Georgia Department of Education. Accelerated Mathematics II Frameworks Student Edition. Unit 3 Statistics

Transcription

1 Accelerated Mathematcs II Frameworks Student Edton Unt 3 Statstcs 2 nd Edton Aprl, 2011

2 Table of Contents Introducton... Error! Bookmark not defned. Task Task Task Task Task Task Task 7: Task Task Task Aprl, 2011 Page 2 of 36

3 Accelerated Mathematcs II Unt 3 Statstcs Students Edton Introducton Data analyss and probablty can be one of the most empowerng strands of mathematcs that you teach because students can use ths strand of mathematcs n ther everyday lfe. Many tmes, students thnk that math s just for math teachers or engneers. If you know of your students nterests then t s pretty easy to show them that statstcs s for everyone. Many students that I teach are really nterested n ther appearances. They often model ther clothes and harstyles after famous celebrtes that they thnk are attractve. In tasks 5-7 of ths framework, students wll use statstcs to determne f move stars are exceptonally beautful when compared to the general populaton. They wll do ths by makng certan measurements of the face and calculatng ther ratos. Students wll explore whether the move star ratos are close to average or whether they are outlers (more than 2 standard devatons away from the mean). Are there any potental Olympans among your students? Should male sports be separated from female sports? In tasks 8 and 9, students wll compare jump dstances of Olympc stars to average people. They wll also explore whether jump dstances are based on leg length, gender, or both. As you read through ths framework, thnk of your own students. If these applcatons do not apply to the nterests of your students, fnd some that wll. If you can collect quanttatve data, then you can analyze t accordng to the methods descrbed n ths framework. Key Standards MA2D1: Usng sample data, students wll make nformal nferences about populaton means and standard devatons a. Pose a queston and collect sample data from at least two dfferent populatons. b. Understand and calculate the means and standard devatons of sets of data c. Use means and standard devatons to compare data sets d. Compare the means and standard devatons of random samples wth the correspondng populaton parameters, ncludng those populaton parameters for normal dstrbutons. Observe that the dfferent sample means vary from one sample to the next. Observe that the dstrbuton of the sample means has less varablty than the populaton dstrbuton. Aprl, 2011 Page 3 of 36

4 RELATED STANDARDS ADDRESSED: MA2P1. Students wll solve problems (usng approprate technology). a. Buld new mathematcal knowledge through problem solvng. b. Solve problems that arse n mathematcs and n other contexts. c. Apply and adapt a varety of approprate strateges to solve problems. d. Montor and reflect on the process of mathematcal problem solvng. MA2P2. Students wll reason and evaluate mathematcal arguments. a. Recognze reasonng and proof as fundamental aspects of mathematcs. b. Make and nvestgate mathematcal conjectures. c. Develop and evaluate mathematcal arguments and proofs. d. Select and use varous types of reasonng and methods of proof. MA2P3. Students wll communcate mathematcally. a. Organze and consoldate ther mathematcal thnkng through communcaton. b. Communcate ther mathematcal thnkng coherently and clearly to peers, teachers, and others. c. Analyze and evaluate the mathematcal thnkng and strateges of others. d. Use the language of mathematcs to express mathematcal deas precsely. MA2P4. Students wll make connectons among mathematcal deas and to other dscplnes. a. Recognze and use connectons among mathematcal deas. b. Understand how mathematcal deas nterconnect and buld on one another to produce a coherent whole. c. Recognze and apply mathematcs n contexts outsde of mathematcs. MA2P5. Students wll represent mathematcs n multple ways. a. Create and use representatons to organze, record, and communcate mathematcal deas. b. Select, apply, and translate among mathematcal representatons to solve problems. c. Use representatons to model and nterpret physcal, socal, and mathematcal phenomena. Formulas and Defntons Census: A census occurs when everyone n the populaton s contacted. Emprcal Rule s as follows: If a dstrbuton s normal, then approxmately 68% of the data wll be located wthn one standard devaton symmetrc to the mean Aprl, 2011 Page 4 of 36

