Question

You have a random sample of 100 population mem- bers from each of two populations. You intend to use sample results to test the following hypotheses: H0: ?1 ? ?2 ? 0 Ha: ?1 ? ?2 ? 0

a. Suppose the sample results are x1 ? 85 and x2 ? 82. Compute the sample test statistic, zstat. Assume the population standard deviations are ?1 ? 9 and ?2 ? 11.

. Using a significance level of .01, show the critical z-scores, zc, for the test and state your decision rule. Report your decision

Answer 1

The provided sample means are shown below: X = 85 X2 = 82 Also, the provided population standard deviations are: 01 = 9 02 = 11 and the sample sizes are ni = 100 and n2 = 100. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: M1 = M2 Ha: My = H2 This corresponds to a two-tailed test, for which a z-test for two population means, with known population standard deviations will be used. (2) Rejection Region 0.01, and the Based on the information provided, the significance level is a = critical value for a two-tailed test is zc = 2.58. The rejection region for this two-tailed test is R = {z: |Z| > 2.58} (3) Test Statistics The Z-statistic is computed as follows: 2 = X1 - X2 Voi/nı tož/n2 85 – 82 92/100 + 112 /100 2.111 (4) Decision about the null hypothesis Since it is observed that | | = 2.111 < zc = 2.58, it is then concluded that the null hypothesis is not rejected. Using the P-value approach: The p-value is p= 0.0348, and since p= 0.0348 > 0.01, it is concluded that the null hypothesis is not rejected. Confidence Interval The 99% confidence interval for pl M2 is -0.661 < M1 – M2 < 6.661. Graphically. p-value = Test 0.0348 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

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