Question

8. listed below are the numbers of years that archbishops and monarchs in a certain country lived after their election or coronation. Treat the values as simple random samples from larger populations. Use a 0.05 significance level to test the claim that the mean longevity for archbishops is less than the mean for monarchs after coronation. Complete parts​ (a) and​ (b) below. All measurements are in years.

Archbishops	18	15	16	15	15	2	14	17	10	15	12	13
	15	14	15	13	17	18	10	17	11	0	23	13
Monarchs	18	19	13	13	17	15	20	16	15	14	17	16

a. Use a 0.05 significance​ level, and test the claim that the mean longevity for archbishops is less than the mean for monarchs after coronation.

What are the null and alternative hypotheses? Assume that population 1 consists of the longevity of archbishops and population 2 consists of the longevity of monarchs.

a. H0: µ1≠µ2

H1: µ1>µ2

b. H0: µ1=µ2

H1: µ1≠µ2

c. H0: µ1=µ2

H1: µ1<µ2

d.H0: µ1≤µ2

H1: µ1>µ2

The test statistic 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 sufficient evidence to support the claim that archbishops have lower mean longevity than monarchs.

B.Reject the null hypothesis. There is not sufficient evidence to support the claim that archbishops have lower mean longevity than monarchs.

C.Reject the null hypothesis. There is sufficient evidence to support the claim that archbishops have lower mean longevity than monarchs.

D.Fail to reject the null hypothesis. There is not sufficient evidence to support the claim that archbishops have lower mean longevity than monarchs.

b. Construct a confidence interval suitable for testing the claim that the mean longevity for archbishops is less than the mean for monarchs after coronation.

___<µ1-µ2<____

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

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

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

Answer 1

For Archbishops

= 13.667, s1 = 4.8245, n1 = 24

For Monarchs

= 16.083, s2 = 2.2344, n2 = 12

Option - C) H₀:

                   H₁:

The test statistic t = ()/sqrt(s1^2/n1 + s2^2/n2)

                             = (13.667 - 16.083)/sqrt((4.8245)^2/24 + (2.2344)^2/12)

                             = -2.05

df = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1 - 1) + (s2^2/n2)^2/(n2 - 1))

   = ((4.8245)^2/24 + (2.2344)^2/12)^2/(((4.8245)^2/24)^2/23 + ((2.2344)^2/12)^2/11)

   = 34

P-value = P(T < -2.05)

             = 0.024

Since the P-value is less than the significance level (0.024 < 0.05), so we should reject the null hypothesis.

Option - C) Reject the null hypothesis. There is sufficient evidence to support the claim that archbishops have lower mean longevity than monarchs.

b) At 95% confidence interval the critical value is t^* = 2.032

The 95% confidence interval for is

() +/- t^* * sqrt(s1^2/n1 + s2^2/n2)

= (13.667 - 16.083) +/- 2.032 * sqrt((4.8245)^2/24 + (2.2344)^2/12)

= -2.416 +/- 2.392

= -4.808, -0.024

Yes, because the confidence interval contains only negative values.