5 95% of the data wll be located wthn two standard devatons symmetrc to the mean 99.7% of the data wll be located wthn three standard devatons symmetrc to the mean Frequency Dstrbuton: Instead of lstng every data pont, a frequency dstrbuton wll lst the value wth ts assocated frequency (number of tmes t s lsted. For example, f the data are 2,2,2,2,2,3,3,3,3,3,3,5,6,6,6 a frequency dstrbuton for the data would be x frequency ( F) = 1 The mean of a frequency dstrbuton, μ, can be found by calculatng n where F s the frequency of the value and n s the sample sze. Note: The sample sze n s the sum of the frequency column. The standard devaton of a frequency dstrbuton can be found by calculatng n Golden Rato: σ = n 2 ( F( ) ) = ϕ = whch s approxmately = n Measures of Center n = 1 Mean: The average =. The symbol for the sample mean s. N The symbol for the populaton mean s μ. Medan: When the data ponts are organzed from least to greatest, the medan s the mddle number. If there s an even number of data ponts, the medan s the average of the two mddle numbers. Mode: The most frequent value n the data set. Measures of Spread (or varablty) Interquartle Range: Q3 Q1 where Q 3 s the 75 th percentle (or the medan of the second half of the data set) and Q 1 s the 25 th percentle (or the medan of the frst half of the data set). Aprl, 2011 Page 5 of 36

6 where Mean Devaton: s each ndvdual data pont, s N the sample mean, and N s the sample sze. Varance: In ths unt, I decded to use the populaton varance throughout. The students have an ntutve understandng of the populaton varance as opposed to the sample varance. The sample varance should be explored n the future. The formula for the populaton varance s as follows: varance : = 1 ( ) Aprl, 2011 Page 6 of 36 n Standard Devaton: The standard devaton s the square root of the varance. The formula for the populaton standard devaton s as follows: standard devaton : n n = 1 2 ( ) Normal Dstrbuton: The standard devaton s a good measure of spread when descrbng a normal dstrbuton. Many thngs n lfe vary normally. Many measurements vary normally such as heghts of men. Most men are around the average heght, but some are shorter and some are taller. The shape of the dstrbuton of men s heghts wll be a bell shape curve. All normal dstrbutons are bell shaped; however, all bell shaped curves are not normal. If a dstrbuton s a normal dstrbuton, then the Emprcal Rule should apply (see Emprcal Rule above). Parameters: These are numercal values that descrbe the populaton. The populaton mean s symbolcally represented by the parameter μ. The populaton standard devaton s symbolcally represented by the parameter σ. Random: Events are random when ndvdual outcomes are uncertan. However, there s a regular dstrbuton of outcomes n a large number of repettons. Sample: A subset, or porton, of the populaton. Samplng Dstrbuton of a Sample Mean: The samplng dstrbuton of a sample mean refers to the dstrbuton of the mean of random samples of a gven sze drawn from the populaton. For example, f the sample sze s 5, then the averages of 5 randomly selected values (per sample) from the populaton are recorded. Ths process s repeated enough tmes to see the shape of the samplng dstrbuton emerge. If the sample sze s small, the shape of the samplng n 2

7 dstrbuton of the sample mean wll be smlar to that of the populaton. If the sample sze s large, the shape of the samplng dstrbuton wll become more normal-lke. The mean of the samplng dstrbuton (the mean of the averages) wll theoretcally be the same as the mean of the populaton regardless of sample sze. The standard devaton of the averages wll be smaller than the populaton standard devaton. Statstcs: These are numercal values that descrbe the sample. The sample mean s symbolcally represented by the statstc. The sample standard devaton s symbolcally represented by the statstc s x. Unt Overvew Students should not have a problem calculatng the standard devaton (except for understandng the symbols n the formula) unless the data s lsted n a frequency dstrbuton. Task #3 s desgned to help students overcome ths problem. Sometmes students do not understand what the standard devaton measures. They just calculate the number and move on. Tasks #2 and #4 s desgned to help students understand why the standard devaton s used and when t s a good measure of spread. The man problem that students mght have wth tasks #5-10 s understandng the concept that the dstrbuton of a sample mean s dfferent than the dstrbuton of ndvdual values. By havng students randomly select dfferent samples and recordng the means of these samples, they should realze that the two dstrbutons are not the same and that the dstrbuton of the mean has less varance. Tasks The remanng content of ths framework conssts of student tasks or actvtes. The frst s ntended to launch the unt. Each actvty s desgned to allow students to buld ther own algebrac understandng through exploraton. The last task s a culmnatng task, desgned to assess student mastery of the unt. There s a student verson, not ncluded n the Student Verson of the unt, as well as a Teacher Edton verson that ncludes notes for teachers and solutons. Aprl, 2011 Page 7 of 36

