Demonstrate Inferences Using Two Samples

Demonstrate Inferences Using Two Samples Student’s Name Institutional Affiliation Demonstrate Inferences Using Two Samples Assignment 5 1 Independent random samples of 64 observations each is chosen from two normal populations with the following means and standard deviations: Population 1 Population 2 µ1 = 12 µ2 = 10 Ϭ1 = 4 Ϭ2 = 3 Let x1 and x2 denote the two sample means. 1 Give the mean and standard deviation of the sampling distribution of x1. The mean of the distribution of sample means is equal to the population mean (12). The standard deviation of the sampling distribution is Ϭ1 /sqrt of n, i.e., 4/sqrt of 64=0.5 2 Give the mean and standard deviation of the sampling distribution of x2. The mean of the distribution of sample means is equal to the population mean (10). The standard deviation of the sampling distribution is Ϭ2 /sqrt of n, i.e., 3/sqrt of 64=0.375 (Aleisa, Al-Ahmad, & Taha, 2011, p. 991). 3 Suppose you were to calculate the difference (x1 - x2) between the sample means. Find the mean and standard deviation of the sampling distribution of (x1 - x2). Mean of the difference is 1 and the standard deviation is 14.2127. 4 Will the statistic (x1 - x2) be normally distributed? Explain. No. This is because the sample size used in the difference is less than and not equal to 30. The larger the sample sizes, the closer to normal it is. 2 In order to compare the means of two populations, independent random samples of 400 observations are selected from each population, with the following results: Sample 1 Sample 2 x1= 5,275 x2 = 5,240 s1 = 150 s2 = 200 5 Use a 95% confidence interval to estimate the difference between the population means (µ1 - µ2). Interpret the confidence interval. The difference x1 -x2 is the unbiased estimator of the difference between the two population means µ1 - µ2 which is = 35. Because population variances are known, we use z value which gives 1.96 for a 95% confidence interval. Using the formula for computing CI difference between population means, the CI will be (10.5, 59.5). The 95% CI means we are 95% confident that the interval contains the true parameter value of µ1 - µ2 = 35. If this process was to be repeated 100 times, then on average 95 of those CI created will contain 35. 6 Test the null hypothesis H0: (µ1 - µ2) = 0 versus the alternative hypothesis Ha: (µ1 - µ2) ≠ 0. Give the significance level of the test, and interpret the result. This is a two-tailed test in z test and the z critical value for a (1-0.95) = 0.05 significance level is (+ or-) 1.96. Using the formula for a pooled variance, s2 will be 176.7766953. The computed z statistic is 35/12.5 = 2.8. The z statistic > z critical and also z statistic > - z critical. Reject the null hypothesis and conclude that the difference between the population means is not equal to 0 and that µ1 is more than µ2. 7 Suppose the test in part b was conducted with the alternative hypothesis Ha: (µ1 - µ2) > 0. How would your answer to part b change? This is a right-tailed test. The z critical value for a 0.05 significance level will be 1.645. The computed z statistic value is 2.8. The z statistic > z critical i.e. (2.8>1.645). Accept the alternative hypothesis and conclude that the mean population is more than 0 that is x1 is more than x2. In part (b) the conclusion is that the difference is not equal to 0 but in (c), it is more than 0. 8 Test the null hypothesis H0: (µ1 - µ2) = 25 versus the alternative hypothesis Ha: (µ1 - µ2) ≠ 25. Give the significance level and interpret the result. Compare your answer to the test conducted in part b. The level of significance is 1-0.95=0.05, and this is a two-tailed test so z critical will be 1.96. The pooled variance is 176.7766953, and thus the z statistic will be 25/12.5 = 2. The z statistic is > z critical, and also it is more than -z critical. Reject the null hypothesis and conclude that the difference in population means it is not equal to 25. 9 What assumptions are necessary to ensure the validity of the inferential procedures applied in parts a – d? Assumptions * Each value is sampled independently from each other value. This means each subject (say x1) provides only one value. * The populations are normally distributed. * The variance of the population is assumed to be equal or the same (assumption of homo...