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.

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: μ_d = 0; H₁: μ_d > 0; right-tailedH₀: μ_d = 0; H₁: μ_d < 0; left-tailed    H₀: μ_d > 0; H₁: μ_d = 0; right-tailedH₀: μ_d = 0; H₁: μ_d ≠ 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.

Answer 1

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.