8 Task 1 Your teacher has a problem and needs your nput. She has to gve one math award ths year to a deservng student, but she can t make a decson. Here are the test grades for her two best students: Bryce: 90, 90, 80, 100, 99, 81 98, 82 Branna: 90, 90, 91, 89, 91, 89, 90, 90 Wrte down whch of the two students should get the math award and dscuss why they should be the one to receve t. Calculate the mean of Bryce s dstrbuton. Calculate the mean devaton, varance, and standard devaton of Bryce s dstrbuton. The formulas for mean devaton, varance, and standard devaton are below. Fll out the table to help you calculate them by hand. mean devaton : standard devaton : for Bryce n = 1 n n = 1 Mean devaton for Bryce: ( ) n varance : 2 n = 1 ( ) n 2 whch s the square root of the varance. ( ) 2 Aprl, 2011 Page 8 of 36

9 Varance for Bryce: Standard devaton for Bryce: = Based on ths nformaton, wrte down whch of the two students should get the math award and dscuss why they should be the one to receve t. What do these measures of spread tell you? Calculate the mean of Branna s dstrbuton. Calculate the mean devaton, varance, and standard devaton of Branna s dstrbuton. for Branna ( ) 2 Mean devaton for Branna: Varance for Branna: Standard devaton for Branna:.5 = What do these measures of spread tell you? Aprl, 2011 Page 9 of 36

10 Task 2 Create a set of 6 data ponts such that the varance and standard devaton s zero. Make a dotplot of the dstrbuton. Create a set of 6 data ponts such that the varance and standard devaton s one. Make a dotplot of the dstrbuton. Create a set of 6 data ponts such that the varance s four and the standard devaton s two. Make a dotplot of the dstrbuton. Create a set of 6 data ponts such that the varance s four and the standard devaton s two and the mean s seven. Make a dotplot of the dstrbuton. Create a set of 6 data ponts such that the varance s sxteen and the standard devaton s four and the mean s ten. Make a dotplot of the dstrbuton. Descrbe the process you used to come up wth your answers. What s the relatonshp between the standard devaton and varance? What does the standard devaton measure?. Aprl, 2011 Page 10 of 36

11 Task 3 2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5 Gven the data, calculate the mean by hand: Cody and Bernce both got the correct answer, but t took Bernce a lot longer. Bernce added all of the values and dvded the sum by 30. Cody told her that there was a qucker way to do t. Can you fgure out how Cody dd t? Show how Cody calculated the mean below. The teacher then asked the students to calculate the standard devaton of the data by hand. 2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5 Carl made a table, lsted every data pont, and used the formula that he learned two days ago, standard devaton : n = 1 ( ) n 2, to calculate the standard devaton. Jessca fnshed the problem n half of the tme that t took Carl because she lsted her data n a frequency dstrbuton and calculated the standard devaton n a slghtly dfferent manner. Show how Jessca calculated the standard devaton below F = frequency Aprl, 2011 Page 11 of 36

12 Jessca then taught Gabe her method to calculate the standard devaton. Gabe dd not get the correct answer. He dd the followng: (2 3.5) + 9(3 3.5) + 5(4 3.5) + 11(5 3.5) 30 What mstake(s) dd Gabe make? Jessca also taught Melody the method to calculate the standard devaton. Melody also dd not get the correct answer. She dd the followng: ( ) + 9( ) + 5( ) + 11( ) 4 What mstake(s) dd Melody make? Jessca also taught Mara the method to calculate the standard devaton. Mara also dd not get the correct answer. She dd the followng: ( ) + 9( ) + 5( ) + 11( ) 30 What mstake(s) dd Mara make? Use Cody s and Jessca s method to calculate the mean and the standard devaton of the frequency dstrbuton below. F Make a formula for fndng the mean of any frequency dstrbuton. Let F stand for the frequency. Make a formula for fndng the standard devaton of a frequency dstrbuton. Aprl, 2011 Page 12 of 36

