-99 RuleLessonWarm up: Recall the 68-95-99 Rule: (These values are approximate.) 68% of the data falls within ______ standard deviations of the mean.95% of the data falls within ______ standard deviations of the mean.99% of the data falls within ______ standard deviations of the mean.Investigation: The 68-95-99 Rule stated that, in a normal distribution, approximately 95% of the data lie within two standard deviations of the mean.Now let’s find the precise z-scores that represent the SYMMETRIC INTERVAL that contains 95% of the data.Use your calculator to determine the area under the standard normal curve between z = -2 and z = 2. What percent of data is in this range?Now use your calculator to determine the area between z = -1.99 and z = 1.99. What percent of data is in this range?Repeat this process by deducting 0.01 from both the upper and lower bound. Continue until you find a range of z-scores for which the area under the standard normal curve is less than 0.95. Which range of z-scores resulted in an area close to 0.95 or 95%?Why is the area not exactly 95%? How could we find an area that is exactly 95%?Example: A headline in a local newspaper reports that "70% of Residents Opposed to Proposed Bylaw." The accompanying article states that the headline was based on an opinion poll the city administration conducted. The article also states that the poll was based on a random survey of 1000 residents and that the results are considered to be "accurate to within three percentage points nineteen times out of twenty." data that have a normal distribution with a mean ? and standard deviation σ, a 95% confidence interval is:? ± 1.96σThis is the range of values that lie within 1.96 standard deviations of the mean. The percent of data that lies in that range is 0.95 or 95%.Note: Round the lower bound down and round the upper bound up.Task: A chocolate bar manufacturing company has found that the mean mass of its chocolate bars is 56 grams and the standard deviation is 2 grams. Construct a 95% confidence interval for the mass of chocolate bars, to the nearest whole number.ResourcesPrinciples of Mathematics - NelsonSection 5.6 (pages 295 - 301)Assessment1. Determine a 95% confidence interval for each set of information given below.a) ? = 50, σ = 2b) ? = 80, σ = 5c) ? = 5.8, σ = 0.02d) ? = 120, σ = 12. A sample of 250 trees in a logging area has a mean diameter of 52 cm, with a standard deviation of 8.5 cm. Determine a 95% confidence interval for this data. [35.34 – 68.66] Glossaryconfidence interval – The interval within which the value of a random variable is estimated to lie with a stated degree of probabilityGlossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.Lesson 5Confidence Intervals in Print and Media 4STAGE 1BIG IDEA: Statistics summarizes data and predict future outcomes in areas such as advertisement, sales, sports and academics. We can make logical assumptions and back them up with numerical data.ENDURING UNDERSTANDINGS: Students will understand …Data can be presented in a misleading fashion.Predictions based on statistics will contain error.ESSENTIAL QUESTIONS: When is it appropriate to use a sample set instead of an entire population?Why would people misrepresent data using statistics?KNOWLEDGE: Students will know …how confidence levels, margin of error and confidence intervals may vary depending on the size of the random samplethe significance of a confidence interval, margin of error or confidence levelSKILLS: Students will be able to …support a position by analyzing statistical data presented in the mediainterpret and explain confidence intervals and margin of error, using examples found in print of electronic mediamake inferences about a population from sample data using given confidence intervalsprovide examples from print or electronic media in which confidence intervals are used to support a particular positionLesson SummaryMake inferences about a population from sample data, using given confidence intervals, and explain the reasoning.Provide examples from print or electronic media in which confidence intervals and confidence levels are used to support a particular position.Interpret and explain confidence intervals and margin of error, using examples found in print or electronic media.Support a position by analyzing statistical data presented in the media. Lesson PlanHookI don't think he got the size of the circles right...LessonLet's look at examples of statistics in the media: [Choose one or more to talk about as a class] through the articles, as a class answer the following questions:How does the margin of error affect the data?What do the confidence intervals used in the data actually mean?Is this data skewed to favour a certain position? Were there enough people sampled to represent the whole population?Overall, was this poll/survey done well? How could something be changed to make it better/worse? Going BeyondHave students create their own misleading and skewed news article to "support" a particular position. Their article should include raw data as well as confidence intervals and/or margin of error.ResourcesPrinciples of Mathematics - NelsonSection 5.6 (page 304)AssessmentHave students find their own examples of confidence intervals and margin of error in the media. Have them interpret and explain the data, then side with a certain position and use the data they found to support their position.Suggested resource(s) is/are:The Edmonton Journal Website Glossarymargin of error - The proportion added to and subtracted from the result to construct the confidence intervalGlossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings. ................
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