AP * Statistics Review. Inference

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1 AP * Statistics Review Iferece Teacher Packet AP* is a trademark of the College Etrace Examiatio Board. The College Etrace Examiatio Board was ot ivolved i the productio of this material. Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

2 Page of Decidig whether to use a test or a cofidece iterval Geerally, use a sigificace test to test a claim. Use a cofidece iterval if you wish to estimate a populatio parameter (μ or p) based o statistics from your sample ( x or ). Oce a cofidece iterval is costructed, you may use it to test claims (where you fail to reject a claim that falls withi the cofidece iterval, ad you reject a claim that falls outside of a cofidece iterval). The α -level of a two-sided hypothesis test is related to the cofidece level of a iterval by C = α. Idetifyig the type of test or iterval to use Is there a sigle umerical variable beig measured for each subject? The we will perform a test for meas. Is there a categorical variable beig measured, ad we are oly cocered with how ofte a sigle respose (a success ) occurs? The we will perform a test for a proportio. For example, if we oly care about the proportio of brow-eyed people, the we ca do a z-test for the proportio of brow-eyed people. Is there a categorical variable beig measured, ad we are cocered with how may resposes fall ito each category? The we will perform a χ -test. For example, if we wat to compare the occurrece of brow, blue, gree, ad grey eyes i two differet groups, we will do a χ -test because we are lookig at multiple categories for the categorical variable eye color. Are we lookig at the relatioship betwee two umerical variables? The we will perform a t-test for the slope of a regressio lie. How may samples? Be careful to idetify the source of each mea or proportio metioed i a problem. If a mea or proportio does ot clearly come from a sample (with a idetifiable sample size ), the it is probably a claim or a populatio proportio which should be used i the ull hypothesis. A two-sample test should have two clearly idetified sample sizes, ad each sample should result i a x or a. Some problems have two lists of umerical data that are liked i some way. For example, they could be pre-test ad post-test scores for a list of studets or temperatures i the su ad temperatures i the shade for a list of days. I these cases, the improvemet (post-test score mius pre-test score) or the differece (temperature i su mius temperature i shade) is the importat variable. These are called matched-pair t tests. Begi by subtractig the two lists of data to obtai oe list of improvemets or differeces. The do a oe sample t-test for a mea. (You will igore the origial two lists after you subtract.) Your ull hypothesis will ofte be that the mea improvemet was zero. For example i the pre- ad Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

3 Page of post-test problem, you might use H0 : μ improvemet = 0 ad H a : μ improvemet > 0, where improvemet is defied as post-test score mius pre-test score. I some χ -tests, there is a clear claim. For example, a compay claims that 50% of the prizes i the popcor boxes are stickers, 0% are rigs, ad 30% are temporary tattoos. I this case, we are comparig the data from oe sample to a claim, usig a χ -test for goodess of fit. The data table for this test cosists of a sigle row of data. I other χ -tests, we are comparig two groups to see if they have the same percetages i each category. For example, we could compare eye colors of a group of me ad a group of wome. I this case, do a χ -test for homogeeity. The data table would have two rows (oe for male ad oe for female) ad multiple colums (for brow eyes, gree eyes, etc.). Rows ad colums may be switched. The data for some χ -tests cosists of two categorical questios asked to a sigle sample of people. For example, we could ask a group of teachers whether they exercise frequetly, ofte, or ever ad whether or ot they missed ay days of school last year due to illess. We would like to see if their aswers to the two questios are idepedet of each other. If they are idepedet, the the proportio who missed school due to illess should be the same for all three exercise categories. I this case, do a χ -test for idepedece. The data table would have two rows (oe for people who missed school due to illess ad oe for those who did ot) ad multiple colums (for the differet exercise categories). Rows ad colums could be switched. The mechaics of the χ - test for idepedece ad the χ -test for homogeeity are exactly the same. What to put i your sigificace test A ull ad a alterative hypothesis (defie the parameter of iterest i words) Note: Always hypothesize about the ukow populatio parameters (μ ad p), ot the sample statistics ( x ad ), which are kow from the data. Idetify the test you are usig ad check the coditios ecessary for doig that test. Formula for the test statistic ( z or t or χ ) Value for the statistic (ca be from calculator if you have writte the formula) ad a shaded picture of the distributio if you have time to draw it The P value (from calculator or table) related to the α -level, plus df for t-tests or the expected values for χ tests Two coclusios: either reject H 0 or do t reject it based o the relatioship of the P-value ad the α level AND write a coclusio about the alterative hypothesis i the cotext of the problem Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