13 Task 4 Under certan condtons, those you wll dscover durng ths actvty, the Emprcal Rule can be used to help you make a good guess of the standard devaton of a dstrbuton. The Emprcal Rule s as follows: For certan condtons (whch you wll dscover n ths actvty), 68% of the data wll be located wthn one standard devaton symmetrc to the mean 95% of the data wll be located wthn two standard devatons symmetrc to the mean 99.7% of the data wll be located wthn three standard devatons symmetrc to the mean For example, suppose the data meets the condtons for whch the emprcal rule apples. If the mean of the dstrbuton s 10, and the standard devaton of the dstrbuton s 2, then about 68% of the data wll be between the numbers 8 and 12 snce 10-2 =8 and 10+2 = 12. We would expect approxmately 95% of the data to be located between the numbers 6 and 14 snce 10-2(2) = 6 and (2) = 14. Fnally, almost all of the data wll be between the numbers 4 and 16 snce 10 3(2) = 4 and (2) = 16. For each of the dotplots below, use the Emprcal Rule to estmate the mean and the standard devaton of each of the followng dstrbutons. Then, use your calculator to determne the mean and standard devaton of each of the dstrbutons. Dd the emprcal rule gve you a good estmate of the standard devaton? For your convenence, there are 100 data ponts for each dotplot. Collecton 1 Dot Plot x Estmated mean: Actual mean: Estmated standard devaton: Actual standard devaton: Dd the emprcal rule help gve you a good estmate of the standard devaton? Aprl, 2011 Page 13 of 36

14 Now that you know what the actual mean and standard devaton, calculate the followng μ σ = and μ + σ =. Locate these numbers on the dotplot above. How many dots are between these numbers? Is ths close to 68%? Do you thnk that the emprcal rule should apply to ths dstrbuton? Collecton 1 Dot Plot x Estmated mean: Actual mean: Estmated standard devaton: Actual standard devaton: Dd the emprcal rule help gve you a good estmate of the standard devaton? Now that you know what the actual mean and standard devaton, calculate the followng μ σ = and μ + σ =. Locate these numbers on the dotplot above. How many dots are between these numbers? Is ths close to 68%? Do you thnk that the emprcal rule should apply to ths dstrbuton? Collecton 1 Dot Plot x Estmated mean: Actual mean: Estmated standard devaton: Actual standard devaton: Dd the emprcal rule help gve you a good estmate of the standard devaton? Now that you know what the actual mean and standard devaton, calculate the followng μ σ = and μ + σ =. Locate these numbers on the dotplot above. How many Aprl, 2011 Page 14 of 36

15 dots are between these numbers? Is ths close to 68%? Do you thnk that the emprcal rule should apply to ths dstrbuton? Collecton 1 Dot Plot x Estmated mean: Estmated standard devaton: Actual mean: Actual standard devaton: Dd the emprcal rule help gve you a good estmate of the standard devaton? Now that you know what the actual mean and standard devaton, calculate the followng μ σ = and μ + σ =. Locate these numbers on the dotplot above. How many dots are between these numbers? Is ths close to 68%? Do you thnk that the emprcal rule should apply to ths dstrbuton? Collecton 1 Dot Plot x Estmated mean: Actual mean: Estmated standard devaton: Actual standard devaton: Dd the emprcal rule help gve you a good estmate of the standard devaton? Now that you know what the actual mean and standard devaton, calculate the followng μ σ = and μ + σ =. Locate these numbers on the dotplot above. How many dots are between these numbers? Is ths close to 68%? Do you thnk that the emprcal rule should apply to ths dstrbuton? Aprl, 2011 Page 15 of 36

16 Collecton 1 Dot Plot x Estmated mean: Actual mean: Estmated standard devaton: Actual standard devaton: Dd the emprcal rule help gve you a good estmate of the standard devaton? Now that you know what the actual mean and standard devaton, calculate the followng μ σ = and μ + σ =. Locate these numbers on the dotplot above. How many dots are between these numbers? Is ths close to 68%? Do you thnk that the emprcal rule should apply to ths dstrbuton? Summary: For whch dstrbutons dd you gve a good estmate of the standard devaton based on the emprcal rule? Whch dstrbutons dd not gve a good estmate of the standard devaton based on the emprcal rule? Whch dstrbutons had close to 68% of the data wthn one standard devaton of the mean? What do they have n common? For whch type of dstrbutons do you thnk the Emprcal rule apples? As you dscovered, the emprcal rule does not work unless your data s bell-shaped. Not all bell-shaped graphs are normal. The next two dotplots are bell-shaped. You wll apply the emprcal rule to determne f the bell-shaped graph s normal or not. Aprl, 2011 Page 16 of 36

