In this problem, assume that the distribution of differences is approximately no

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.

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?

(b) What sampling distribution will you use? What assumptions are you making?

The Student's t. We assume that d has an approximately normal distribution.The standard normal. We assume that d has an approximately normal distribution.    The standard normal. We assume that d has an approximately uniform distribution.The Student's t. 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. (Round your answer to four decimal places.)


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

(e) State your conclusion in the context of the application.

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.    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.

Weather Station 1 2 3 4 5 January 130 122 125 64 78 April 111 101 99 88 61

Explanation / Answer

Given that,
population mean(u)=125
sample mean, x =64
standard deviation, s =78
number (n)=
null, H0: Ud = 0
alternate, H1: Ud < 0
level of significance, = 0.01
from standard normal table,right tailed t /2 =3.747
since our test is right-tailed
reject Ho, if to > 3.747
we use Test Statistic
to= d/ (S/n)
Where
Value of S^2 = [ di^2 – ( di )^2 / n ] / ( n-1 ) )
d = ( Xi-Yi)/n) = 11.8
We have d = 11.8
Pooled variance = Calculate value of Sd= S^2 = Sqrt [ 2343-(59^2/5 ] / 4 = 20.29
to = d/ (S/n) = 1.3
Critical Value
The Value of |t | with n-1 = 4 d.f is 3.747
We got |t o| = 1.3 & |t | =3.747
Make Decision
Hence Value of |to | < | t | and Here we Do not Reject Ho
p-value :right tail - Ha : ( p > 1.3004 ) = 0.13166
hence value of p0.01 < 0.13166,here we do not reject Ho

ANSWERS
---------------
null, H0: Ud = 0
alternate, H1: Ud < 0
test statistic: 1.3
critical value: reject Ho, if to > 3.747
decision: Do not Reject Ho
p-value: 0.13166

H0: d = 0; H1: d > 0; right-tailed
The standard normal. We assume that d has an approximately normal distribution.

At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
Fail to reject the null hypothesis, there is insufficient evidence to claim average peak wind gusts are higher in January

X Y X-Y (X-Y)^2 130 111 19 361 122 101 21 441 125 99 26 676 64 88 -24 576 78 61 17 289 59 2343

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