Chapter 9 Inferences from Two Samples. Section 9-1. Overview. Overview. Requirements. Section 9-2 Inferences About Two Proportions Key Concept

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1 Chapter 9 Ifereces from Two Samples 9- Overview 9- Ifereces About Two Proportios 9-3 Ifereces About Two Meas: Idepedet Samples 9-4 Ifereces from Matched Pairs 9-5 Comparig Variatio i Two Samples Sectio 9- Overview Overview There are may importat ad meaigful situatios i which it becomes ecessary to compare two sets of sample data. This chapter exteds the same methods itroduced i Chapters 7 ad 8 to situatios ivolvig two samples istead of oly oe. Slide Slide Sectio 9- Ifereces About Two Proportios Key Cocept This sectio presets methods for usig two sample proportios for costructig a cofidece iterval estimate of the differece betwee the correspodig populatio proportios, or testig a claim made about the two populatio proportios. Slide 3 Requiremets. We have proportios from two idepedet simple radom samples.. For each of the two samples, the umber of successes is at least 5 ad the umber of failures is at least 5. Slide 4 Notatio for Two Proportios For populatio, we let: p = populatio proportio = size of the sample x = umber of successes i the sample p = x (the sample proportio) q = p The correspodig meaigs are attached to p,, x, p. ad q, which come from populatio. Slide 5 Pooled Sample Proportio The pooled sample proportio is deoted by p ad is give by: p = x x We deote the complemet of p by q, so q = p Slide 6

2 Test Statistic for Two Proportios For H 0 : p = p H : p p, H : p < p, H : p > p z = ( p p ) ( p p ) pq pq Test Statistic for Two Proportios - cot For H 0 : p = p H : p p, H : p < p, H : p > p where p p = 0 (assumed i the ull hypothesis) p = x ad p x = Slide 7 p = x x ad q = p Slide 8 Test Statistic for Two Proportios - cot P-value: Use Table A-. (Use the computed value of the test statistic z ad fid its P-value by followig the procedure summarized by Figure 8-6 i the text.) Critical values: Use Table A-. (Based o the sigificace level α, fid critical values by usig the procedures itroduced i Sectio 8- i the text.) Example: For the sample data listed i the Table below, use a 0.05 sigificace level to test the claim that the proportio of black drivers stopped by the police is greater tha the proportio of white drivers Slide 9 Slide 0 Example: For the sample data listed i the previous Table, use a 0.05 sigificace level to test the claim that the proportio of black drivers stopped by the police is greater tha the proportio of white drivers = 00 x = 4 p = x = 4 = = 400 x = 47 p = x = 47 = H 0 : p = p, H : p > p p = x x = 4 47 = q = = Example: For the sample data listed i the previous Table, use a 0.05 sigificace level to test the claim that the proportio of black drivers stopped by the police is greater tha the proportio of white drivers = 00 x = 4 p = x = 4 = = 400 x = 47 p = x = 47 = z = ( ) 0 ( )(0.8935) ( )(0.8935) z = 0.64 Slide Slide

3 Example: For the sample data listed i the previous Table, use a 0.05 sigificace level to test the claim that the proportio of black drivers stopped by the police is greater tha the proportio of white drivers = 00 x = 4 p = x = 4 = = 400 x = 47 p = x = 47 = z = 0.64 This is a right-tailed test, so the P- value is the area to the right of the test statistic z = The P-value is 0.6. Because the P-value of 0.6 is greater tha the sigificace level of α = 0.05, we fail to reject the ull hypothesis. Example: For the sample data listed i the previous Table, use a 0.05 sigificace level to test the claim that the proportio of black drivers stopped by the police is greater tha the proportio of white drivers z = 0.64 = 00 x = 4 p = x = 4 = = 400 x = 47 p = x = 47 = Because we fail to reject the ull hypothesis, we coclude that there is ot sufficiet evidece to support the claim that the proportio of black drivers stopped by police is greater tha that for white drivers. This does ot mea that racial profilig has bee disproved. The evidece might be strog eough with more data. Slide 3 Slide 4 Example: For the sample data listed i the previous Table, use a 0.05 sigificace level to test the claim that the proportio of black drivers stopped by the police is greater tha the proportio of white drivers = 00 x = 4 p = x = 4 = = 400 x = 47 p = x = 47 = Cofidece Iterval Estimate of p - p ( p p ) E < ( p p ) < ( p p ) E where E = z α/ p q p q Slide 5 Slide 6 Example: For the sample data listed i the previous Table, fid a 90% cofidece iterval estimate of the differece betwee the two populatio proportios. = 00 x = 4 p = x = 4 = = 400 x = 47 p = x = 47 = E = z α/ E =.645 E = p q p q (.)(.88)(0.05)(0.895) Example: For the sample data listed i the previous table, use a 0.05 sigificace level to test the claim that the proportio of black drivers stopped by the police is greater tha the proportio of white drivers who are stopped. = 00 ( ) < ( p p ) < ( ) x = < ( p p ) < p = x = 4 = = 400 x = 47 p = x = 47 = Slide 7 Slide 8