17 Make a frequency dstrbuton for the dotplot below. Calculate the mean and the standard devaton of the dstrbuton. Collecton 1 Dot Plot x Mark the mean on your dotplot above. Calculate the followng μ σ = and μ + σ =. Mark these ponts on the x- axs of the dotplot. How many data ponts are between these values? Calculate the followng μ 2σ = and μ + 2σ =. Mark these ponts on the x-axs of the dotplot. How many data ponts are between these values? Calculate the followng μ 3σ = and μ + 3σ =. Mark these ponts on the x-axs of the dotplot. How many data ponts are between these values? Is t lkely that ths sample s from a normal populaton? Explan. Outlers are values that are beyond two standard devatons from the mean n ether drecton. Whch values from the data would be consdered to be outlers? Aprl, 2011 Page 17 of 36

18 Make a frequency dstrbuton for the dotplot below. Calculate the mean and the standard devaton of the dstrbuton. Collecton 1 Dot Plot x Mark the mean on your dotplot above. Calculate the followng μ σ = and μ + σ =. Mark these ponts on the x- axs of the dotplot. How many data ponts are between these values? Calculate the followng μ 2σ = and μ + 2σ =. Mark these ponts on the x-axs of the dotplot. How many data ponts are between these values? Calculate the followng μ 3σ = and μ + 3σ =. Mark these ponts on the x-axs of the dotplot. How many data ponts are between these values? Is t lkely that ths sample s from a normal populaton?. Outlers are values that are beyond two standard devatons from the mean n ether drecton. Whch values from the data would be consdered to be outlers? Aprl, 2011 Page 18 of 36

19 Task 5 Student name and gender Top of head to chn Top of head to pupl Pupl to nosetp Pupl to Lp Wdth of nose Outsde dstance between eyes Background nformaton for the unt: Eucld of Alexandra (300 B.C.) defned the golden rato n hs book, Elements. Snce then, artsts and archtects who deem ths rato as beng the most aesthetcally pleasng rato have used t as a bass for ther art and buldngs. It s thought that Leonardo da Vnc may have used the golden rectangle (havng sdes that are n the golden rato) when pantng the face of the Mona Lsa. The dmensons of Salvador Dal s pantng, Sacrament of the Last Supper, are also equal to the Golden Rato. The Greeks used the golden rato n buldng the Parthenon n Athens. Throughout tme, psychologsts have tred to determne what humans consder to be beautful. Gustav Theodor Fechner conducted an experment durng the 1860 s and found that students preferred rectangular shapes that had the golden rato (approxmately 1.62). Snce then, smlar experments have had conflctng results. Some psychologsts thnk that humans who have facal feature ratos closest to the golden rato are deemed as the most beautful. Other psychologsts thnk that the people wth the most average measurements n ther facal features are consdered to be the most beautful. Stll others beleve that people who are not average (have hgher cheek bones, thnner jaw, and larger eyes than normal) are deemed as the most beautful. Through the use of statstcs, and usng our class as a sample, we wll nvestgate the average dmensons of the face and calculate ther ratos. What queston are we tryng to answer wth our nvestgaton? Make the followng measurements for yourself and each student n your group. You should use centmeters to be a lttle more accurate for the small areas of your face. Aprl, 2011 Page 19 of 36

20 Wdth of head Harlne to pupl Nosetp to chn Lps to chn Length of lps Nosetp to lps Top of head to chn/wdth of head Top of head to pupl/pupl to Lp nose tp to chn/lps to chn nose tp to chn/pupl to nosetp wdth of nose/nosetp to lps outsde dst. Bet. Eyes/harlne to pupl length of lps/wdth of nose Were any of your ratos close to beng golden? Dd anyone n your group have ratos close to the golden rato? Pool the class data. You should round your ratos to the nearest hundredth. Make 7 dotplots of the class dstrbuton below. Fnd the mean and standard devaton of each dotplot. Descrbe the shape of the dstrbuton for each rato. Summary: Dd any student n your class have facal ratos that were close to beng golden? Whch ratos had an average close to beng golden? Aprl, 2011 Page 20 of 36

