Question

Find a 95 percent confidence interval for the difference between means, where n1 = 50, n2 = 36, X1 = 80, X2 = 75, sı2 = 5, and s22 = 3. Assume unequal variances. Essay Toolbar navigation B 1 U s 를 들 를 А

Answer 1

Solution:

We need to construct the 95% confidence interval for the difference between the population means 11 - Ho, for the case that the population standard deviations are not known. The following information has been provided about each of the samples: 80 2.236 50 Sample Mean 1 (X1) = Sample Standard Deviation 1 (si) = Sample Size 1 (N.) = Sample Mean 2 (X) = Sample Standard Deviation 2 (52) = Sample Size 2 (N2) = 75 1.732 36 Based on the informationa provided, we assume that the population variances are unequal, so then the number of degrees of freedom is computed as follows: (sini + si df si na) 2 = 85.582 ( 82) 72 +1 7, +1 The critical value for a = 0.05 and df = 85.582 degrees of freedom is te = t1-a/2:n–1 = 1.988. The corresponding confidence interval is computed as shown below: Since we assume that the population variances are unequal, the standard error is computed as follows: se = si n2 ni = 2.2362 50 1.7322 36 = 0.428 Now, we finally compute the confidence interval: CI (#1 - X2 - tex se, X-X2 + te X se) (80 - 75 - 1.988 x 0.428, 80 - 75 +1.988 x 0.428) (4.149,5.851) Therefore, based on the data provided, the 95% confidence interval for the difference between the population means 11 - 12 is 4.149<H1 - 12 <5.851, which indicates that we are 95% confident that the true difference between population means is contained by the interval (4.149,5.851).

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The provided sample means are shown below. X1 = 15.4 X2 = 14.5 Also, the provided sample standard deviations are: Si = 2 89 = 2.28 and the sample sizes are ni = 35 and ng = 18. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: Mi se Ha: Li > H2 This corresponds to a right-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used (2) Rejection Region Based on the information provided, the significance level is a = 0.01, and the degrees of freedom are df = 30.712. In fact, the degrees of freedom are computed as follows, assuming that the population variances are unequal: Hence, it is found that the critical value for this right-tailed test is te = 2.454, for a = 0.01 and df = 30.712 The rejection region for this right-tailed test is R= {t:t> 2.454) (3) Test Statistics Since it is assumed that the population variances are unequal, the t-statistic is computed as follows: X X2 t= V+ 15.4 - 14.5 = 1.418 V 35 + (4) Decision about the null hypothesis Since it is observed that t = 1.418 <te = 2.454, it is then concluded that the null hypothesis is not rejected. Using the P-value approach: The p-value is p = 0.0832, and since p = 0.0832 > 0.01, it is concluded that the null hypothesis is not rejected. (5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean Mi is greater than P2, at the 0.01 significance level.