(Same question just could not fit all into one photo) ### Understanding P-Value and Hypothesis Testing Through Sampling DistributionsThis diagram illustrates different sampling distributions with shaded areas representing P-values in hypothesis testing. Each graph is a normal distribution curve ranging from -4 to 4 on the x-axis.1. **Graph (a)**: The shaded area is on the left tail, within the range around the mean of the distribution, representing a significance level or P-value area.2. **Graph (b)**: The shaded area is on both tails (left and right), showing a two-tailed test, where significance is being tested on extremes of both ends of the distribution.3. **Graph (c)**: Similar to graph (a), the shaded area is again on the left tail but covers a slightly larger area, indicating a larger significance region.4. **Graph (d)**: The shaded area is on the right tail, contrasting graph (a), marking significance on the right extreme of the distribution.### Hypothesis Testing Outcomes(d) Conclusion options based on test results:- \[\] 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 statistically significant.- \[\] At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.(e) Application context conclusions:- \[\] 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 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.This section provides foundational insights into evaluating statistical significance using hypothesis tests at a 0.01 significance level, applied to real-world data, such as wind gusts in January. In this problem, we assume 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 | 120 | 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 \( \alpha = 0.01 \). (Let \( d = \text{January} - \text{April} \).)#### (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? - \( H_0: \mu_d = 0 \); \( H_1: \mu_d \neq 0 \); two-tailed - \( H_0: \mu_d = 0 \); \( H_1: \mu_d > 0 \); right-tailed - \( H_0: \mu_d = 0 \); \( H_1: \mu_d < 0 \); left-tailed - \( H_0: \mu_d > 0 \); \( H_1: \mu_d \leq 0 \); right-tailed *(Selected option: \( H_0: \mu_d = 0 \); \( H_1: \mu_d > 0 \); 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 Student’s t. We assume that \( d \) has an approximately normal distribution.-

There are two dependent samples which are January and April.We have to test whether the peak wind…

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. 1 2 3 4 5 139 120 126 64 78 108 113 102 88 61 Weather Station January April LUSE SALT Does this information indicate that the peak wind gusts are higher in January than in April? Use a = 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? Ho: H = 0; H₁: Hg 0; two-tailed O Ho: Hd = 0; H₁: H > 0; right-tailed O Hoi Hd=0; H₁: Hg <0; left-tailed O Hoi Ha>0; H₁: H = 0; right-tailed (b) What sampling distribution will you use? What assumptions are you making? O The Student's t. We assume that d has an approximately uniform distribution. O The Student's t. We assume that d has an approximately normal distribution. O The standard normal. We assume that d has an approximately uniform distribution. O The standard normal. 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. O P-value > 0.250 O 0.125 < P-value < 0.250 O 0.050 < P-value < 0.125 O 0.025 < P-value < 0.050 0.005

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P