Question

# In this problem, assume that the distribution of differences is approximately normal. Note: For degrees of...

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.

At five weather stations on Trail Ridge Road in Rocky Mountain National Park, the peak wind gusts (in miles per hour) for January and April are recorded below.

 Weather Station 1 2 3 4 5 January 140 138 122 64 78 April 96 114 113 88 61

Does this information indicate that the peak wind gusts are higher in January than in April? Use α = 0.01. (Let d = January − April.)

(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; right-tailed

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

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

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 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. (Round your answer to four decimal places.)

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 = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

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

Fail to reject the null hypothesis, there is sufficient evidence to claim average peak wind gusts are higher in January.

Fail to reject the null hypothesis, there is insufficient evidence to claim average peak wind gusts are higher in January.

Reject the null hypothesis, there is sufficient evidence to claim average peak wind gusts are higher in January.

Reject the null hypothesis, there is insufficient evidence to claim average peak wind gusts are higher in January.

The statistical software output for this problem is:

Hence,

a) Level of significance = 0.01

H0: μd = 0; H1: μd > 0; right-tailed

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

Test statistic = 1.258

c) P - value = 0.1385

d) At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

e) Fail to reject the null hypothesis, there is insufficient evidence to claim average peak wind gusts are higher in January.

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