Chapter 8 Slide 1 Inferences from Two Samples 8-1 Overview 8-2 Inferences about Two Proportions 8-3 Inferences about Two Means: Independent Samples 8-4 Inferences about Matched Pairs 8-5 Comparing Variation in Two Samples Chapter 8, Triola, Elementary Statistics , MATH 1342
Slide 2 Section 8-1 & 8-2 Overview and Inferences about Two Proportions Created by Erin Hodgess, Houston, Texas Chapter 8, Triola, Elementary Statistics , MATH 1342
Overview (p.438) Slide 3 There are many important and meaningful situations in which it becomes necessary to compare two sets of sample data. Chapter 8, Triola, Elementary Statistics , MATH 1342
Inferences about Slide 4 Two Proportions Assumptions (p.439) 1. We have proportions from two independent simple random samples. 2. For both samples, the conditions np ≥ 5 and nq ≥ 5 are satisfied. Chapter 8, Triola, Elementary Statistics , MATH 1342
Notation for Slide 5 Two Proportions For population 1, we let: p 1 = population proportion n 1 = size of the sample x 1 = number of successes in the sample ^ p 1 = x 1 (the sample proportion) n 1 Corresponding meanings are attached ^ ^ ^ ^ q 1 = 1 – p 1 to p 2 , n 2 , x 2 , p 2 . and q 2 , which come from population 2. Chapter 8, Triola, Elementary Statistics , MATH 1342
Pooled Estimate of Slide 6 p 1 and p 2 � The pooled estimate of p 1 and p 2 is denoted by p. x 1 + x 2 p = n 1 + n 2 � q = 1 – p Chapter 8, Triola, Elementary Statistics , MATH 1342
Test Statistic for Slide 7 Two Proportions (p.441) For H 0 : p 1 = p 2 , H 0 : p 1 = p 2 , H 0 : p 1 = p 2 H 1 : p 1 ≠ p 2 , H 1 : p 1 < p 2 , H 1 : p 1 > p 2 where p 1 – p 2 = 0 (assumed in the null hypothesis) x 1 x 2 ^ ^ p 1 p 2 and = = n 1 n 2 x 1 + x 2 q = 1 – p p = and n 1 + n 2 Chapter 8, Triola, Elementary Statistics , MATH 1342
Test Statistic for Slide 8 Two Proportions (p.441) For H 0 : p 1 = p 2 , H 0 : p 1 = p 2 , H 0 : p 1 = p 2 H 1 : p 1 ≠ p 2 , H 1 : p 1 < p 2 , H 1 : p 1 > p 2 ^ ^ ( p 1 – p 2 ) – ( p 1 – p 2 ) z = pq pq + n 2 n 1 Chapter 8, Triola, Elementary Statistics , MATH 1342
Example: For the sample data listed in Table 8-1, use a 0.05 significance level to test Slide 9 the claim that the proportion of black drivers stopped by the police is greater than the proportion of white drivers who are stopped. (p.441) Chapter 8, Triola, Elementary Statistics , MATH 1342
Example: For the sample data listed in Table 8-1, use a 0.05 significance level to test Slide 10 the claim that the proportion of black drivers stopped by the police is greater than the proportion of white drivers who are stopped. n 1 = 200 H 0 : p 1 = p 2 , H 1 : p 1 > p 2 x 1 = 24 ^ p = x 1 + x 2 = 24 + 147 = 0.106875 p 1 = x 1 = 24 = 0.120 n 1 + n 2 200+1400 n 1 200 q = 1 – 0.106875 = 0.893125. n 2 = 1400 x 2 = 147 ^ p 2 = x 2 = 147 = 0.105 n 2 1400 Chapter 8, Triola, Elementary Statistics , MATH 1342
Example: For the sample data listed in Table 8-1, use a 0.05 significance level to test Slide 11 the claim that the proportion of black drivers stopped by the police is greater than the proportion of white drivers who are stopped. n 1 = 200 z = (0.120 – 0.105) – 0 x 1 = 24 (0.106875)(0.893125) + (0.106875)(0.893125) 200 1400 ^ p 1 = x 1 = 24 = 0.120 n 1 200 z = 0.64 n 2 = 1400 x 2 = 147 ^ p 2 = x 2 = 147 = 0.105 n 2 1400 Chapter 8, Triola, Elementary Statistics , MATH 1342
Example: For the sample data listed in Table 8-1, use a 0.05 significance level to test Slide 12 the claim that the proportion of black drivers stopped by the police is greater than the proportion of white drivers who are stopped. n 1 = 200 (0.120 – 0.105) – 0.040 < ( p 1 – p 2 ) < (0.120 – 0.105) + 0.040 –0.025 < ( p 1 – p 2 ) < 0.055 x 1 = 24 ^ p 1 = x 1 = 24 = 0.120 n 1 200 n 2 = 1400 x 2 = 147 ^ p 2 = x 2 = 147 = 0.105 n 2 1400 Chapter 8, Triola, Elementary Statistics , MATH 1342
Example: For the sample data listed in Table 8-1, use a 0.05 significance level to test Slide 13 the claim that the proportion of black drivers stopped by the police is greater than the proportion of white drivers who are stopped. n 1 = 200 x 1 = 24 ^ p 1 = x 1 = 24 = 0.120 n 1 200 n 2 = 1400 x 2 = 147 ^ p 2 = x 2 = 147 = 0.105 n 2 1400 Chapter 8, Triola, Elementary Statistics , MATH 1342
Example: For the sample data listed in Table 8-1, use a 0.05 significance level to test Slide 14 the claim that the proportion of black drivers stopped by the police is greater than the proportion of white drivers who are stopped. n 1 = 200 z = 0.64 x 1 = 24 This is a right-tailed test, so the P- ^ p 1 = x 1 = 24 = 0.120 value is n 1 200 the area to the right of the test statistic z = 0.64. The P -value is 0.2611. n 2 = 1400 Because the P -value of 0.2611 is x 2 = 147 greater than the significance level of α ^ p 2 = x 2 = 147 = 0.105 = 0.05, we fail to reject the null n 2 hypothesis. 1400 Chapter 8, Triola, Elementary Statistics , MATH 1342