4 Page 3 of Phrasig to put i your coclusios Example: A seed maufacturer claims that at least 97% of their seeds will germiate. You suspect that the germiatio rate is less, so you buy a radom selectio of these seeds to test this claim. You calculate a P-value based o the hypotheses H 0 : p = 0.97 ad H a : p < 0. 97, where p is the germiatio rate of all seeds sold by this compay. Whe P < α, reject H 0. First, draw a mathematical coclusio about H 0 : Sice the P-value of 0.07 is less tha the α -level of 0.05, reject H 0. A value as extreme as my sample s germiatio rate should oly occur.7% of the time by radom chace if the compay s claim is true. The, write a coclusio about H a i the cotext of the problem: We ca coclude that the germiatio rate of the seeds is sigificatly lower tha the 97% claimed by the compay. Whe P > α, do ot reject H 0. First, draw a mathematical coclusio about H 0 : Sice the P-value of 0.09 is greater tha the α -level of 0.05, there is ot sufficiet evidece to reject H 0. A value as extreme as my sample s germiatio rate would occur 0.9% of the time by radom chace if the compay s claim is true. The, write a coclusio about H a i the cotext of the problem: We caot coclude that the germiatio rate of the seeds is sigificatly lower tha the 97% claimed by the compay. Note that we ever accept H 0. How to check the coditios ecessary to do the tests for meas ad proportios Always check to see if the sample is idepedet: radomly take from the populatio of iterest ad that the populatio is at least 0 times the sample size. What you are studyig Mea Oe Sample What you kow Use this Why? s (stadard deviatio of sample) Graph the sample if < 40 to see if it is approx. ormal x t = μ properties of t distributio s Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

5 Proportio Mea (Note: It is uusual to kow σ of the populatio.) Mea (Note: It is uusual to kow σ of the populatio.) Review of Iferece Page 4 of po > 0 ( po) > 0 Use the claimed proportio here. σ populatio is ormal σ populatio may ot be ormal, but > 30 z = x μ z = σ x μ z = σ p0 p0 ( p0 ) you ca approximate the biomial distributio by the ormal distributio because you ca always use z scores for ormal distributios Cetral Limit Theorem says that distributios get more ormal as icreases What you are studyig Differece of meas (two idepedet samples) Two Samples What you kow Use this Why? s ad s (stadard deviatio of samples), graph each oe to see if it is approx. ormal if < 40 t = x s x s + properties of t distributio Differece of two proportios Differece of two depedet meas Differece of meas (two id. samples) (Note: It is uusual to kow σ.) > 5 ( ) > 5 > 5 ( ) > 5 Use the pooled here. THIS IS MATCHED PAIRS!!! σ ad σ populatio is ormal z = p p p ( p ) + X where p = + X + Fid the differece betwee each pair ad do a oe sample t test. z = x x σ + σ you ca approximate the biomial distributio by the ormal distributio. We are oly iterested i oe piece of data for each subject; usually the improvemet or differece (after-before). because you ca always use z scores for ormal distributios Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