21 Task 6 Your teacher collected the ratos and recorded the dotplot of the rato: top of head to chn / wdth of head for all of hs/her classes. Take a moment and copy t on your paper. Calculate the mean and standard devaton of ths dstrbuton. Descrbe the shape. Top of head to chn / Wdth of head mean: S.D.: Compare ths dstrbuton to your class dstrbuton from yesterday. We are now gong to assume that ths dstrbuton s the dstrbuton of all people n our populaton. We are gong to nvestgate the dstrbuton of the mean for a random sample of 5 people from the populaton dstrbuton. Each student wll repeat the followng process 4 tmes: Use your calculator to choose your sample of sze 5 by httng the followng keys and makng the followng commands: Math, PRB, RANDINT(1, populaton sze, 5). If you have repeated numbers, that s okay. Go to the dotplot for the populaton and count the dots from left to rght to locate the numbers your calculator chose. Record the assocated ratos for these 5 ponts. Calculate the mean of these 5 ratos and record the mean. You should round your means to the nearest hundredth. Repeat ths process 3 more tmes for a new sample of 5 students. Record your 4 averages n a dotplot on the board. Aprl, 2011 Page 21 of 36

22 Draw the dotplot below. Label the horzontal axs as the Dstrbuton of the mean for the rato (Top of head to chn / Wdth of head) for n = 5 (sample sze of 5). Calculate the mean of the sample means. How does ths mean compare to the populaton mean? Calculate the standard devaton of the sample means. How does ths standard devaton compare to the populaton standard devaton. If you were a researcher, you would most lkely obtan a sample one tme. If you were to randomly choose a sample of 5 people, what s the probablty that the sample mean of those 5 people s the same as the populaton mean? (Hnt, count the dots on your dotplot). What happens to the dstrbuton when the sample sze ncreases? We are gong to nvestgate the dstrbuton of the mean for a random sample of 15 people from the populaton dstrbuton. Each student wll repeat the followng process 4 tmes: Use your calculator to choose your sample of sze 5 by httng the followng keys and makng the followng commands: Math, PRB, RANDINT(1, populaton sze, 15). If you have repeated numbers, that s okay. Go to the dotplot for the populaton and count the dots from left to rght to locate the numbers your calculator chose. Record the assocated ratos for these 15 ponts. Calculate the mean of these 15 ratos and record the mean. You should round your means to the nearest hundredth. Repeat ths process 3 more tmes for a new sample of 15 students. Record your 4 averages n a dotplot on the board. Draw the dotplot below. Label the horzontal axs as the Dstrbuton of the mean for the rato (Top of head to chn / Wdth of head) for n = 15 (sample sze of 15). Aprl, 2011 Page 22 of 36

23 Calculate the mean of the sample means. How does ths mean compare to the populaton mean? Calculate the standard devaton of the sample means. How does ths standard devaton compare to the populaton standard devaton. If you were a researcher, you would most lkely obtan a sample one tme and then proceed to do your research on that sample. If you were to randomly choose a sample of 15 people, what s the probablty that the sample mean of those 15 people s the same as the populaton mean? (Hnt, count the dots on your dotplot). Fll out the table for the followng: Dstrbuton Of ndvdual data ponts n the populaton Of the mean for sample sze = 5 Of the mean for sample sze = 15 Shape of the dstrbuton Mean of the Dstrbuton Standard Devaton of the Dstrbuton Probablty that the mean of one sample s the same as the mean of the dstrbuton Based on the values n the table above, what can you conclude? Aprl, 2011 Page 23 of 36

24 Task 7: Based on our answers to task 5 and 6, whch ratos of our class are close to beng golden? Do you thnk that beautful move stars or models would have facal ratos that are closer to the golden rato? What queston are we tryng to answer wth our nvestgaton? Move star/model name and gender Top of head to chn Top of head to pupl Pupl to nosetp Pupl to Lp Wdth of nose Outsde dstance between eyes Wdth of head Harlne to pupl Nosetp to chn Lps to chn Length of lps Nosetp to lps Make the followng measurements for at least two move stars or models, one male and one female, who are beautful n your opnon. You should use centmeters to be a lttle more accurate for the small areas of ther face. Top of head to chn/wdth of head Top of head to pupl/pupl to Lp Aprl, 2011 Page 24 of 36

25 nose tp to chn/lps to chn nose tp to chn/pupl to nosetp wdth of nose/nosetp to lps outsde dst. Bet. Eyes/harlne to pupl length of lps/wdth of nose Were any of ther ratos close to beng golden? Dd anyone n your group have ratos close to the golden rato? Pool the class data. You should round your ratos to the nearest hundredth. Make 7 dotplots of the class dstrbuton below. Fnd the mean and standard devaton of each dotplot. Descrbe the shape of the dstrbuton for each rato. Copy the dotplots of the move stars/models next to the dotplots of the students from task #5. Fll n the table below to compare the move star/model dstrbutons to the class dstrbutons. Rato Top of head to chn/wdth of head Top of head to pupl/pupl to Lp nose tp to chn/lps to chn nose tp to chn/pupl to nosetp wdth of nose/nosetp to lps Shape of the dstrbuton Class Move Star Mean of the Dstrbuton Class Move Star Standard Devaton of the Dstrbuton Class Aprl, 2011 Page 25 of 36 Move Star Percent of ndvduals who had the golden rato n the dstrbuton Class Move Star

