Unit 29: Inference for Two-Wy Tbles Prerequisites Unit 13, Two-Wy Tbles is prerequisite for this unit. In ddition, students need some bckground in significnce tests, which ws introduced in Unit 25. Additionl Topic Coverge Additionl coverge of inference for two-wy tbles cn be found in The Bsic Prctice of Sttistics, Chpter 23, Two Ctegoricl Vribles: The Chi-Squre Test. Activity Description Students should work in smll groups on this ctivity. The ctivity consists of three prts. The first prt provides justifiction for the formul for computing the expected cell counts for chi-squre tbles. Students cn work on Prt I on their own or it could be prt of lecture/ clss discussion. Prts II nd III involve two different structures for dtsets, both of which re pproprite for the chi-squre nlysis covered in this unit. Here re the two dt structures: (1) subjects from single smple re clssified ccording to two ctegoricl vribles nd (2) subjects from multiple smples (drwn from different popultions) re clssified ccording to single ctegoricl vrible. In the ltter cse, which smple cn be thought of s the second ctegoricl vrible. In the first cse, chi-squre test for independence is performed; in the second cse, chi-squre test for homogeneity is performed. The chi-squre test sttistics nd the nlyses re the sme for both situtions. So, in this unit, we hve put little emphsis on distinguishing between these two situtions. Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 1
Mterils For Prt III, bgs of t lest two different types of M&Ms re needed. Lrge-sized bgs were used for the smple dt, with the exception of the M&Ms minis, for which medium bg ws purchsed. In ddition, students will need pper pltes or bowls to contin the M&Ms while they re being counted. Prt I: Introduction Assumption of Independence nd Expected Count Formul Prt I provides n explntion of the expected counts formul used in chi-squre test of independence. Students need to be fmilir with the Multipliction Rule from Unit 19, Probbility Models. This prt could be pproched either s n ctivity or s prt of n informl lecture tht introduces the topic of this ctivity. It could lso be skipped nd students could move directly to Prt II. Prt II: Single Smple, Clssified on Two Ctegoricl Vribles For this prt, students will need to collect dt from people. The clss could serve s the smple, or perhps combine this clss with nother clss, or hve students dd their friends to the smple. Students will need to clssify ech individul in the smple by gender nd eye color. An esy wy to collect the dt is to drw tble on the bord. Ech student should come up to the bord nd put tlly line in the pproprite box for gender nd eye color. After students hve completed their entries, numbers cn replce the tlly mrks. Students cn then copy the tble from the bord nd begin work on Prt II. Prt III: Multiple Smples, Clssified on One Ctegoricl Vrible Students should work in groups to collect the dt on the M&Ms colors. Agin, you my wnt to put chrt on the bord nd hve students enter their results for ech color s they finish sorting their M&Ms into colors. Once the dt re collected, groups will need copy of the clss dt. Since the resulting two-wy tble is quite lrge, group members should be encourged to divide up the work of computing the expected cell counts. The color distribution of M&Ms differs by types nd hs chnged over the yers. You cn write to Mrs, the mkers of M&Ms, for the ltest color distribution in its cndies. Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 2
