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 139 124 126 64 78 April 108 113 102 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-tailedH0: μd = 0; H1: μd < 0; left-tailed    H0: μd > 0; H1: μd = 0; right-tailedH0: μd = 0; H1: μd ≠ 0; two-tailed

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

The standard normal. We assume that d has an approximately uniform distribution.The standard normal. We assume that d has an approximately normal 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 > 0.2500.125 < P-value < 0.250    0.050 < P-value < 0.1250.025 < P-value < 0.0500.005 < P-value < 0.025P-value < 0.005

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 reject the null hypothesis and conclude the data are not 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 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.

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

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.    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.

Ans:

 January April d 1 139 108 31 2 124 113 11 3 126 102 24 4 64 88 -24 5 78 61 17 d-bar 11.8 sd 21.371

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:

t=(11.8-0)/(21.371/SQRT(5))

t=1.235

c)df=5-1=4

p-value=P(t>1.235)=tdist(1.235,4,1)=0.1423

0.125 < P-value < 0.250

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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