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.

In environmental studies, sex ratios are of great importance. Wolf
society, packs, and ecology have been studied extensively at
different locations in the U.S. and foreign countries. Sex ratios
for eight study sites in northern Europe are shown below.

Location of Wolf
Pack |
% Males
(Winter) |
% Males
(Summer) |

Finland | 60 | 63 |

Finland | 62 | 51 |

Finland | 64 | 63 |

Lapland | 55 | 48 |

Lapland | 64 | 55 |

Russia | 50 | 50 |

Russia | 41 | 50 |

Russia | 55 | 45 |

It is hypothesized that in winter, "loner" males (not present in
summer packs) join the pack to increase survival rate. Use a 5%
level of significance to test the claim that the average percentage
of males in a wolf pack is higher in winter. (Let *d* =
winter − summer.)

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;
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

*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 standard normal. We assume that *d* has an
approximately uniform distribution.

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 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.

*P*-value > 0.250

0.125 < *P*-value <
0.250

0.050 < *P*-value < 0.125

0.025 < *P*-value < 0.050

0.005 < *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 = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we fail to 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 insufficient evidence to claim that the average percentage of male wolves in winter is higher.

Fail to reject the null hypothesis, there is sufficient evidence to claim that the average percentage of male wolves in winter is higher.

Reject the null hypothesis, there is insufficient evidence to claim that the average percentage of male wolves in winter is higher.

Reject the null hypothesis, there is sufficient evidence to claim that the average percentage of male wolves in winter is higher.

Answer #1

0.000 | hypothesized value | |

56.375 | mean % Males (Winter) | |

53.125 | mean % Males (Summer) | |

3.250 | mean difference (% Males (Winter) - % Males (Summer)) | |

7.146 | std. dev. | |

2.527 | std. error | |

8 | n | |

7 | df | |

1.286 | t | |

.1196 | p-value (one-tailed, upper) | |

-2.725 | confidence interval 95.% lower | |

9.225 | confidence interval 95.% upper | |

5.975 | margin of error |

a)

*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.

Sample test statistic= 3.250

c)

0.125 < *P*-value < 0.250

d)

At the α = 0.05 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 that the average percentage of male wolves in winter is higher.

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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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
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the closest d.f. that is smaller. In
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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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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
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