Question

In order to compare the means of two​ populations, independent random samples of 400 observations are selected from each​ population, with the results found in the table to the right. Complete parts a through e below.

Sample 1  

overbar x = 5,305

s1= 154

Sample 2

overbar x = 5,266

s2 = 199

a. Use a​ 95% confidence interval to estimate the difference between the population means (mu 1 - mu 2). Interpret the confidence interval.

The confidence interval is (__,__).

(Round to one decimal place as needed.)

Interpret the confidence interval. Select the correct answer below.

A. We are​ 95% confident that each of the population means falls outside of the confidence interval.

B. We are​ 95% confident that the difference between the population means falls outside of the confidence interval.

C. We are​ 95% confident that each of the population means is contained in the confidence interval.

D. We are​ 95% confident that the difference between the population means falls in the confidence interval.

b. 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. Use alphaα=0.05.

What is the test​ statistic?

z = __ (Round 2 decimal places)

What is the observed significance​ level, or​ p-value?

p-value = __ (Round 3 decimal places)

Interpret the results. Choose the correct answer below.

A. Do not reject H0. There is not sufficient evidence that the population means are different.

B.Do not reject H0. There is sufficient evidence that the population means are different.

C.Reject H0. There is sufficient evidence that the population means are different.

D.Reject H0. There is not sufficient evidence that the population means are different.

c. Suppose the test in part b was conducted with the alternative hypothesis Ha: μ1−μ2>0. How would your answer to part b​ change? Select the correct choice below and fill in the answer box within your choice. (Round to three decimal places)

A. The test statistic would be ___​, and the null hypothesis would not be rejected in favor of the new alternative hypothesis.

B. The observed significance​ level, or​ p-value, would be ___​, and the null hypothesis would be rejected in favor of the new alternative hypothesis.

C. The observed significance​ level, or​ p-value, would be ___, and the null hypothesis would not be rejected in favor of the new alternative hypothesis.

D. The test statistic would be ___, and the null hypothesis would be rejected in favor of the new alternative hypothesis.

d. Test the null hypothesis H0: μ1−μ2=27 versus Ha: μ1−μ2≠27. Give the significance level and interpret the result. Use α=0.05. Compare your answer to the test conducted in part b.

What is the test​ statistic?

z = ___ (round two decimal places)

What is the observed significance​ level, or​ p-value?

p-value = ___ (Round 3 decimal places)

Interpret the results. Choose the correct answer below.

A. Reject H0. There is sufficient evidence to conclude that (μ1−μ2) is not equal to 27.

B. Do not reject H0. There is sufficient evidence to conclude that (μ1−μ2) is not equal to 27.

C. Reject H0. There is not sufficient evidence to conclude that (μ1−μ2) is not equal to 27.

D. Do not reject H0. There is not sufficient evidence to conclude that (μ1−μ2) is not equal to 27.

Compare your answer to the test conducted in part b. Choose the correct answer below.

A. The test in part b supported the hypothesis that the means are not different. The test in part d supported the hypothesis that the difference is 27.

B.The test in part b supported the hypothesis that the means are different. The test in part d supported the hypothesis that the difference is not

27.

C.The test in part b supported the hypothesis that the means are not different. The test in part d supported the hypothesis that the difference is not 27.

D.The test in part b supported the hypothesis that the means are different. The test in part d supported the hypothesis that the difference is

27.

e. What assumptions are necessary to ensure the validity of the inferential procedures applied in parts a-d​?

A. One must assume that the two samples are small.

B. One must assume that the two samples are dependent random samples.

C. One must assume that the two samples are independent random samples.

D. One must assume that the​ z-distribution is approximately normal.

Answer 1

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