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 smalier. 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 January 37 122128GA78 108 115 100l 88 61 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 altemate hypotheses. Will you use a left-tailed, right tailed, or two-talled test? O Ho' μο" 0; H1: μ. 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 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 standard normal, We assume that d has an approximately uniform distribution. What is the value of the z- or t-statistic? (Round your answer to three decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. -2 (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 a? 0 At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. O At the α-001 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. 0 At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α-001 level, we reject the null hypothesis and conclude the data are statistically significant. (e) State your conclusion in the context of the application. O Fail to reject the null hypothesis, there is insufficient evidence to claim average peak wind gusts are higher in January. O Fail to reject the null hypothesis, there is sufficient evidence to claim average peak wind gusts are higher in January. O 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.

Answer 1

paired test

Sample #1	Sample #2	difference , Di =sample1-sample2	(Di - Dbar)²
137	108	29	309.76
122	115	7	19.36
128	100	28	275.56
64	88	-24	1253.16
78	61	17	31.36

	sample 1	sample 2	Di	(Di - Dbar)²
sum =	529	472	57	1889.200

mean of difference ,    D̅ =ΣDi / n =   11.400
std dev of difference , Sd =    √ [ (Di-Dbar)²/(n-1) =    21.7325

a)

Level of Significance ,    α =    0.01

Ho :   µd=   0
Ha :   µd >   0
right tailed

b) the student's t .we assume that d has normal distribution

mean of difference ,    D̅ =ΣDi / n =   11.400                  
                          
std dev of difference , Sd =    √ [ (Di-Dbar)²/(n-1) =    21.7325                  
                          
std error , SE = Sd / √n =    21.7325   / √   5   =   9.7191      
                          
t-statistic = (D̅ - µd)/SE = (   11.4000   -   0   ) /    9.7191   =   1.173

graph 1 is correct graph

d)

Degree of freedom, DF=   n - 1 =    4  
  
p-value =        0.1529   [excel function: =t.dist.rt(t-stat,df) ]
Conclusion:     p-value>α , Do not reject null hypothesis      

answer: option b)

e)

option a)