The Video Solutions 1. Dr. Prdis Sbeti investigtes the nonstop evolutionry rms rce between our bodies nd the infectious microorgnisms tht invde nd inhbit them. In other words, she investigtes connections between genotypes nd protections from infectious diseses. Her work on Lss fever is still in its erly stges. 2. Sickle cell nemi hemoglobin muttion, HbS. 3. H0 : No ssocition betweeen mlri nd HbS. H : Assocition between mlri nd HbS. 4. Expected count = (row totl)(column totl). grnd totl 5. We reject the null hypothesis nd conclude tht there is n ssocition between the HbS gene nd mlri. Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 3
Unit Activity: Associtions With Color Solutions Prt I: Introduction Assumption of Independence nd Expected Count Formul 1.. P(DEM nd femle) = P(DEM) P(femle) = 196 246 0.1929 b. Expected number = 196 246 ( ) ( = 196 )( 246) 96.432 c. Expected count = (196)(246)» 96.432 d. P(DEM nd mle) = P(DEM)P(mle) = 196 254 0.1991 Expected number = 196 254 Expected count = (196)(254)» 99.57 ( ) = (196)(254) 99.57 2.. Expected Mle Femle Politicl DEM (Blue) 96.43 99.57 196 Preference GOP (Red) 91.02 93.98 185 Color IND (White) 58.55 60.45 119 246 254 ( ) 2 ( ) 2 b. χ 2 Activity 107 Solutions 96.43 2 89 99.57 = 96.43 99.57 df = (3 1)(2 1) = 2; p 0.02... ( 56 60.45) 2 60.45 7.825 Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 4
c. There is sufficient evidence to reject the null hypothesis. There is ssocition between these two vribles. In other words, they re dependent. 3.. Smple dt will be used to provide smple nswers. Gender Eye Color Count Blue Brown Other Mle 8 20 6 Femle 4 16 12 34 32 b. H0 : No ssocition between gender nd eye color. H : Assocition between gender nd eye color. c. Smple nswer: d. Smple nswer: 12 36 18 66 ( χ 2 = 8 6.18 ) 2 6.18 Activity Solutions 3 Eye Color Count Blue Brown Other Gender Mle 8 20 6 34 6.18 18.55 9.27 Femle 4 16 12 32 5.82 17.45 8.73 12 36 18 66 Activity Solutions 3c ( 20 18.55) 2... 18.55 ( 12 8.73) 2 8.73 3.72 ; df = 2 p 0.151. There is insufficient evidence to reject the null hypothesis. In other words, there is no strong evidence to suggest tht there is n ssocition between eye color nd gender. 4.. Smple dt (will be used for smple nswers) (See next pge...): Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 5
Type 1 Type 2 Type 3 Type 4 Count Drk Regulr Penut Mini Green 112 109 41 228 490 Blue 188 160 39 203 590 Color Yellow Ornge 75 141 91 47 210 123 36 187 423 487 Red 81 62 20 221 384 Brown 59 84 30 100 273 b. H0 : No ssocition Activity between Solutions 4M&M type nd color distribution. H : Assocition between M&M type nd color distribution. c. Smple nswer: 656 629 213 1149 2647 Color Type 1 Type 2 Type 3 Type 4 Count Drk Regulr Penut Mini Green Blue Yellow Ornge Red Brown 112 188 75 141 81 59 109 160 91 123 62 84 41 39 47 36 20 30 228 203 210 187 221 100 121.4 146.2 104.8 120.7 95.2 67.7 116.4 140.2 100.5 115.7 91.2 64.9 39.4 47.5 34 39.2 30.9 22 212.7 256.1 183.6 211.4 166.7 118.5 490 590 423 487 384 273 656 629 213 1149 2647 Activity Solutions 4c d. χ 2 100.3 ; df = (6 1)(4 1) = 15; p 0 There is n ssocition between M&Ms type nd color distribution. In other words, Different types of M&Ms hve different color distributions. Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 6
