Prepared by Lloyd R. Jaisingh A PowerPoint Presentation Package to Accompany Applied Statistics in Business & Economics, 4th edition David P. Doane and Lori E. Seward Prepared by Lloyd R. Jaisingh

Two-Sample Hypothesis Tests Chapter 10 Chapter Contents 10.1 Two-Sample Tests 10.2 Comparing Two Means: Independent Samples 10.3 Confidence Interval for the Difference of Two Means, 1 - 2 10.4 Comparing Two Means: Paired Samples 10.5 Comparing Two Proportions 10.6 Confidence Interval for the Difference of Two Proportions, 1 - 2 10.7 Comparing Two Variances

Two-Sample Hypothesis Tests Chapter 10 Chapter Learning Objectives LO10-1: Recognize and perform a test for two means with known 1 and 2. LO10-2: Recognize and perform a test for two means with unknown 1 and 2. LO10-3: Recognize paired data and be able to perform a paired t test. LO10-4: Explain the assumptions underlying the two-sample test of means. LO10-5: Perform a test to compare two proportions using z. LO10-6: Check whether normality may be assumed for two proportions. LO10-7: Use Excel to find p-values for two-sample tests using z or t. LO10-8: Carry out a test of two variances using the F distribution. LO10-9: Construct a confidence interval for μ1− μ2 or π1− π2 (optional).

What is a Two-Sample Test LO10-1 10.1 Two-Sample Tests Chapter 10 LO10-1: Recognize and perform a test for two means with known σ1 and σ2. What is a Two-Sample Test A Two-sample test compares two sample estimates with each other. A one-sample test compares a sample estimate against a non-sample benchmark. Basis of Two-Sample Tests The logic of two-sample tests is based on the fact that two samples drawn from the same population may yield different estimates of a parameter due to chance. If the two sample statistics differ by more than the amount attributable to chance, then we conclude that the samples came from populations with different parameter values.

10.2 Comparing Two Means: Independent Samples LO10-1 LO10-2 10.2 Comparing Two Means: Independent Samples Chapter 10 LO10-4 LO10-1: Recognize and perform a test for two means with known σ1 and σ2. Format of Hypotheses The hypotheses for comparing two independent population means µ1 and µ2 are: LO10-2: Recognize and perform a test for two means with unknown σ1 and σ2. LO10-4: Explain the assumptions underlying the two-sample test of means.

10.2 Comparing Two Means: Independent Samples LO10-1 LO10-2 10.2 Comparing Two Means: Independent Samples Chapter 10 LO10-4 Summary for the Test Statistic If the population variances 12 and 22 are known, then use the normal distribution. If population variances are unknown and estimated using s12 and s22, then use the Students t distribution. Table 10.1

10.3 Confidence Interval for the Difference of Two Means 1 - 2 Chapter 10 LO10-9 LO10-9: Construct a confidence interval for 1 − 2 or 1 - 2 (optional).

10.4 Comparing Two Means: Paired Samples LO10-3 10.4 Comparing Two Means: Paired Samples Chapter 10 LO10-3: Recognize paired data and be able to perform a paired t test. Paired t Test Paired data typically come from a before/after experiment. In the paired t test, the difference between x1 and x2 is measured as d = x1 – x2 The mean and standard deviation for the differences d are given below. The test statistic is just for a one-sample t-test. Apply one-sample t-test here.

10.5 Comparing Two Proportions LO10-5 10.5 Comparing Two Proportions Chapter 10 LO10-5: Perform a test to compare two proportions using z. Testing for Zero Difference: 1 = 2 To compare two population proportions, 1, 2, use the following hypotheses Assuming normality with n ≥ 10 and n(1- ) ≥ 10 for both samples.. LO10-6: Check whether normality may be assumed for two proportions.

10.5 Comparing Two Proportions LO10-5 10.5 Comparing Two Proportions Chapter 10 Testing for Non-Zero Difference

10.6 Confidence Interval for the Difference of Two Proportions 1 - 2 Chapter 10 If the confidence interval does not include 0, then we reject the null hypothesis.

10.7 Comparing Two Variances LO10-8 Chapter 10 LO10-8: Carry out a test of two variances using the F distribution. Format of Hypotheses To test whether two population means are equal, we may also need to test whether two population variances are equal.

10.7 Comparing Two Variances LO10-8 Chapter 10 The F Test The test statistic is the ratio of the sample variances: If the variances are equal, this ratio should be near unity: F = 1.

10.7 Comparing Two Variances LO10-8 Chapter 10 The F Test If the test statistic is far below 1 or above 1, we would reject the hypothesis of equal population variances. The numerator s12 has degrees of freedom df1 = n1 – 1 and the denominator s22 has degrees of freedom df2 = n2 – 1. The F distribution is skewed with the mean > 1. and its mode < 1.

10.7 Comparing Two Variances LO10-8 Chapter 10 The F Test: Critical Values   EXCEL’s F Test

10.7 Comparing Two Variances LO10-8 Chapter 10 Assumptions of the F Test The F test assumes that the populations being sampled are normal. It is sensitive to non-normality of the sampled populations. MINITAB reports both the F test and an alternative Levene’s test and p-values.