26 outsde dst. Bet. Eyes/harlne to pupl length of lps/wdth of nose In a paragraph, compare the move star/model dstrbutons to the class dstrbutons. Make sure that you dscuss center, shape and spread. If there are any dscrepances, dscuss possble reasons why they exst. What s the answer to the queston that was posed at the begnnng of ths task? Aprl, 2011 Page 26 of 36

27 Task 8 Background Informaton: The standng long jump s an athletc event that was featured n the Olympcs from 1900 to In performng the standng long jump, the sprnger stands at a lne marked on the ground wth hs feet slghtly apart. The athlete takes off and lands usng both feet, swngng hs arms and bendng hs knees to provde forward drve. In Olympc rules, the measurement taken was the longest of three tres. The jump must be repeated f the athletes falls back or uses a step at take-off. The men s record for the standng long jump s 3.71 meters. The women's record s 2.92 m (9 ft 7 n). Men and women are often separated n the Olympcs and also n hgh school sports for varous reasons. In terms of the standng long jump, do you thnk that they are separated due to the reason that men n general jump farther than women. Or, do you thnk that men can jump farther because they are generally taller and have longer legs than women? Pose a queston to nvestgate: Collect Data: Jump three tmes accordng to the method descrbed above. Record the class nformaton n the table below. Gender Length of leg from wast to the floor Jump dstance.best of 3 jumps Aprl, 2011 Page 27 of 36

28 Compare Males to Females: Make a back to back stem and leaf plot. Calculate the mean and standard devaton for the male and the female jump dstances. Compare the two dstrbutons. Compare Short to Tall: Dvde your class data by heght and not by gender. Make a back to back stem and leaf plot. Calculate the means and standard devatons. Compare the two dstrbutons. Make a scatterplot of (length of leg, length of jump). Ft a least squares regresson lne to the data. What s the correlaton coeffcent? Jump Dst. Length of Leg Is there an assocaton between leg length and jump dstance? Aprl, 2011 Page 28 of 36

29 Task 9 Your teacher has collected the data from all of hs/her Accelerated Mathematcs II classes. Assume that the populaton s All students n your Teacher s Accelerated Geometry/Advanced Algebra classes. Your class wll represent a sample of these students. Is your class a random sample? Explan. Compare Males to Females: Make a back to back stem and leaf plot for all GPS Accelerated Mathematcs II students n your teacher s classes. Calculate the mean and standard devaton for the male and the female jump dstances for the populaton. Compare the two dstrbutons. Compare your class dstrbuton from yesterday to the populaton dstrbutons. Is your class representatve of the populaton? Populaton mean Class mean Class standard devaton Males Females Populaton standard devaton Recall from yesterday that the men s record for the standng long jump s 3.71 meters, and the women's record s 2.92 m (9 ft 7 n). How many populaton standard devatons s 3.71 meters away from the male s populaton mean? How many populaton standard devatons s 2.92 meters away from the women s populaton mean? If a value s more than 2 standard devatons above or below the mean of a normal dstrbuton, then that value s consdered to be an outler. Is 3.71 meters an outler for men? Is 2.92 meters an outler for women? Random Samples: Use the randint feature on your calculator to select a random sample of 15 dfferent males from the populaton of all males n your teacher s GPS Geometry classes. Calculate the mean jump dstances of these 15 randomly selected males. Aprl, 2011 Page 29 of 36

30 Pool the class data. Each student should record ther sample mean on a class dotplot. Draw the class dotplot below: Mean jump dstances of males for a random sample of sze 15 What s the mean of the sample means? What s the standard devaton of the sample means? Now use the randint feature on your calculator to select a random sample of 15 dfferent females from the populaton of all females n your teacher s Accelerated Mathematcs II classes. Calculate the mean jump dstances of these 15 randomly selected females. Pool the class data. Each student should record ther sample mean on a class dotplot. Draw the class dotplot below: Mean jump dstances of females for a random sample of sze 15 What s the mean of the sample means? What s the standard devaton of the sample means? Compare the mean and standard devaton of the sample means to the populaton parameters. Fll n the table to help you make the comparson of the populaton mean, the class mean, and the mean of the random sample means. Aprl, 2011 Page 30 of 36

