Question

A study was done on skull sizes of humans during different time periods. The results are shown in the table. Assume that the two samples are independent simple random samples selected from normally distributed​ populations, and do not assume that the population standard deviations are equal. Complete parts​ (a) and​ (b) below.

	4000 B.C.	A.D 150
u	u1	u2
n	25	25
x	131.44 mm	135.18 mm
s	5.03	5.03

a. Use a 0.01 significance​ level, and test the claim that the mean skull breadth in 4000 B.C is less than the mean skull breadth in A.D 150.

What are the null and alternative​ hypotheses?

The test​ statistic, t, is ____ ​(Round to two decimal places as​ needed.)

The​ P-value is ______ ​(Round to three decimal places as​ needed.)

State the conclusion for the test.

A. Fail to reject the null hypothesis. There is not sufficient evidence to support the claim that the mean skull breadth was less in 4000 B.C. than A.D. 150.

B.Rejectthe null hypothesis. There is not sufficient evidence to support the claim that the mean skull breadth was less in 4000 B.C. than A.D. 150.

C. Fail to reject the null hypothesis. There is sufficient evidence to support the claim that the mean skull breadth was less in 4000 B.C. than A.D. 150.

D. Reject the null hypothesis. There is sufficient evidence to support the claim that the mean skull breadth was less in 4000 B.C. than A.D. 150.

b. Construct a confidence interval suitable for testing the claim that the mean skull breadth in 4000 B.C is less than the mean skull breadth in A.D 150.

___ < u₁-u₂ < ____

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

Does the confidence interval support the conclusion of the​ test?

(Yes or No), because the confidence interval contains (only positive values or only negative values or zero).

Answer 1

a) (1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​

Ha: μ1​ < μ2​

This corresponds to a left-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

t= -2.63

DEGREES OF FREEDOM= 48

P value using table we have

p value=0.006

Since P value SMALLER than the 0.01 level of significance therefore SIGNIFICANT

Decision: REJECT NULL HYPOTHESIS H0.

CONCLUSION: Reject the null hypothesis.There is sufficient evidence to support the claim that the mean skull breadth was less in 4000 B.C. than A.D. 150.

Solution B: Standard error of mean= 1.42

t critical= 2.68

Margin of error= 3.8056

Difference of means= -3.74

So the 99% confidence interval is −7.556<μ1​−μ2​<0.076.

No, the confidence interval does Not support the conclusion of the​ test.because the confidence interval contains zero.