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 |
139 | 124 | 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 *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;
right-tailed*H*_{0}: *μ*_{d}
= 0; *H*_{1}: *μ*_{d} ≠ 0;
two-tailed

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

The standard normal. 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 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.

*P*-value > 0.2500.125 < *P*-value <
0.250 0.050 < *P*-value <
0.1250.025 < *P*-value < 0.0500.005 <
*P*-value < 0.025*P*-value < 0.005

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 reject the null hypothesis
and conclude the data are not 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 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.

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

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 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.

Answer #1

Ans:

January | April | d | |

1 | 139 | 108 | 31 |

2 | 124 | 113 | 11 |

3 | 126 | 102 | 24 |

4 | 64 | 88 | -24 |

5 | 78 | 61 | 17 |

d-bar | 11.8 | ||

sd | 21.371 |

a) level of significance=0.01

H0: μd = 0; H1: μd > 0; right-tailed

b)

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

Test statistic:

t=(11.8-0)/(21.371/SQRT(5))

t=1.235

c)df=5-1=4

p-value=P(t>1.235)=tdist(1.235,4,1)=0.1423

**0.125 < P-value < 0.250**

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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