STATISTICS ELEMENTARY MARIO F. TRIOLA Chapter 8 Inferences from Two Samples MARIO F. TRIOLA EIGHTH EDITION

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

8-1 Overview There are many important and meaningful situations in which it becomes necessary to compare two sets of sample data. page 438 of text Examples in the discussion

8-2 Inferences about Two Means: Independent and Large Samples

Two Samples: Independent 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. Text will use the wording ‘matched pairs’. Example at bottom of page 438- 439

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. page 439

Test Statistic for Two Means: Independent and Large Samples Hypothesis Tests Test Statistic for Two Means: Independent and Large Samples (x1 - x2) - (µ1 - µ2) z = 1. 2 2 2 + n1 n2

Test Statistic for Two Means: Independent and Large Samples 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.

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. Example on page 440 of text

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

Coke Versus Pepsi

Coke Versus Pepsi Claim: 1  2 Ho : 1 = 2 H1 : 1  2  = 0.01 Reject H0 Fail to reject H0 Reject H0 Z = - 2.575 Z = 2.575 1 -  = 0 or Z = 0

Test Statistic for Two Means: Independent and Large Samples Coke Versus Pepsi Test Statistic for Two Means: Independent and Large Samples (x1 - x2) - (µ1 - µ2) z = 1. 2 2 2 + n1 n2

Test Statistic for Two Means: Independent and Large Samples Coke Versus Pepsi Test Statistic for Two Means: Independent and Large Samples z = (0.81682 - 0.82410) - 0 0.0075707 2 0.005701 2 + 36 36 = - 4.63

Coke Versus Pepsi Claim: 1  2 Ho : 1 = 2 H1 : 1  2  = 0.01 Reject H0 Fail to reject H0 Reject H0 Z = - 2.575 Z = 2.575 sample data: z = - 4.63 1 -  = 0 or Z = 0

Coke Versus Pepsi Claim: 1  2 Ho : 1 = 2 H1 : 1  2  = 0.01 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. Reject H0 Fail to reject H0 Reject H0 Further explanation of interpretation is given in text. The magnitude of the difference is the weights is not anything that consumers would notice. Also this test simply indicates the Coke ingredients weigh less which does not indicate that there is less volume of the product. Reject Null Z = - 2.575 Z = 2.575 sample data: z = - 4.63 1 -  = 0 or Z = 0

Confidence Intervals (x1 - x2) - E < (µ1 - µ2) < (x1 - x2) + E page 442 of text

Confidence Intervals (x1 - x2) - E < (µ1 - µ2) < (x1 - x2) + E 1 2 2 2 where E = z + n1 n2