31 Males Females Populaton mean Class mean Mean of sample means Populaton standard devaton Class standard devaton Standard devaton of sample means Whch came closer to the populaton mean.the class mean or the mean of the sample means? Why? Whch had the smallest standard devaton the populaton dstrbuton, the class dstrbuton, or the dstrbuton of the sample means? Why? Is there a relatonshp between heght and jump dstance? Make a scatterplot of (length of leg, length of jump) for the students n all of your teacher s GPS Geometry classes. Ft a least squares regresson lne to the data. What s the correlaton coeffcent? Jump Dst. Length of Leg Compare ths scatterplot to yesterday s scatterplot of the class dstrbuton. Is there a stronger or weaker assocaton or s t about the same? Aprl, 2011 Page 31 of 36

32 Based on your scatterplot, do you thnk that jump dstance depends on the length of a person s leg? Ft a least squares regresson lne to your data. What s the equaton? Use ths equaton to predct the length of a person s leg who could jump 3.71 meters (male world record). Do you thnk that the Olympan s legs were ths long? Explan. Aprl, 2011 Page 32 of 36

33 Task 10 We have looked at samplng dstrbutons taken from normal dstrbutons (golden rato and jump dstances). Today we wll examne a samplng dstrbuton from a non-normal dstrbuton. What was the prce of your last harcut/style/color? Collect class data and draw dotplot of the dstrbuton below. Prce of your last harcut Descrbe the shape of the dstrbuton. Calculate the mean and the standard devaton: Calculate the medan and the Interquartle range: Are there any outlers? Whch s a better measure of center for the class data? Why? Is your harcut prce closer to the mean prce or the medan prce? Aprl, 2011 Page 33 of 36

34 Perform a smulaton 50 tmes to determne whether the mean or medan s a better measure of center. Your teacher wll randomly select a student wth hs/her calculator. If you are randomly selected, tell your teacher f your harcut s closer to the mean or medan prce for your class. You may be called on more than once. You may never be selected. It s random. Tally the responses below. Class Tally: Your ndvdual harcut prce s closer to the.. Mean: Medan: What percent of the 50 smulatons were closer to the mean? What percent of the 50 smulatons were closer to the medan? If you randomly pck a student n your class, s t more lkely that ther prce was closer to the mean or medan? Whch s a better measure of center? Dstrbuton of the Mean: Use the RandInt feature on your calculator to randomly select 5 harcuts. Fnd the average prce of these 5 harcuts. Repeat the process 2 more tmes. What are the average prces for your other two smulatons? Pool your class data. Record the class data on the dotplot below. Dstrbuton of the average prce of a harcut sample sze 5 Aprl, 2011 Page 34 of 36

35 How does the shape of ths dstrbuton compare to the dotplot of ndvdual prces that you graphed earler? Are there any outlers? What s the mean of the dstrbuton of the average cost for sample sze 5? How does t compare to the mean of the dstrbuton of ndvdual prces? Would the mean be good to use for the dstrbuton of the average? Explan. What s the standard devaton of the average cost for sample sze 5? How does ths compare to the standard devaton of ndvdual prces? Use the RandInt feature on your calculator to randomly select 12 harcuts. Fnd the average prce of these 12 harcuts. Repeat the process 2 more tmes. What are the average prces for your other two smulatons? Pool your class data. Record the class data on the dotplot below. Dstrbuton of the average prce of a harcut sample sze 12 How does the shape of ths dstrbuton compare to the dotplot of ndvdual prces that you graphed earler? Are there any outlers? Aprl, 2011 Page 35 of 36

36 What s the mean of the dstrbuton of the average cost for sample sze 12? How does t compare to the mean of the dstrbuton of ndvdual prces? Would the mean be good to use for the dstrbuton of the average? Explan. What s the standard devaton of the average cost for sample sze 12? How does ths compare to the standard devaton of ndvdual prces? Fll n the table below wth the statstcs that you have obtaned. Dstrbuton Shape of the dstrbuton Mean of the Dstrbuton Standard Devaton of the Dstrbuton Of ndvdual data ponts n the populaton Of the mean for sample sze = 5 Of the mean for sample sze = 12 Summarze your fndngs below n a short paragraph. Aprl, 2011 Page 36 of 36