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10 Chapter Two-Sample Hypothesis Tests Two-Sample Tests Comparing Two Means: Independent Samples Comparing Two Means: Paired Samples Comparing Two Proportions Comparing Two Variances McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Two-Sample Tests 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 nonsample benchmark. • 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. • Two samples that are drawn from the same population may yield different estimates of a parameter due to chance. 10-2 Comparing Two Means: Independent Samples Format of Hypotheses • The hypotheses for comparing two independent population means µ1 and µ2 are: 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. 10-3 Comparing Two Means: Independent Samples; Paired Samples Paired t Test Table 10.1 • 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 10-4 Comparing Two Means: Paired Samples • The mean d and standard deviation sd of the sample of n differences are calculated with the usual formulas for a mean and standard deviation. Apply the 1-sample t-test to these differences. • Comparing Two Proportions Testing for Zero Difference: 1 = 2 • To compare two population proportions, 1, 2, use the following hypotheses 10-5 Comparing Two Proportions Pooled Proportion • If H0 is true, there is no difference between 1 and 2, so the samples are pooled (averaged) into one “big” sample to estimate the common population proportion. 10-6 Comparing Two Proportions Test Statistic • The test statistic for the hypothesis 1 = 2 may also be written as: Assuming normality with n > 10 and n(1-) > 10 for both samples. 10-7 Comparing Two Proportions Analogy to Confidence Intervals • The confidence interval for 1 – 2 without pooling the samples is: • If the confidence interval does not include 0, then we reject the null hypothesis. 10-8 Comparing Two Proportions Testing for Non-Zero Differences • Testing for equality is a special case of testing for a specified difference D0 between two proportions. • If the hypothesized difference D0 is non-zero, the test statistic is: calc 10-9 Comparing Two Variances Format of Hypotheses • • To test whether two population means are equal, we may also need to test whether two population variances are equal. The hypotheses may be stated as 10-10 Comparing Two Variances Format of Hypotheses • An equivalent way to state these hypotheses would be to use ratios since the variance can never be less than zero and it would not make sense to take the difference between two variances. 10-11 Comparing Two Variances 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 10-12 Comparing Two Variances 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 1 = n1 – 1 and the denominator s22 has degrees of freedom 2 = n2 – 1. • Critical values for the F test are denoted FL (left tail) and FR (right tail). 10-13 Comparing Two Variances The F Test • • A right-tail critical value FR may be found from Appendix F using 1 and 2 degrees of freedom. FR = F1, 2 A left-tail critical value FR may be found by reversing the numerator and denominator degrees of freedom, finding the critical value from Appendix F and taking its reciprocal: FL = 1/F2, 1 10-14 Comparing Two Variances The F Test • • • Reject the null hypothesis if Fcalc > FR for a right-tail test for a given . Reject the null hypothesis if Fcalc < FL for a left-tail test for a given . Reject the null hypothesis if Fcalc > FR or if Fcalc < FL for a two-tail test. Here we use /2 to obtain the critical values. 10-15