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.

Weather
Station |
1 |
2 |
3 |
4 |
5 |

January |
140 | 138 | 122 | 64 | 78 |

April |
96 | 114 | 113 | 88 | 61 |

Does this information indicate that the peak wind gusts are
higher in January than in April? Use *α* = 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?

*H*_{0}: *μ*_{d} = 0;
*H*_{1}: *μ*_{d} > 0;
right-tailed

*H*_{0}: *μ*_{d} = 0;
*H*_{1}: *μ*_{d} < 0;
left-tailed

*H*_{0}: *μ*_{d} = 0;
*H*_{1}: *μ*_{d} ≠ 0;
two-tailed

*H*_{0}: *μ*_{d} > 0;
*H*_{1}: *μ*_{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 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 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. (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 *α*?

At the *α* = 0.01 level, we fail to reject the null
hypothesis and conclude the data are 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.

At the *α* = 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.

Fail to 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.

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.

Answer #1

The statistical software output for this problem is:

Hence,

a) Level of significance = 0.01

*H*_{0}: *μ*_{d} = 0;
*H*_{1}: *μ*_{d} > 0;
right-tailed

b) The Student's *t*. We assume that *d* has an
approximately normal distribution.

Test statistic = 1.258

c) P - value = 0.1385

d) At the *α* = 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.

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.
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approximately normal. Note: For degrees of freedom
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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.
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