6 Differece of meas (two id. samples) (Note: It is uusual to kow σ.) σ ad σ populatio may ot be ormal, but > 30 ad > 30 Review of Iferece Page 5 of z = x x σ + σ Cetral Limit Theorem says that distributios get ormal as icreases How to check the coditios for the chi-squared tests Data must be couts (ot averages or proportios). Data i sample are idepedet (chose radomly ad < 0% of the populatio) Groups are large eough that all expected values 5. How to check the coditios for the t- test for the slope of a regressio lie The scatterplot must look liear. There must be o patter i the residual plot (errors are idepedet). The residual plot has a costat spread (errors have costat variability). Histogram of residuals is approximately ormal. Iferece usig cofidece itervals I geeral, you must put the followig three thigs i a cofidece iterval problem:. Idetify the iterval you will use ad check the coditios ecessary to use the iterval.. Calculate the iterval. 3. Iterpret the iterval i the cotext of the problem. The coditios we must check are the same as for the associated sigificace tests (as show i the table above), with oe exceptio. Whe performig sigificace tests for oe or two proportios, you check to see that is large eough by examiig p o ad ( p o ) where po is the claimed proportio which appears i the ull hypothesis. Sice there is o claim i a cofidece iterval problem, use the sample proportio i these checks. For oe-sample itervals, we require that ˆ p ad ( ) be over 0. I two-sample itervals, we require that ˆp, ( ), ˆp, ad ( ) ˆ p be over 5. Also, i proportio cofidece itervals, we must use the sample proportio to calculate the stadard error. Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

7 Page 6 of Formulas: Oe mea (σ ukow) Differece i two meas x ± t (σ ukow) ( x ) * s s * x ± t + Oe mea * σ (σ kow; uusual case) x ± z Differece i two meas (σ kow; uusual case) ( x x ) ± z * s σ σ + Oe proportio ( ) Differece i two proportios ( ) * ± z ± z * ( ) ( ) + Good cofidece iterval coclusios Make sure to state these i the cotext of the questio. I am C% cofidet that my iterval captures the populatio value μ or p. C out of 00 itervals costructed usig this method would capture the populatio value μ or p. Bad cofidece iterval coclusios Avoid makig these statemets: C% of the x values or values would fall i my iterval. C% of the data is i my iterval. There is a C% chace that the populatio value μ or p is i my iterval. Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

8 Page 7 of Multiple Choice Questios o Iferece. A study foud that 63 of radomly selected me ad 30 out of 65 radomly selected wome prefer cats to dogs. You wat to test the hypothesis that wome like cats more. Choose the correct hypotheses ad pooled p ˆ. (A) H 0 : p M = p F ; H A : p M < p F ; p ˆ =.4 (B) H 0 : p M = p F ; H A : p M > p F ; p ˆ =.49 (C) H 0 : p M = p F ; H A : p M < p F ; p ˆ =.49 (D) H 0 : p M = p F ; H A : p M > p F ; p ˆ =.4 (E) H 0 : p M < p F ; H A : p M > p F ; p ˆ =.4. A idepedet testig lab obtaied radom samples of ew haloge bulbs ad stadard icadescet bulbs made by the same compay to establish the compay s claim that, o average, the haloge bulb lasts loger tha the icadescet oe. Which test would you use? (A) a matched pair t test (B) a t-test for the differece i two meas (C) a z-test for the differece i two proportios (D) a t-test for the slope of the regressio lie (E) a χ -test for homogeeity 3. A certai populatio follows a ormal distributio with mea μ ad stadard deviatio σ. You costruct a 95% cofidece iterval for μ ad fid it to be. ± 0.9. Which of the followig is true? (A) I a test of the hypotheses H o : μ=., H A : μ., H O would be rejected at the.05 level. (B) I a test of the hypotheses H o : μ=.9, H A : μ.9, H O would be rejected at the.05 level. (C) I a test of the hypotheses H o : μ=.9, H A : μ.9, H O would be rejected at the.05 level. (D) I a test of the hypotheses H o : μ=0, H A : μ 0, H O would be rejected at the.05 level. (E) A coclusio about hypotheses caot be made from a cofidece iterval. Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

