Question

Reserve Problems Chapter 10 Section 1 Problem 1 Consider the hypothesis test Ho: Mi - uz = 0 against H: Mi - H2 + 0 samples below: I 35 38 31 32 33 30 31 29 39 38 30 38 36 31 39 32 36 39 II 32 29 33 32 31 29 30 36 31 34 30 30 30 32 34 34 Variances: 0 = 3.9, 02 = 0.6. Use a = 0.05. (a) Test the hypothesis and find the P-value. Find the test statistic. Round your answers to four decimal places (e.g. 98.7654). ZO P-value = (b) Explain how this test could be conducted with a confidence interval. Round your answers to three decimal places (e.g. 98.765). <Hi - U23 We are 95% confident that value 1 exceeds value 2 by between (c) What is the power of the test for the true difference in means of 2? Round your answer to three decimal places (e.g. 98.765). The power of the test is 1-B =

Answer 1

FOR THIS HYPOTHESIS TEST WE CONDUCT Z-TEST FOR TWO MEANS , WITH POPULATION MEAN TO BE KNOWN.

The provided sample means are shown below: X = 34.278 X2 = 31.687 Also, the provided population standard deviations are: 01 = 3.9 09 = 0.6 and the sample sizes are ni 18 and no = 16. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho:μι = μ2 Ha : HY = 12 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 Based on the information provided the significance level is a = 0.5, and the critical value for a two- tailed test is ze = 0.67. The rejection region for this two-tailed test is R = {:: 1| > 0.67} (3) Test Statistics The Z-statistic is computed as follows: -X Voni +one 34.278 - 31.687 3.92/18 +0.62/16 2.782

(4) Decision about the null hypothesis Since it is observed that :| = 2.782 >= 0.67, it is then concluded that the null hypothesis is rejected Using the P-value approach: The p-value is p = 0.0054, and since p=0.0054 <0.5, it is concluded that the null hypothesis is rejected. Confidence interval The 50% confidence interval for Mi - H9 is 1.963 <H1 - H2 <3.219. Graphically p-value = Test 0.0054 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -4.0 -35 -3.0 -25 -2.0 -1.5 -10 -0.5 0.0 0.5 1.0 1.5 20 25 3.0 3.5 4.0

THE VALUE OF Z = 2.782

THE P-VALUE = 0.0054

THE 95% CONFIDENCE INTERVAL = 1.963 TO 3.219