Question

In order to compare the means of two​ populations, independent random samples of 395 observations are selected from each​ population, with the results found in the table to the right. Complete parts a through e below.	Sample 1	Sample 2
	x1=5,294	x2=5,261
	s1 =153	s2=197

a. Use a​ 95% confidence interval to estimate the difference between the population means

left parenthesis mu 1 minus mu 2 right parenthesisμ1−μ2.

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 the difference between the population means falls in the confidence interval.

B. We are​ 95% confident that each of 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 outside of 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 α=0.05.

What is the test​ statistic?

zequals=_____

​(Round to two decimal places as​ needed.)

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

​p-value= _____

​(Round to three decimal places as​ needed.)

Interpret the results. Choose the correct answer below.

A.Reject Upper H 0H0. There is not sufficient evidence that the population means are different.

B.Do not reject Upper H 0H0. There is sufficient evidence that the population means are different.

C.Do not reject Upper H 0H0. There is not sufficient evidence that the population means are different.

D.Reject Upper H 0H0. There is sufficient evidence that the population means are different.

c. Suppose the test in part b was conducted with the alternative hypothesis Upper H Subscript a Baseline : left parenthesis mu 1 minus mu 2 right parenthesis greater than 0Ha: 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 as​ needed.)

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

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

d. Test the null hypothesis Upper H 0 : left parenthesis mu 1 minus mu 2 right parenthesis equals 26H0: μ1−μ2=26 versus Upper H Subscript a Baseline : left parenthesis mu 1 minus mu 2 right parenthesis not equals 26Ha: μ1−μ2≠26.

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 to two decimal places as​ needed.)

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

​p-value=

​(Round to three decimal places as​ needed.)

Interpret the results. Choose the correct answer below.

A.Do not reject Upper H 0H0. There is not sufficient evidence to conclude that (μ1−μ2) is not equal to 26.

B.Reject Upper H 0H0. There is sufficient evidence to conclude that (μ1−μ2) is not equal to 26.

C.Do not reject Upper H 0H0. There is sufficient evidence to conclude that (μ1−μ2) is not equal to 26.

D.Reject Upper H 0H0. There is not sufficient evidence to conclude that (μ1−μ2) is not equal to 26.

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 different. The test in part d supported the hypothesis that the difference is not 26.

B.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 26.

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 26.

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 26.

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 dependent random samples.

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

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

Since sample sizes are large so we can use z critical values instead of t critical values

(a)

Answers: (8.4, 57.6)

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

(b)

Correct option:

D.Reject Upper H 0H0. There is sufficient evidence that the population means are different.

(c)

Correct option:

B.The test statistic would be 2.63 and the null hypothesis would be rejected in favor of the new alternative hypothesis.

(d)

Correct option : A.Do not reject Upper H 0H0. There is not sufficient evidence to conclude that (μ1−μ2) is not equal to 26.

B.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 26.

(e)

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