9 Page 8 of 4. Which of the followig is a coditio for choosig a t-iterval rather tha a z-iterval whe costructig a cofidece iterval for the mea of a populatio? (A) The stadard deviatio of the populatio is ukow. (B) There is a outlier i the sample data. (C) The sample may ot have bee a simple radom sample. (D) The populatio is ot ormally distributed. (E) The data are liked so a matched-pairs test is ecessary. 5. You wat to see whether or ot high school chages childre s educatioal plas. You take a radom sample of 6 th graders ad of th graders ad ask them whether they pla to get a job right after high school, go to college, or get a advaced degree. Which test do you perform? (A) a χ test for homogeeity (B) a two-sample z-test for proportios (C) a matched pair t-test (D) a χ test for goodess of fit (E) a t-test for the slope of the regressio lie 6. The Ceters for Disease Cotrol report a survey of radomly chose Americas age 45 ad older, which foud that 5 of 00 me ad 80 of 78 wome suffered from some form of arthritis. You wat to estimate the differece i the proportios of me ad wome over 45 who have arthritis with a 95% cofidece iterval. What stadard error will you use? (A) 0.09 (B) (C) 0.05 (D) (E) A two-sided hypothesis test for a populatio mea is sigificat at the % level of sigificace. Which of the followig is ecessarily true? (A) The 99% cofidece iterval of the mea cotais 0. (B) The 99% cofidece iterval of the mea does ot cotai 0. (C) The 99% cofidece iterval of the mea cotais the hypothesized mea. (D) The 99% cofidece iterval of the mea does ot cotai the hypothesized mea. (E) The 99% cofidece iterval is ot useful here. Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

10 Page 9 of 8. Which of the followig is ot a characteristic of the χ distributio? (A) Its shape is based o the sample size. (B) It is skewed to the right. (C) It approaches a ormal distributio as the degrees of freedom icrease. (D) It ca ever take o egative values. (E) It is always used for oe-sided sigificace tests. 9. Which of the followig would be the most appropriate for measurig the associatio betwee geder ad favorite color based o a radom sample of subjects? (A) (B) (C) (D) (E) a two-sample t-test a correlatio coefficiet a χ -test for idepedece a oe-sample z-test for a proportio a t-test for the slope of the regressio lie 0. Sixty seior accout executives were classified ito three groups, labeled A, B, ad C. There were 6 i group A, 9 i group B ad 5 i group C. At the.05 sigificace level, we would like to test if is it reasoable to coclude that the proportio of the populatio that falls ito each group is the same. Which of the followig is a correct coclusio? (A) Reject H 0. The proportio i the three groups is ot sigificatly differet. (B) Reject H 0. The proportio i the three groups is ot the same. (C) Do ot reject H 0. The proportio i the three groups is ot sigificatly differet. (D) Do ot reject H 0. The proportio i the three groups is ot the same. (E) We caot perform a sigificace test because there are three groups. Use the followig iformatio to aswer questios ad. A oe sample t test yields a t statistic of.089. The sample size was 6.. The alterative hypothesis was i the form H a : μ > Is there sigificat evidece at the α =.05 level to reject the ull hypothesis? (A) No, because the P-value is betwee 0.05 ad 0.0. (B) No, because the P-value is betwee 0.05 ad (C) No, because the sample mea was sigificatly above (D) Yes, because the P-value is betwee 0.05 ad 0.0. (E) Yes, because the P-value is betwee 0.05 ad Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

11 Page 0 of. If the alterative hypothesis was H a : μ istead, would you reject the ull hypothesis at the α = 0.05 level? (A) No, because the P-value is betwee 0.05 ad 0.0. (B) No, because the P-value is betwee 0.05 ad (C) No, because the sample mea was sigificatly above (D) Yes, because the P-value is betwee 0.05 ad 0.0. (E) Yes, because the P-value is betwee 0.05 ad Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

