Question

In this problem, assume that the distribution of differences is approximately normal. Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more "conservative" answer. In the following data pairs, A represents the cost of living index for utilities and B represents the cost of living index for transportation. The data are paired by metropolitan areas in the United States. A random sample of 46 metropolitan areas gave the following information. A:  90  85  86 108  83 101  91 125 105 B: 101  91 104 102 109 111  93 116 115 A: 118 133 104  84  80  77  90  92  90 B: 106  95 110 112 105  93 119  99 109 A: 106  95 110 112 105  93 119  99 109 B:  96 109 103 107 103 102 101  86  94 A: 109 113  90 121 120  85  91  91  97 B:  88 100 104 119 116 104 121 108  86 A:  95 115  99  86  88 106  80 108  90  87 B: 100  83  88 103  94 125 115 100  96 127 Do the data indicate that the U.S. population mean cost of living index for utilities is less than that for transportation in these areas? Use α = 0.05. (Let d = A − B.) (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? H0: μd = 0; H1: μd > 0; right-tailed H0: μd = 0; H1: μd < 0; left-tailed H0: μd = 0; H1: μd ≠ 0; two-tailed H0: μd < 0; H1: μd = 0; left-tailed (b) What sampling distribution will you use? What assumptions are you making? The standard normal. We assume that d has an approximately normal distribution. The standard normal. We assume that d has an approximately uniform distribution. The Student's t. We assume that d has an approximately uniform distribution. The Student's t. We assume that d has an approximately normal distribution. What is the value of the sample test statistic? (Round your answer to three decimal places.) (c) Find (or estimate) the P-value. P-value > 0.250 0.125 < P-value < 0.250 0.050 < P-value < 0.125 0.025 < P-value < 0.050 0.005 < P-value < 0.025 P-value < 0.005 Sketch the sampling distribution and show the area corresponding to the P-value. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. (e) State your conclusion in the context of the application. Reject the null hypothesis, there is insufficient evidence to claim that the mean cost of living index for utilities is less than that for transportation. Fail to reject the null hypothesis, there is insufficient evidence to claim that the mean cost of living index for utilities is less than that for transportation. Reject the null hypothesis, there is sufficient evidence to claim that the mean cost of living index for utilities is less than that for transportation. Fail to reject the null hypothesis, there is sufficient evidence to claim that the mean cost of living index for utilities is less than that for transportation.

Answer 1

(a) What is the level of significance?

α = 0.05

State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

H0: μd = 0; H1: μd < 0; left-tailed

(b) What sampling distribution will you use? What assumptions are you making?

The Student's t. We assume that d has an approximately normal distribution.

What is the value of the sample test statistic? (Round your answer to three decimal places.)

-1.760

(c) Find (or estimate) the P-value.

0.050 < P-value < 0.125

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

(e) State your conclusion in the context of the application.

Reject the null hypothesis, there is sufficient evidence to claim that the mean cost of living index for utilities is less than that for transportation.

99.152	mean A
103.652	mean B
-4.500	mean difference (A - B)
17.345	std. dev.
2.557	std. error
46	n
45	df
-1.760	t
.0426	p-value (one-tailed, lower)