Exercise Solutions 1.. There were two cells with expected counts less thn 1. The guidelines cll for ll expected counts to be greter thn 1. Also, there were 7 cells with expected counts below 5. Tht mens tht round 39% of the cells hve expected counts under 5. The guidelines stte tht no more thn 20% of the cells should hve expected counts less thn 5. b. See solution to (c). c. Bsed on the completed tble below, ll expected counts were greter thn 1. Two expected counts were below 5, which is just under 17% of the cells. So, the expected counts in the tble below meet the guidelines. Energy Drinks None One Two Three Observed Observed Observed Observed 57 11 4 1 Environment 144 44 13 8 598 160 36 34 Count Expected Expected Expected Expected Frm 52.55 14.14 3.49 2.83 Country 150.44 40.48 9.98 8.10 City 596.01 160.38 39.54 32.08 799 215 53 43 73 209 828 1110 Ex. Solution 1(c ) d. This is 4 3 tble; df = (4 1)(3 1) = 6. The chi-squre test sttistic is clculted below: ( 57 52.55) 2 ( 144 150.44) 2 ( 598 596.01) 2 χ 2 = 52.55 150.44 596.01 ( 11 14.14 ) 2 ( 44 40.48) 2 ( 160 160.38) 2 14.14 40.48 160.38 ( 4 3.49 ) 2 ( 13 9.98) 2 ( 36 39.54) 2 3.49 9.98 39.54 ( 1 2.83) 2 ( 8 8.10 ) 2 ( 34 32.08) 2 2.83 8.10 32.08 4.268 e. p 0.64. (See re under density curve below.) There is insufficient evidence to reject the null hypothesis. We found no cler evidence of n ssocition between 12 th -grde students consumption of energy drinks nd their growing-up environment. Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 7
Chi-squre Density Curve, df = 6 0.6405 0 4.268 2 Χ 2.. Gender is the explntory vrible. We would like to use gender to explin how students rte their intelligence compred to their peers. b. H0 : No ssocition between gender nd intelligence rting. H : Assocition between gender nd intelligence rting. c. Count Femle Gender Mle Intelligence Below Averge Averge Above Averge 437 456 2243 1643 4072 4593 448.5 444.5 1951.7 1934.3 4351.8 4343.2 6752 6692 893 3886 8665 13444 d. df = (2 1)(3 1) = 2 χ 2 = ( 437 448.5) 2 448.5 ( 456 444.5) 2 444.5 124.1 ( 2243 1951.7) 2 1951.7 ( 1643 1934.3) 2 1934.3 ( 4072 4351.8) 2 4351.8 ( 4593 4313.2) 2 4313.2 (Answers my vry somewht depending on the number of decimls used in the expected cell count.) Ex. Solution 2c e. p 0. Reject the null hypothesis. There is sttisticlly significnt difference between how mles nd femles rte their intelligence compred to their peers. (In other words, there is n ssocition between gender nd intelligence rting.) Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 8
3.. H0 : No ssocition between intelligence rting nd verge grdes. H : Assocition between intelligence rting nd verge grdes. b. Intelligence Averge Grde Count A B C or Below Above Averge Below 2886 1335 305 4044 1881 416 1387 585 164 2894.9 1323 308 4055.8 1853.6 431.6 1366.2 624.4 145.4 8317 3801 885 4526 6341 2136 13003 Exercise Solution 3b c. df = (3 1)(3 1) = 4 χ 2 = (2886 2894.9)2 2894.9... (164 145.4)2 145.4 6.35 As shown below, p 0.174 Chi-Squre Density Curve, df = 4 0.1745 0 6.35 2 Χ d. We would expect to see vlue from chi-squre distribution with df = 4 s or more extreme thn 6.35 roughly 17.4% of the time. So, this is somewht common occurrence. It does not provide strong evidence ginst the null hypothesis. Generlly strong evidence mens tht the percentge should be below 5%. 4.. H0 : No ssocition between gender nd hours worked/week. H : Assocition between gender nd hours worked/week. b. χ 2 = 12.705 ; p = 0.005 < 0.05. Therefore, the results re significnt. There is n ssocition between gender nd hours worked per week. (Note: The prcticl significnce is nother mtter nd cnnot be determined by p-vlue.) Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 9
c. The biggest discrepncy in work ptterns is tht higher percentge of mles did not work (43.52%) compred to femles (40.59%). Furthermore, in every ctegory of hours worked/ week, there is higher percentge of femles thn mles. Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 10