Example: For the sample data listed in Table 8-1, use a 0.05 significance level to test Slide 15 the claim that the proportion of black drivers stopped by the police is greater than the proportion of white drivers who are stopped. n 1 = 200 z = 0.64 x 1 = 24 Because we fail to reject the null ^ p 1 = x 1 = 24 = 0.120 hypothesis, we conclude that there is not n 1 200 sufficient evidence to support the claim that the proportion of black drivers n 2 = 1400 stopped by police is greater than that for x 2 = 147 white drivers. This does not mean that ^ p 2 = x 2 = 147 = 0.105 racial profiling has been disproved. The n 2 evidence might be strong enough with 1400 more data. Chapter 8, Triola, Elementary Statistics , MATH 1342
Confidence Interval Slide 16 Estimate of p 1 - p 2 ^ ^ ^ ^ ( p 1 – p 2 ) – E < ( p 1 – p 2 ) < ( p 1 – p 2 ) + E ^ ^ ^ ^ p 1 q 1 p 2 q 2 E = z α/ 2 + where n 2 n 1 Chapter 8, Triola, Elementary Statistics , MATH 1342
Slide 17 Example: For the sample data listed in Table 8-1, find a 90% confidence interval estimate of the difference between the two population proportions. (p.444) ^ ^ ^ ^ p 1 q 1 p 2 q 2 n 1 = 200 E = z α/ 2 + n 2 n 1 x 1 = 24 ^ (.12)(.88)+(0.105)(0.895) p 1 = x 1 = 24 = 0.120 E = 1.645 n 1 200 200 1400 n 2 = 1400 E = 0.400 x 2 = 147 ^ p 2 = x 2 = 147 = 0.105 n 2 1400 Chapter 8, Triola, Elementary Statistics , MATH 1342
Slide 18 Section 8-3 Inferences about Two Means: Independent Samples Created by Erin Hodgess, Houston, Texas Chapter 8, Triola, Elementary Statistics , MATH 1342
Definitions Slide 19 Two Samples: Independent The sample values selected from one population are not related or somehow paired with the sample values selected from the other population. If the values in one sample are related to the values in the other sample, the samples are dependent. Such samples are often referred to as matched pairs or paired samples. Chapter 8, Triola, Elementary Statistics , MATH 1342
Assumptions (p.453) Slide 20 1. The two samples are independent. 2. Both samples are simple random samples. 3. Either or both of these conditions are satisfied: The two sample sizes are both large (with n 1 > 30 and n 2 > 30) or both samples come from populations having normal distributions. Chapter 8, Triola, Elementary Statistics , MATH 1342
Hypothesis Tests Slide 21 Test Statistic for Two Means: ( x 1 – x 2 ) – ( µ 1 – µ 2 ) t = s 2 s 1 2 2 . + n 2 n 1 Chapter 8, Triola, Elementary Statistics , MATH 1342
Hypothesis Tests Slide 22 Test Statistic for Two Means: Degrees of freedom: In this book we use this estimate: df = smaller of n 1 – 1 and n 2 – 1. P -value: Refer to Table A-3. Use the procedure summarized in Figure 7-6. Critical values: Refer to Table A-3. Chapter 8, Triola, Elementary Statistics , MATH 1342
McGwire Versus Bonds (p.455) Slide 23 Data Set 30 in Appendix B includes the distances of the home runs hit in record-setting seasons by Mark McGwire and Barry Bonds. Sample statistics are shown. Use a 0.05 significance level to test the claim that the distances come from populations with different means. McGwire Bonds 70 73 n 418.5 403.7 x 45.5 30.6 s Chapter 8, Triola, Elementary Statistics , MATH 1342
McGwire Versus Bonds Slide 24 Chapter 8, Triola, Elementary Statistics , MATH 1342
McGwire Versus Bonds Slide 25 Claim: μ 1 ≠ μ 2 H o : μ 1 = μ 2 H 1 : μ 1 ≠ μ 2 α = 0.05 n 1 – 1 = 69 n 2 – 1 = 72 df = 69 t .025 = 1.994 Chapter 8, Triola, Elementary Statistics , MATH 1342
McGwire Versus Bonds Slide 26 Test Statistic for Two Means: ( x 1 – x 2 ) – ( µ 1 – µ 2 ) t = 2 2 s 1 s 2 . + n 2 n 1 Chapter 8, Triola, Elementary Statistics , MATH 1342
McGwire Versus Bonds Slide 27 Test Statistic for Two Means: (418.5 – 403.7 ) – 0 t = 30.6 2 45.5 2 + 70 73 = 2.273 Chapter 8, Triola, Elementary Statistics , MATH 1342
McGwire Versus Bonds Slide 28 Claim: μ 1 ≠ μ 2 H o : μ 1 = μ 2 H 1 : μ 1 ≠ μ 2 α = 0.05 Figure 8-2 Chapter 8, Triola, Elementary Statistics , MATH 1342
McGwire Versus Bonds Slide 29 Claim: μ 1 ≠ μ 2 There is significant evidence to support the claim that there is a difference between the H o : μ 1 = μ 2 mean home run distances of Mark McGwire H 1 : μ 1 ≠ μ 2 and Barry Bonds. α = 0.05 Reject Null Figure 8-2 Chapter 8, Triola, Elementary Statistics , MATH 1342
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