428 CHAPTER 8 Hypothesis Testing with Two Samples Testing the Difference Between Means (Independent Samples, s1 and s2 Unknown) 8.2 What You Should Learn How to perform a two-sample t-test for the difference between two means m1 and m2 using independent samples with s1 and s2 unknown The Two-Sample t@Test for the Difference Between Means The Two-Sample t-Test for the Difference Between Means In Section 8.1, you learned how to test the difference between means when both population standard deviations are known. Both population standard deviations are not known in many real-life situations. In this section, you will learn how to use a t@test to test the difference between two population means m1 and m2 using independent samples from each population when s1 and s2 are unknown. These conditions are necessary to perform such a test: (1) the population standard deviations are unknown, (2) the samples are randomly selected, (3) the samples are independent, and (4) the populations are normally distributed or each sample size is at least 30. When these conditions are met, the sampling distribution for the difference between the sample means x1 - x2 is approximated by a t@distribution with mean m1 - m2. So, you can use a two-sample t@test to test the difference between the population means m1 and m2. The standard error and the degrees of freedom of the sampling distribution depend on whether the population variances s1 2 and s2 2 are equal, as shown in the next definition. A two-sample t@test is used to test the difference between two population means m1 and m2 when (1) s1 and s2 are unknown, (2) the samples are random, (3) the samples are independent, and (4) the populations are normally distributed or both n1 Ú 30 and n2 Ú 30. The test statistic is x1 - x2, and the standardized test statistic is t = 1x1 - x22 - 1m1 - m22 sx 1 -x2 . Variances are equal: If the population variances are equal, then information from the two samples is combined to calculate a pooled estimate of the standard deviation nS. ns = B1n1 - 12s1 2 + 1n 2 - 12s2 2 n1 + n2 - 2 The standard error for the sampling distribution of x1 - x2 is sx 1 -x2 = ns# A1 n1 + 1 n2 Variances equal and d.f. = n1 + n2 - 2. Variances are not equal: If the population variances are not equal, then the standard error is sx 1 -x2 = Bs2 1 n1 + s2 2 n2 Variances not equal and d.f. = smaller of n1 - 1 and n2 - 1. Two-Sample t-Test for the Difference Between Means Study Tip To perform the two-sample t-test described at the right, you will need to know whether the variances of two populations are equal. In this chapter, each example and exercise will state whether the variances are equal. You will learn to test for differences between two population variances in Chapter 10.
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