Chpter 9: Inferences bsed on Two smples: Confidence intervls nd tests of hypotheses 9.1 The trget prmeter : difference between two popultion mens : difference between two popultion proportions : rtio of two popultion vrinces 9. Compring two popultion mens: independent smpling Cse 1, Lrge smples Conditions required for vlid lrge smple: 1. two smple nd selected from two independent popultion,. nd n 30 x x Smpling distribution of ( 1 ) is pproximtely norml with: Men: x1 x µ = ( ) 1 Stndrd error: σ ( x1 x) = + n n σ σ 1 (1- α ) 100% Confidence intervl for ( µ 1 µ ) (the difference of two popultion mens): Interpret: We re (1- α ) 100% confident tht the true between these two popultions will flls in this intervl. Exmple1, IETSTUY, To investigte the effect of new low-ft diet on weight loss, two rndom smples of 100 people ech re selected. One group of 100 is plced on the low-ft diet, while the other group with regulr diet. For ech person, the mount of weight lost (or gined) in 3-week period is recorded. iet Low-ft diet (1) Regulr diet () Weight loss 8, 1, 13,.., 10 (100 observtions) 6, 14, 4,, 8 (100 observtions) 1
Q: Form 95% confidence intervl for ( µ 1 µ ), the difference between the popultion men weight losses for the two diets. Interpret the result. Group Sttistics IET N Men Std. evition Std. Error Men WTLOSS LOWFAT 100 9.31 4.668.467 REGULAR 100 7.40 4.035.404 How cn we mke inference bsed on (1- α ) 100% Confidence intervl for ( µ )? If the confidence intervl, it implies tht there is difference between these two popultion mens; If the confidence intervl, it implies tht there is difference between these two popultion mens. Exmple: A confidence intervl for ( µ ) is (-10, 4), wht inference cn we mke? A confidence intervl for ( µ ) is (-18, -9), wht inference cn we mke? A confidence intervl for ( µ ) is (3.6, 1.4), wht inference cn we mke?
Hypothesis test for ( µ ): Criticl-vlue pproch: 1.. significnce level α 3. test sttistic: 4. rejection region : when H 1 µ 0 when H < 0 when H > 0 5. conclusion: if the vlue of test sttistic flls in R.R, reject H 0,nd conclude tht t α level, there is sufficient evidence to conclude H is true. if the vlue of test sttistic does not fll in R.R, do not reject H 0, nd conclude tht t α level, there is insufficient evidence to conclude P-vlue pproch: 1. H µ = 0 1 0 H µ ( or µ µ < or µ µ > ) 1 0 1 0 1 0. significnce level α ( x1 x) 0 3. test sttistic z0 = S1 S + n n 1 H is true. 4. p-vlue = when H 1 µ 0; when H < 0; when H > 0. 5. Conclusion: if p-vlue is α, reject H 0, nd conclude tht t α level, there is sufficient evidence to conclude H is true. If p-vlue is no less thnα, H 0, nd conclude tht t α level, there is insufficient evidence to conclude H is true. 3
Exmples for compring two popultion mens: independent, lrge-smples: Exmple1, IETSTUY, To investigte the effect of new low-ft diet on weight loss, two rndom smples of 100 people ech re selected. One group of 100 is plced on the low-ft diet, while the other group with regulr diet. For ech person, the mount of weight lost (or gined) in 3-week period is recorded. iet Low-ft diet (1) Regulr diet () Weight loss 8, 1, 13,.., 10 (100 observtions) 6, 14, 4,, 8 (100 observtions). At =0.05, conduct test of hypothesis to determine whether the men weight loss for low-ft diet is different from tht of regulr diet. b. At =0.05, conduct test of hypothesis to determine whether the men weight loss for low-ft diet is greter thn tht of regulr diet. 4
SPSS output for IETSTUY, Group Sttistics IET N Men Std. evition Std. Error Men WTLOSS LOWFAT 100 9.31 4.668.467 REGULAR 100 7.40 4.035.404 Independent Smples Test Levene's Test for Equlity of Vrinces t-test for Equlity of Mens 95% Confidence F Sig. t df Sig. (-tiled) Men ifference Std. Error ifference Intervl of the ifference Lower Upper WTLOSS Equl vrinces ssumed Equl vrinces not ssumed 1.367.44 3.095 198.00 1.910.617.693 3.17 3.095 193.940.00 1.910.617.693 3.17 Cse, Smll smples with equl vrinces Conditions required for vlid smll smple: 1. The two smples re nd selected from the two trget popultion,. Both smpled popultions hve distributions tht re pprox., 3. The popultion vrinces re. ( σ 4. Smple size is smll ( ). = σ ) 1 Since these two popultions hve equl vrince, ( σ = σ ), it is resonble to construct 1 for use in confidence intervls nd test sttistics. 5
