SECTION 8.1 Testing the Difference Between Means (Independent Samples, s1 and s2 Known) 421 Picturing the World In a recent year, there were 116,898 public school teachers in Georgia and 113,327 in Ohio. In a survey that year, 200 public school teachers in each of these states were asked to report their salary. The results are shown below. It is claimed that the mean salary in Ohio is greater than the mean salary in Georgia. (Adapted from National Education Association) Georgia x1 = $60,578 n1 = 200 Ohio x2 = $61,406 n2 = 200 Determine a null hypothesis and alternative hypothesis for this claim. When the conditions on the preceding page are met and the sampling distribution for x1 - x2 is a normal distribution, you can use the z@test to test the difference between two population means m1 and m2. The standardized test statistic takes the form of z = 1Observed difference2 - 1Hypothesized difference2 Standard error . As you read the definition and guidelines for a two-sample z@test, note that if the null hypothesis states m1 = m2, m1 … m2, or m1 Ú m2, then m1 = m2 is assumed and the hypothesized difference m1 - m2 is equal to 0. A two-sample z@test can be used to test the difference between two population means m1 and m2 when these conditions are met. 1. Both s1 and s2 are known. 2. The samples are random. 3. The samples are independent. 4. The populations are normally distributed or both n1 Ú 30 and n2 Ú 30. The test statistic is x1 - x2. The standardized test statistic is z = 1x1 - x22 - 1m1 - m22 sx 1 -x2 where sx 1 -x2 = Bs 2 1 n1 + s 2 2 n2 . Two-Sample z-Test for the Difference Between Means Using a Two-Sample z-Test for the Difference Between Means (Independent Samples, S1 and S2 Known) In Words In Symbols 1. Verify that s1 and s2 are known, the samples are random and independent, and either the populations are normally distributed or both n1 Ú 30 and n2 Ú 30. 2. State the claim mathematically State H0 and Ha. and verbally. Identify the null and alternative hypotheses. 3. Specify the level of significance. Identify a. 4. Determine the critical value(s). Use Table 4 in Appendix B. 5. Determine the rejection region(s). 6. Find the standardized test statistic z = 1x1 - x22 - 1m1 - m22 sx 1 -x2 and sketch the sampling distribution. 7. Make a decision to reject or fail to If z is in the rejection region, reject the null hypothesis. then reject H0. Otherwise, fail to reject H0. 8. Interpret the decision in the context of the original claim. GUIDELINES A hypothesis test for the difference between means can also be performed using P@values. Use the guidelines above, skipping Steps 4 and 5. After finding the standardized test statistic, use Table 4 in Appendix B to calculate the P@value. Then make a decision to reject or fail to reject the null hypothesis. If P is less than or equal to a, then reject H0. Otherwise, fail to reject H0.
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