4 Sectio 9-3 Ifereces About Two Meas: Idepedet Samples Key Cocept This sectio presets methods for usig sample data from two idepedet samples to test hypotheses made about two populatio meas or to costruct cofidece iterval estimates of the differece betwee two populatio meas. Slide 9 Part : Idepedet Samples with σ ad σ Ukow ad Not Assumed Equal Slide 0 Defiitios Two samples are idepedet if the sample values selected from oe populatio are ot related to or somehow paired or matched with the sample values selected from the other populatio. Two samples are depedet (or cosist of matched pairs) if the members of oe sample ca be used to determie the members of the other sample. 4 Requiremets. σ a σ are ukow ad o assumptio is made about the equality of σ ad σ.. The two samples are idepedet. 3. Both samples are simple radom samples. 4. Either or both of these coditios are satisfied: The two sample sizes are both large (with > 30 ad > 30) or both samples come from populatios havig ormal distributios. Slide Slide Hypothesis Test for Two Meas: Idepedet Samples Hypothesis Test - cot Test Statistic for Two Meas: Idepedet Samples t = (x x ) (µ µ ). s s Degrees of freedom: P-values: I this book we use this simple ad coservative estimate: df = smaller of ad. Refer to Table A-3. Use the procedure summarized i Figure 8-6. Critical values: Refer to Table A-3. Slide 3 Slide 4

5 McGwire Versus Bods Sample statistics are show for the distaces of the home rus hit i record-settig seasos by Mark McGwire ad Barry Bods. Use a 0.05 sigificace level to test the claim that the distaces come from populatios with differet meas. McGwire Versus Bods - cot Below is a Statdisk plot of the data McGwire Bods x s Slide 5 Slide 6 McGwire Versus Bods - cot Claim: µ µ H o : µ = µ H : µ µ α = 0.05 = 69 = 7 df = 69 t.05 = McGwire Versus Bods - cot Test Statistic for Two Meas: t = (x x ) (µ µ ) s s Slide 7 Slide 8 McGwire Versus Bods - cot Test Statistic for Two Meas: McGwire Versus Bods - cot Claim: µ µ H o : µ = µ t = ( ) H : µ µ α = 0.05 =.73 Slide 9 Slide 30

6 McGwire Versus Bods - cot Cofidece Itervals Claim: µ µ H o : µ = µ H : µ µ α = 0.05 There is sigificat evidece to support the claim that there is a differece betwee the mea home ru distaces of Mark McGwire ad Barry Bods. (x x ) E < (µ µ ) < (x x ) E Reject the Null Hypothesis where E = tα/ α/ s s Slide 3 Slide 3 McGwire Versus Bods Cofidece Iterval Method Usig the data give i the precedig example, costruct a 95% cofidece iterval estimate of the differece betwee the mea home ru distaces of Mark McGwire ad Barry Bods. E = t α/ α/ E =.994 E = 3.0 s s Slide 33 6 McGwire Versus Bods Cofidece Iterval Method - cot Usig the data give i the precedig example, costruct a 95% cofidece iterval estimate of the differece betwee the mea home ru distaces of Mark McGwire ad Barry Bods. ( ) 3.0 < (µ µ ) < ( ) < (µ µ ) < 7.8 We are 95% cofidet that the limits of.8 ft ad 7.8 ft actually do cotai the differece betwee the two populatio meas. Slide 34 Part : Alterative Methods Idepedet Samples with σ ad σ Kow. Slide 35 Slide 36

7 Requiremets. The two populatio stadard deviatios are both kow.. The two samples are idepedet. 3. Both samples are simple radom samples. 4. Either or both of these coditios are satisfied: The two sample sizes are both large (with > 30 ad > 30) or both samples come from populatios havig ormal distributios. Hypothesis Test for Two Meas: Idepedet Samples with σ ad σ Both Kow z = (x x ) (µ µ ) σ σ P-values ad critical values: Refer to Table A-. Slide 37 Slide 38 Cofidece Iterval: Idepedet Samples with σ ad σ Both Kow (x x ) E < (µ µ ) < (x x ) E 7 Methods for Ifereces About Two Idepedet Meas where E = zα/ α/ σ σ Slide 39 Figure 9-3 Slide 40 Requiremets Assume that σ = σ ad Pool the Sample Variaces.. The two populatio stadard deviatios are ot kow, but they are assumed to be equal. That is σ = σ.. The two samples are idepedet. 3. Both samples are simple radom samples. Slide 4 4. Either or both of these coditios are satisfied: The two sample sizes are both large (with > 30 ad > 30) or both samples come from populatios havig ormal distributios. Slide 4

8 Hypothesis Test Statistic for Two Meas: Idepedet Samples ad σ = σ Cofidece Iterval Estimate of µ µ : Idepedet Samples with σ = σ Where. t = (x x ) (µ µ ) s = ( ) ( -) ( ) ( ) ad the umber of degrees of freedom is df = - s Slide 43 (x x ) E < (µ µ ) < (x x ) E where E = tα/ ad umber of degrees of freedom is df = - Slide 44 Strategy Uless istructed otherwise, use the followig strategy: Assume that σ ad σ are ukow, do ot assume that σ = σ, ad use the test statistic ad cofidece iterval give i Part of this sectio. (See Figure 9-3.) 8 Slide 45