420 CHAPTER 8 Hypothesis Testing with Two Samples It is important to remember that when you perform a two-sample hypothesis test using independent samples, you are testing a claim concerning the difference between the parameters in two populations, not the values of the parameters themselves. For a two-sample hypothesis test with independent samples, 1. the null hypothesis H0 is a statistical hypothesis that usually states there is no difference between the parameters of two populations. The null hypothesis always contains the symbol …, =, or Ú. 2. the alternative hypothesis Ha is a statistical hypothesis that is true when H0 is false. The alternative hypothesis contains the symbol 7, ≠, or 6. DEFINITION To write the null and alternative hypotheses for a two-sample hypothesis test with independent samples, translate the claim made about the population parameters from a verbal statement to a mathematical statement. Then, write its complementary statement. For instance, for a claim about two population parameters m1 and m2, some possible pairs of null and alternative hypotheses are eH0: m1 = m2 Ha: m1 ≠ m2 , e H0: m1 … m2 Ha: m1 7 m2 , and e H0: m1 Ú m2 Ha: m1 6 m2 . Regardless of which hypotheses you use, you always assume there is no difference between the population means 1m1 = m22. Two-Sample z@Test for the Difference Between Means In the remainder of this section, you will learn how to perform a z@test for the difference between two population means m1 and m2 when the samples are independent. These conditions are necessary to perform such a test. 1. The population standard deviations are known. 2. The samples are randomly selected. 3. The samples are independent. 4. The populations are normally distributed or each sample size is at least 30. When these conditions are met, the sampling distribution for x1 − x2, the difference of the sample means, is a normal distribution with mean and standard error as shown in the table below and the figure at the left. In Words In Symbols The mean of the difference of the sample means is the assumed difference between the two population means. When no difference is assumed, the mean is 0. Mean = mx 1 -x2 = mx 1 - mx 2 = m1 - m2 The variance of the sampling distribution is the sum of the variances of the individual sampling distributions for x1 and x2. The standard error is the square root of the sum of the variances. Standard error = sx 1 -x2 = 2sx 1 2 + sx 2 2 = Bs 2 1 n1 + s 2 2 n2 1 − 2 x1 − x2 σ μ μ Sampling Distribution for x1 − x2 x1 − x2 Study Tip You can also write the null and alternative hypotheses as shown below. bH0: m1 - m2 = 0 Ha: m1 - m2 ≠0 bH0: m1 - m2 … 0 Ha: m1 - m2 7 0 bH0: m1 - m2 Ú 0 Ha: m1 - m2 6 0
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