(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 gusts are higher in January than in April or not.This is Matched pairs t-test. JanuaryApril13910812011312610264887861(a). The level of significance is = 0.01.Null Hypothesis:The peak wind gusts are same in January as in April.Alternative Hypothesis:The peak wind gusts are higher in January than in April.Rejection Rule:If p-value (=0.01), then reject the null hypothesis.(b). The Student’s t. We assume that d has an approximately normal distribution.EXCEL (Data Analysis) Procedure:Step 1: In EXCEL sheet, enter January in one column and enter April in one column.Step 2: In Data, select Data Analysis and choose t test: Paired Two-samples for Mean.Step 3: In Input Variable 1 Range: select January.Step 4: In Input Variable 2 Range: select April.Step 5: Select Lables. Enter Alpha as 0.01. Click Ok.EXCEL (Data Analysis) output: t-Test: Paired Two Sample for MeansJanuaryAprilMean105.494.4Variance1057.8436.3Observations55Pearson Correlation0.75999Hypothesized Mean Difference0df4t Stat1.144964P(T<=t) one-tail0.158038t Critical one-tail3.746947P(T<=t) two-tail0.316076t Critical two-tail4.604095The value of the sample test statistic is t = 1.145. (c). p-value = 0.1580The estimate of the p-value is 0.125 < P-value < 0.250.Since p-value (=0.1580) > (=0.01), then fail to reject .(d). Decision:At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.(e). conclusion:Fail to reject the null hypothesis, there is insufficient evidence to claim average peak wind gusts are higher in January.(a). 0.01(b). The Student’s t. We assume that d has an approximately normal distribution.The value of the sample test statistic is t = 1.145.(c). 0.125 < P-value < 0.250.(d). At the = 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.