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 135 122 128 64 78 April 108 115 102 88 61 Does this information indicate that the peak wind gusts are higher in January than in April? Use ? = 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: ?d = 0; H1: ?d < 0; left-tailedH0: ?d > 0; H1: ?d = 0; right-tailed H0: ?d = 0; H1: ?d > 0; right-tailedH0: ?d = 0; H1: ?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 normal distribution.The Student's t. We assume that d has an approximately normal distribution. The Student's t. We assume that d has an approximately uniform distribution.The standard normal. We assume that d has an approximately uniform 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

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Description: 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-val

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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	135	122	128	64	78
April	108	115	102	88	61

Does this information indicate that the peak wind gusts are higher in January than in April? Use ? = 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: ?_d = 0; H₁: ?_d < 0; left-tailedH₀: ?_d > 0; H₁: ?_d = 0; right-tailed H₀: ?_d = 0; H₁: ?_d > 0; right-tailedH₀: ?_d = 0; H₁: ?_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 normal distribution.The Student's t. We assume that d has an approximately normal distribution. The Student's t. We assume that d has an approximately uniform distribution.The standard normal. We assume that d has an approximately uniform 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 ??

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

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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