(1- )100% confidence intervl for ( µ 1 µ ): t α with df = n1+ n. Hypothesis test for ( µ 1 µ ): 1. H0 = 0 H µ ( or µ µ < or µ µ > ) 1 0 1 0 1 0. level of significnce α ; 3. test sttistic: t = with df = 4. rejection region : when H 1 µ 0 when H < 0 when H > 0; 5. conclusion: if the vlue of test sttistic flls in R.R, H 0,nd conclude tht t α level, there is sufficient evidence to conclude H is true. if the vlue of test sttistic does not fll in R.R, H 0, nd conclude tht t α level, there is insufficient evidence to conclude H is true. 6
Exmples for compring two popultion mens: independent, smll-smples: Exmple1: REAING, Suppose we wish to compre new method of teching reding to slow lerners to the current stndrd method. The response vrible is the reding test score fter 6 months. slow lerners re rndomly selected, 10 re tught by the new method, 1 by the stndrd method. The test score is listed below. New method (1) 80, 80, 79, 81, 76, 66, 71, 76, 70, 85 Stndrd method () 79, 6, 70, 68, 73, 76, 86, 73, 7, 68, 75, 66. Use 95% confidence intervl to estimte the true men difference between the test score for the new method nd the stndrd method. Interpret the intervl. b. Conduct test of hypothesis to determine whether the stndrd method leds to lower test score thn new method. Useα = 0.05. 7
SPSS output for REAING, Group Sttistics METHO N Men Std. evition Std. Error Men SCORE NEW 10 76.40 5.835 1.845 ST 1 7.33 6.344 1.831 Independent Smples Test Levene's Test for Equlity of Vrinces t-test for Equlity of Mens 95% Confidence F Sig. t df Sig. (-tiled) Men ifference Std. Error ifference Intervl of the ifference Lower Upper SCORE Equl vrinces ssumed Equl vrinces not ssumed.00.967 1.55 0.136 4.067.60-1.399 9.533 1.564 19.769.134 4.067.600-1.360 9.493 Cse 3, Smll smples with unequl vrince: Conditions: 1. two smples re rndomly nd independently selected from the two trget popultion,. both smpled popultions re pprox. norml, 3. the popultions vrince re not equl ( σ Procedure is on textbook P4-43. σ ). 1 8
9.3 Compring two popultion mens: pired difference experiments Two smpling compring: Exmple1: To investigte the effect of new teching method on reding. 1. Rndomly select slow lerner students, 10 re ssigned to new method, while the other 1 re ssigned to the stndrd method, the response vrible is the reding test score fter 6 months. (independent smpling). 8 pirs slow lerner re selected, not rndomly, two lerners in ech pir with the similr reding IQs; in ech pir, one use new method, the other one use stndrd method, then the pired test score difference could be used to mke inference bout ( µ ). Exmple: To investigte the effect of new protein diet on weight loss. 1. Two rndom smples of 100 people ech re selected. One group of 100 is plced on the low-ft diet, while the other group with regulr diet. For ech person, the mount of weight lost (or gined) in 3-week period is recorded. (independent smpling). FA rndomly choose five individuls with regulr diet nd record their weight (in pounds), then instruct them to follow the protein diet for three weeks. At the end of this period, their weights re recorded gin. The pired weight differences between these two diets could be used to mke inference bout ( µ ). (two subjects in ech pir with similr level, then ssign tretments, to see the effect) Pired difference experiment: Ech pir hs two experimentl units, re pired nd the re nlyzed. : mking comprisons within groups of similr experimentl units. Pired difference experiment is simple exmple of rndomized block design. (Red textbook P43, 433) Exmple dt (NEW PROTEIN IET): Person Weight before (1) Weight fter () 1 148 141 193 188 3 186 183 4 195 189 5 0 198 Note: The vrible we re interested is. 9
Inference bsed on pired difference (lrge smple): Conditions required for vlid lrge-smple inference bout µ : 1. A rndom smple of is selected from the trget popultion of differences;. The smple size is n. Pired difference (1- )100% confidence intervl for µ µ 1 µ = ( ): Pired difference Test of hypothesis for µ = ( µ ) : 1.. significnce level α ; x 0 3. test sttistic: z = σ n 4. rejection region: Z > when H : Z α Z < Z α when H : Z > when H : Z α 5. conclusion. Exmples of mking inference bsed on pired difference (lrge smple): Exmple, To investigte which supermrket (A or B) hs the lower prices in town, gency rndomly selected 100 items common to ech of the two supermrkets nd recorded the prices chrged by ech supermrket. The summry results re provided below. x =.09 x = 1.99 x = 0.10 A B S = 0.4 S = 0.19 S = 0.03 A B. Form 95% confidence intervl for µ = µ A µ B. Interpret the result. 10