12 Page of Free Respose Questios o Iferece. A fitess traier wats to kow if her weight-liftig program ca quickly improve upper body stregth i older people. To fid out, she has a group of radomly selected people over 55 years old do push-ups for 90 secods ad couts the umber each ca do. After these people participate i her weightliftig program for 3 weeks, she tests them agai i the same way. Here are the results: Perso Before After Does the program help? Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

13 Page of. There are two mai dog parks i Dallas, oe ear White Rock Lake ad oe ear dowtow. The dowtow dog park is smaller ad is located udereath several highway overpasses. There are may apartmets, towhomes, ad lofts earby. The White Rock Lake dog park is larger ad provides a place for dogs to swim i the lake. The eighborhoods earby are a mix of sigle family homes with some apartmets. Jessica believes that sice the dowtow dog park is ear may apartmets, may of the dogs there will be smaller breeds, while the White Rock Lake park will attract larger, more active breeds. I order to test this assertio, she chooses radom times durig a moth to visit each park. She categorizes the dogs there by size. Toy (< 0 lbs) Small ( 0 lbs) Medium (-50 lbs) Large (5-00 lbs) Giat (over 00 lbs) Dowtow White Rock Lake Does the breed distributio for the dowtow dog park differ sigificatly from the White Rock Lake dog park at the α = 0.05 level? Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

14 Page 3 of 3. Are female or male studets more likely to atted college outside their home state? I order to fid out, radom samples of male ad female college-boud high school seiors were take i the Dallas/Fort Worth metropolita area. I September followig their high school graduatios, the studets i the samples were cotacted to see if they were attedig college i Texas or outside of it. (Studets who were ot attedig college were elimiated from the study.) The results are summarized i the followig table. Attedig college outside of Texas Attedig college i Texas female 39 male 94 a) Write ad iterpret a 95% cofidece iterval for the differece i proportio of male ad female studets attedig college outside of Texas. b) Based oly o your cofidece iterval, does the data from the radom samples idicate that there is a differece i proportios of male ad female studets attedig college outside of Texas? Justify your aswer. Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

15 Page 4 of Key to Iferece Multiple Choice A We are tryig to prove that p F > pm, ad the pooled p ˆ = = B We are comparig two meas, that of haloge bulbs ad that of icadescet bulbs. 3. D The cofidece iterval is (0.,.0). Zero is ot i this iterval, so we reject that claim. A 95% cofidece level correspods to a α -level of 0.05 for a two-sided test. 4. A Use a z test if σ is kow; use a t test if we must approximate σ usig s from the sample. 5. A We are comparig two groups o their aswer to a categorical questio. 6. C Stadard error = p ( p ) p ( p ) + = 0.05 (Aswer choice B icorrectly uses the pooled, which would be correct i a sigificace test, but ot a cofidece iterval.) 7. D Whe a test is sigificat, Ho was rejected. The claim (or hypothesized mea) was NOT i the iterval. 8. C The t distributio approaches the ormal distributio as icreases, but the χ distributio is always skewed. 9. C We are lookig at oe group s aswers to two questios with categorical aswers. 0. C The expected value for each group is 0. The value of the χ statistics is 3., ad the P-value of the test is 0.. This is higher tha ay α level, so we do ot reject H o, which says that the groups are the same. We ca t say that the groups differ sigificatly.. E The df = 5, so the P value is 0.07, which is less tha 0.05, so we reject H o.. A For a two-sided test, double the P value to 0.054, which is greater tha 0.05, so we do ot reject H. o Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

16 Page 5 of Rubric for Iferece for Free Respose. Solutio Part : Idetify the correct test by ame or formula, subtract to get the improvemet, check coditios. Sice the two lists are before ad after results for the same people, this is a xdiff 0 matched pair t test. OR t = s diff Subtract after before to get each participat s improvemet: Perso Before After Improvemet Check coditios for oe-sample t test: The data are idepedet because the participats were radomly chose ad is less tha 0% of the populatio of adults over 55. AND The data is early ormal because a examiatio of the dotplot of the differeces shows a uimodal graph with o outliers: - X 0 X XX XXX 3 XX 4 XX 5 X Part : Write hypotheses, idetifyig the parameter of iterest. H o : μimprovemet = 0 H a : μimprovemet > 0 where μ = the true mea improvemet i umber of push-ups doe improvemet Part 3: Perform the test, usig correct mechaics, icludig value of the t statistic, the degrees of freedom, ad the P value. x μ.67 0 t = = = 4.9 s.749 With df =, P( x >.67) = P( t > 4.9) = improveme t Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

