Question

 

The data for a random sample of 8 paired observations are shown in the table to the right.

a. What are the appropriate null and alternative hypotheses to test whether the mean for population 2 is larger than that for population​ 1?

b. Conduct the test identified in part a using alpha equals 0.01

c. Find a 99​% confidence interval for mu Subscript d. Interpret this result.

d. What assumptions are necessary to ensure the validity of this analysis?

			Pair 1 Population 1 Population 2 1 20 22 2 56 63 3 51 55 4 15 20 5 34 36 6 56 60 7 45 45 8 46 47
			What are the hypotheses for mu Subscript d equals​(mu 1 - mu 2)? A.H0​: mu Subscript d < 0 Ha​: mu Subscript d = 0 B. H0​:mu Subscript d = 0 Ha​:mu Subscript d > C.H0​: mu Subscript d not equals 0 Ha​:mu Subscript d = 0 D.H0​: mu Subscript d >0 Ha​:mu Subscript d =0 E.H0​: mu Subscript d =0 Ha​:mu Subscript d not equals 0 F.H0​: mu Subscript d =0 Ha​:mu Subscript d < 0 b. Identify the rejection region for testing the hypotheses from part a. _______ Select the correct choice below and fill in the answer box to complete your choice. ​(Round to three decimal places as​ needed.) A. t < ______ B. t > ______ C. |t| > _____ Calculate the test statistic. t = ______ ​(Round to three decimal places as​ needed.) Give the appropriate conclusion for the test. Reject H0. Since the test statistic is in the rejection​ region, there is sufficient evidence to conclude that the mean for population 2 is greater than that for population 1. c. The confidence interval for mu Subscript d is (___ , ___) (Round to two decimal places as​ needed.) Interpret this result. The interval __________0, which _________evidence that the mean for population 2 is greater than that for population 1. d. What conditions are required for the test results to be​ valid? Select all that apply. A.The variances of the two populations are approximately the same. B.The populations being sampled are approximately normal. C.The population of differences is approximately normal. D.No assumptions are required.

Answer 1

= (-2 + (-7) + (-4) + (-5) + (-2) + (-4) + 0 + (-1))/8 = -3.125

s_d = sqrt(((-2 + 3.125)^2 + (-7 + 3.125)^2 + (-4 + 3.125)^2 + (-5 + 3.125)^2 + (-2 + 3.125)^2 + (-4 + 3.125)^2 + (0 + 3.125)^2 + (-1 + 3.125)^2)/7) = 2.2952

a) Option - F) H₀: = 0

                      H₁: < 0

b) At alpha = 0.01, the critical value is t_{0.01, 7} = -2.998

Reject H₀, if t < -2.998

Option - A) t < -2.998

The test statistic t = ( - D)/(s_d/)

                            = (-3.125 - 0)/(2.2952/)

                            = -3.851

Reject H₀. Since the test statistic is in the rejection region, there is sufficient evidence to conclude that the mean for population 2 is greater than that for population 1.

c) At 99% confidence interval the critical value is t^* = 3.5

The 99% confidence interval for is

+/- t^* * s_d/

= -3.125 +/- 3.5 * 2.2952/

= -3.125 +/- 2.84

= -5.965, -0.285

= -5.97, -0.29

The interval doesn't contain 0, which gives sufficient evidence that the mean for population 2 is greater than that for population 1.

d) Option - B) The populations being sampled are approximately normal.