b. Conduct test of hypothesis to determine whether the men price for supermrket B is cheper thn tht for supermrket B? Useα = 0.05. 11
Inference bsed on pired difference (smll smple): Conditions required for vlid smll-smple inference bout µ : 1. A of difference is selected from the trget popultion of differences;. The popultion of differences is pproximtely distributed; 3. The smple size n < 30. 1. (1- )100% confidence intervl for Pired difference µ µ 1 µ = ( ):, t α with df =. Test of hypothesis for Pired difference µ µ 1 µ 1. H = 0 0 = ( ): H ( or H < or H > ) 0 0 0. significnce level α ; 3. test sttistic: t = 4. rejection region: when H 0 5. conclusion. when H < 0 when H > 0 1
Exmples of mking inference bsed on pired difference (smll smple): Exmple 1, NEW PROTEIN IET: To investigte new protein diet on weight-loss, FA rndomly choose five individuls nd record their weight (in pounds), then instruct them to follow the protein diet for three weeks. At the end of this period, their weights re recorded gin. person Weight before (1) Weight fter () 1 148 141 193 188 3 186 183 4 195 189 5 0 198. Clculte 95% confidence intervl for the difference between the men weights before nd fter the diet is used. Interpret the intervl. b. o the dt provide sufficient evidence tht the protein diet hs effect on the weight loss? Useα = 0.05. (p-vlue =? ) 13
SPSS output for Exmple1, NEW PROTEIN IET, Pired Smples Sttistics Men N Std. evition Std. Error Men Pir 1 W1 184.80 5 1.347 9.547 W 179.80 5.354 9.997 Pired Smples Test Pired ifferences t df Sig. (-tiled) Std. Error 95% Confidence Intervl Men Std. evition Men of the ifference Pir 1 W1 - W 5.000 1.581.707 3.037 6.963 7.071 4.00 Lower Upper SPSS output for Exmple, PAIRESCORES, Pired Smples Sttistics Men N Std. evition Std. Error Men Pir 1 NEW 76.00 8 6.98.449 ST 71.63 8 7.009.478 Pired Smples Test Pired ifferences t df Sig. (-tiled) Std. Error 95% Confidence Intervl Men Std. evition Men of the ifference Pir 1 NEW - ST 4.375 1.685.596.966 5.784 7.344 7.000 Lower Upper 14
Exmple, PAIRESCORES, To investigte the effect of new teching method on improving reding test score, 8 pirs slow lerner re selected, not rndomly, two lerners in ech pir with the similr reding IQs; in ech pir, one use new method, the other one use stndrd method. Then fter 6 months, the test scores re recorded. Pir New method (1) Stndrd method () 1 77 7 74 68 3 8 76 4 73 68 5 87 84 6 69 68 7 66 61 8 80 76. Construct 95% confidence intervl to estimte the difference of men test scores between new method nd stndrd method. Interpret the result b. o the dt provide sufficient evidence tht the new method leds to higher test scores thn the stndrd method? Useα = 0.05. (p-vlue =? ) 15
9.4 Compring two popultion proportions: independent smpling Conditions required for vlid lrge-smple inferences bout ( p1 p) : 1. The two smples re rndomly nd independently selected from the two trget popultions.. The smple size n 1 nd n re both lrge. (This condition will be stisfied if both.) Under lrge smple size, by the Centrl Limit Theorem, the smpling distribution of ( pˆˆ1 p) is pproximtely norml with: men: stndrd devition: ( p p ): Lrge smple 100(1- )% confidence intervl for 1 ( p p ): Lrge smple test of hypothesis bout 1 1.. Level of significnce α ; 3. Test sttistic: 4. rejection region : Z < Z or Z > Z when H : p1 p 0 α α Z < Z α when H : p1 p < 0 H : p p > 0; Z > Z α when 1 5. conclusion. 16
Exmples for compring two popultion proportions (lrge-smple): Exmple: Smoking Survey, Suppose the Americn cncer Society rndomly smples 1500 dults in 1995 nd then smpled 1750 dults in 005 to do smoking survey to determine whether there ws evidence tht the percentge of smokers hd decresed. 1995 (1) 005 () n 1 = 1500 n = 1750 x 1 = 555 x = 578 efine: : the true proportion of dult smokers in 1995; : the true proportion of dult smokers in 005.. Give point estimte of the percentge difference of dult smokers between 1995 nd 005? b. o the dt indicte tht the proportion of dult smokers decresed over this 10-yer period? Useα = 0.05. c. Form 95% confidence intervl for ( p1 p) to estimte the extent of the decrese. Interpret it. 17