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ELEMENTARY Chapter 8 MARIO F. TRIOLA STATISTICS Inferences from Two Samples EIGHTH Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION 1 Chapter 8 Inferences from Two Samples 8-1 Overview 8-2 Inferences about Two Means: Independent and Large Samples 8-3 Inferences about Two Means: Matched Pairs 8-4 Inferences about Two Proportions 8-5 Comparing Variation in Two Samples 8-6 Inferences about Two Means: Independent and Small Samples Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 2 8-1 Overview There are many important and meaningful situations in which it becomes necessary to compare two sets of sample data. Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 3 8-2 Inferences about Two Means: Independent and Large Samples Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 4 Definitions 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. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 5 Assumptions 1. The two samples are independent. 2. The two sample sizes are large. That is, n1 > 30 and n2 > 30. 3. Both samples are simple random samples. Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6 Hypothesis Tests Test Statistic for Two Means: Independent and Large Samples z = (x1 - x2) - (µ1 - µ2) 2. 1 n1 2 2 + n 2 Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 7 Hypothesis Tests Test Statistic for Two Means: Independent and Large Samples and If and are not known, use s1 and s2 in their places. provided that both samples are large. P-value: Use the computed value of the test statistic z, and find the P-value by following the same procedure summarized in Figure 7-8. Critical values: Based on the significance level , find critical values by using the procedures introduced in Section 7-2. Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 8 Coke Versus Pepsi Data Set 1 in Appendix B includes the weights (in pounds) of samples of regular Coke and regular Pepsi. Sample statistics are shown. Use the 0.01 significance level to test the claim that the mean weight of regular Coke is different from the mean weight of regular Pepsi. Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 9 Coke Versus Pepsi Data Set 1 in Appendix B includes the weights (in pounds) of samples of regular Coke and regular Pepsi. Sample statistics are shown. Use the 0.01 significance level to test the claim that the mean weight of regular Coke is different from the mean weight of regular Pepsi. Regular Coke Regular Pepsi n 36 36 x 0.81682 0.82410 s 0.007507 0.005701 Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 10 Coke Versus Pepsi Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 11 Coke Versus Pepsi Claim: 1 2 Ho : 1 = 2 H1 : 1 2 = 0.01 Reject H0 Z = - 2.575 Fail to reject H0 1 - = 0 Reject H0 Z = 2.575 or Z = 0 Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 12 Coke Versus Pepsi Test Statistic for Two Means: Independent and Large Samples z = (x1 - x2) - (µ1 - µ2) 2. 1 n1 2 2 + n 2 Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 13 Coke Versus Pepsi Test Statistic for Two Means: Independent and Large Samples z = (0.81682 - 0.82410) - 0 0.0075707 2 36 + 0.005701 2 36 = - 4.63 Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 14 Coke Versus Pepsi Claim: 1 2 Ho : 1 = 2 H1 : 1 2 = 0.01 Reject H0 sample data: z = - 4.63 Z = - 2.575 Fail to reject H0 1 - = 0 Reject H0 Z = 2.575 or Z = 0 Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 15 Coke Versus Pepsi Claim: 1 2 Ho : 1 = 2 H1 : 1 2 There is significant evidence to support the claim that there is a difference between the mean weight of Coke and the mean weight of Pepsi. = 0.01 Reject H0 Fail to reject H0 Reject H0 Reject Null sample data: z = - 4.63 Z = - 2.575 1 - = 0 Z = 2.575 or Z = 0 Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 16 Confidence Intervals (x1 - x2) - E < (µ1 - µ2) < (x1 - x2) + E Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 17 Confidence Intervals (x1 - x2) - E < (µ1 - µ2) < (x1 - x2) + E where E = z 2 1 n1 2 2 + n 2 Chapter 8. Sections 8.1 and 8.2 Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 18