17 Page 6 of Part 4: Usig the calculatios, write a coclusio i the cotext of the problem. Sice the P value of is less tha ay reasoableα, we have evidece to reject H o. We ca coclude that the improvemet is sigificatly above zero; the participats did improve the umber of push-ups they could do. Scorig Each part is essetially correct (E), partially correct (P), or icorrect (I). Part is essetially correct if the studet correctly idetifies the test, subtracts to fid the improvemet, checks for idepedece, ad graphs the improvemets to show that they are uimodal ad symmetric. Part is partially correct if the studet correctly does two or three of those. Part is icorrect if the studet does oly zero or oe of those.. Part is essetially correct if the studet correctly gives both hypotheses ad idetifies the parameter. Part is partially correct if the studet does oly oe of those. Part 3 is essetially correct if the studet correctly gives the value of the t statistic, the degrees of freedom, ad the P-value. Part 3 is partially correct if the studet gives oly oe or two of these. Part 4 is essetially correct if the studet correctly liks the P-value to the alpha-level i order to reject H O AND gives the coclusio (that the program does help) i cotext. Part 4 is partially correct if the studet gives oly oe of these coclusios. To assig a score to this questio let a E = poit, a P = 0.5 poits, ad a I = 0 poits. Sum the total poits for the studet s score. If a studet has a half poit, look at the questio holistically to determie if the score should be rouded up or trucated. 4 Complete Respose 3 Substatial Respose Developig Respose Miimal Respose Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

18 Page 7 of. Solutio Part : Idetify the correct test by ame or formula, check coditios ecessary to do test. Sice we are comparig two differet radom samples o multiple categories, we will do a χ test for homogeeity OR Check the coditios for a χ test: The data are couts. The two samples are idepedet. Each expected value is at least 5. Part : Write hypotheses. H O : The White Rock Lake dog park ad the dowtow dog park have the same distributio of dogs by size (are homogeeous). H A : The White Rock Lake dog park ad the dowtow dog park do ot have the same distributio of dogs by size. Part 3: Perform the test, usig correct mechaics, icludig value of the χ statistic, the degrees of freedom, the expected values, ad the P value. Expected values are i paretheses: Toy (< 0 lbs) Small ( 0 lbs) Medium (-50 lbs) Large (5-00 lbs) Giat (over 00 lbs) Dowtow 39 (34) 7 (68) 0 (85) 89 (07) (9) White Rock Lake 77 (8) 58 (6) 88 (04) 75 (57) 5 (44) df ( rows )( ) = ( )( 5 ) = 4 ( O ) ( 39 34) ( 7 68) ( 0 85) ( 5 44) = colums E χ = E P = 34 ( χ > 3.) = = 3. Part 4: Usig the calculatios, write a coclusio i the cotext of the problem. Sice the P-value of is less tha the α-level of 0.05, we ca reject H O. Based o these samples, the White Rock Lake dog park ad the dowtow dog park have a sigificatly differet distributio of dogs by size. Scorig Parts, 3, ad 4 ca be essetially correct (E), partially correct (P), or icorrect (I). Part ca be essetially correct (E) or icorrect (I). Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