Review Questions Solutions 1.. H0 : No ssocition between hbitt use nd eel species. H : Assocition between hbitt use nd eel species. b. Hbitt Use Count Spotted Purplemouth G S B 127 99 264 116 67 161 142.8 97.5 249.7 100.2 68.5 175.3 243 166 425 490 344 834 c. Here re the clcultions Review Questions for the chi-squre Solutions 1btest sttistic: χ 2 = ( 127 142.8) 2 142.8 ( 67 68.5) 2 68.5 6.28 ( 116 100.2) 2 100.2 ( 264 249.7) 2 249.7 ( 99 97.5) 2 97.5 ( ) 2 161 175.3 175.3 The degrees of freedom re: df = (3 1)(2 1) = 2. Using softwre, p 0.043. Since p < 0.05, we reject the null hypothesis nd conclude tht there is n ssocition between hbitt use nd mory eel species. d. Column percentges re more pproprite. The explntory vrible is the eel species. So, we should compre the conditionl distributions of hbitt use for ech species of mory eel. Spotted Purplemouth G 25.9% 33.7% Hbitt S 20.2% 19.5% Use B 53.9% 40.8% 100% 100% We lern tht mjority Review (53.9%) Questions of the Solutions spotted 1d mory eels were found in border hbitts compred to only 46.8% of the purplemouth mory eels. Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 11
2.. Eductionl ttinment is the explntory vrible nd voting is the response vrible. We expect tht person s highest eductionl ttinment will shed light on whether or not they voted in the 2012 elections. b. H0 : No ssocition between eduction nd voting. H : Assocition between eduction nd voting. c. Highest Eductionl Attinment Voted Nov. 2012 Count Yes No Not HS Grd 57 64 Expected 84.5 36.5 HS Grd/No College 227 163 Expected 272.3 117.7 Some College/Associte's 303 51 Expected 254.1 109.9 Bchelor's or Higher 303 51 Expected 247.1 106.9 121 390 364 354 858 371 1229 ( 57 84.5) 2 d. χ 2 = 84.5 ( 64 36.5) 2... 51 106.9 36.5 106.9 Review Questions Solutions 2c ( ) 2 100.1 df = (4 1)(2 1) = 3; p 0.000 Since p < 0.5, the results re significnt. There is reltionship between these two vribles. e. Since the explntory vrible is highest eductionl ttinment, the chrt below represents grphiclly the conditionl distributions of voting for ech level of highest eductionl ttinment. 100 90 85.6 80 74.5 70 Percent 60 50 40 52.9 47.1 41.8 58.2 30 25.5 20 14.4 10 0 Voted Nov. No Yes No Yes No Yes Eduction Not HS Grd HS Grd/No College Some College/Assoc. Percent within levels of Highest Eductionl Attinment No Yes Bchelor s or higher As the level of highest eductionl ttinment increses, so does the prticiption in voting. More educted people re more likely to vote thn those who re not educted. Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 12
3.. Energy Shots Consumed Per Dy Count Femle Mle 896 938 None 1834 888.89 945.11 63 70 Less thn one 133 64.46 68.54 16 19 One 35 16.96 18.04 5 16 Two 21 10.18 10.82 7 5 Three 12 5.82 6.18 1 0 Four 1 0.48 0.52 4 4 Five or Six 8 3.88 4.12 4 7 Seven or more 11 5.33 5.67 996 1059 2055 b. No, the guidelines re not stisfied. There re two cells tht hve counts below 1 (0.48 nd Review Questions Solutions 3 0.52). In ddition, there re 4 cells with counts less thn 5, which is 25% of the cells. c. Smple nswer (students my decide to combine different ctegories): Energy Shots Consumed Per Dy Count Femle Mle 896 938 None 1834 888.9 945.1 79 89 One or Less 168 81.4 86.6 12 21 Two or Three 33 16 17 9 11 Four or more 20 9.7 10.3 996 1059 2055 d. Smple nswer is bsed on smple nswer to (c): χ 2 = 2.282 ; p 0.52. Review Questions Solutions 3c There is insufficient evidence to reject the null hypothesis. There is insufficient evidence to indicte tht there is linkge between mounts of energy drink shots consumed nd gender. Unit 29: Inference for Two-Wy Tbles Fculty Guide Pge 13