19 Page 8 of Part is essetially correct if the studet correctly idetifies the test, metios idepedece, ad states that the expected values are over 5. Part is partially correct if the studet correctly does oe or two of those.. Part is essetially correct if the studet correctly gives both hypotheses i words. Part 3 is essetially correct if the studet correctly gives the value of the χ statistic, the degrees of freedom, the expected values, ad the P-value. Part 3 is partially correct if the studet gives oly two or three of these. Part 3 is icorrect if the studet gives oly zero or oe of these. Part 4 is essetially correct if the studet correctly liks the P-value to the alpha-level i order to reject H O AND gives the coclusio (that the sizes of the dogs at each park differ) i cotext. Part 4 is partially correct if the studet gives oly oe of these coclusios. To assig a score to this questio let a E = poit, a P = 0.5 poits, ad a I = 0 poits. Sum the total poits for the studet s score. If a studet has a half poit, look at the questio holistically to determie if the score should be rouded up or trucated. 4 Complete Respose 3 Substatial Respose Developig Respose Miimal Respose Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

20 Page 9 of 3. Solutio Part : Idetify the correct iterval by ame or formula, check coditios. This is a two-sample z iterval for the differece i two proportios OR * ( ) ( ) ( ) ˆ p ± z +. Check coditios for two-proportio z iterval: The data are idepedet because the participats were radomly chose ad each is less tha 0% of the populatio of college-boud high school male or female seiors i the Dallas/Fort Worth area. AND The sample sizes are large eough because = (450) 0.69 = > 0 or 5 ( ) ( ) = (450)( 0.73) = 39 > 0 or 5 ˆ = (306)( 0.307) = 94 > 0 or 5 ( ) = (306)( 0.693) = > 0 or 5 p Part : Calculate the iterval. (This may be doe i either order, male-female or femalemale.) 94 p ˆ ˆ M = = pf = = * ( ) ( ˆ ) ˆ ( ˆ M pm pf pf ) ˆ M pf ± z + M ( 0.693) 0.69( 0.73) = ( ) ± = ± = ( 0.08, 0.04) = (.8%, 0.4%) Part 3: Iterpret the iterval. F Based o these samples, I am 95% cofidet that the iterval (-.8%, 0.4%) captures the true differece betwee the populatio proportio of male DFW studets who atted college outside Texas ad the populatio proportio of female DFW studets who atted college outside Texas. OR Based o these samples, I am 95% cofidet that the true differece betwee the populatio proportio of male DFW studets who atted college outside Texas ad the populatio proportio of female DFW studets who atted college outside Texas is betwee -.8% ad 0.4%. Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

21 Page 0 of Part 4: Sice 0 is i the 95% cofidece iterval, zero is a plausible value for the differece i proportios, pm pf. The evidece shows o sigificat differece betwee the proportio of male studets attedig college outside of Texas ad the proportio of female studets attedig college outside of Texas. Scorig Each of the four parts ca be essetially correct (E) or icorrect (I). Part is essetially correct if the iterval is idetified ad the studet commets o both idepedece ad large sample size. The miimum amout ecessary is a idicatio that the umber of successes ad failures for both samples is over 0 (or 5) AND a metio of idepedece (or idepedece with a check mark). The studet does ot have to repeat the fact that the samples are radom. Part ca be essetially correct eve if there is a idetifiable mior arithmetic error. Part 4 is essetially correct if the studet states that zero is ot i the iterval ad liks this to either a 95% cofidece level or a 5% sigificace level. Part 4 is icorrect if the studet says o without justificatio or if the studet says o because zero is ot i the iterval. 4 Complete Respose (4E) All four parts essetially correct 3 Substatial Respose (3E) Three parts essetially correct Developig Respose (E) Two parts essetially correct Miimal Respose (E) Oe part essetially correct Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit:

22 Page of AP Statistics Exam Coectios The list below idetifies free respose questios that have bee previously asked o the topic of Iferece o the AP Statistics Exam. These questios are available from the CollegeBoard ad ca be dowloaded free of charge from AP Cetral. Free Respose Questios 00 Questio 5 00 Questio Questio 6 Copyright 009 Layig the Foudatio, Ic., Dallas, TX. All rights reserved. Visit: