Question

# In this problem, assume that the distribution of differences is approximately normal. Note: For degrees of...

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?

H0: μd = 0; H1: μd < 0; left-tailed

H0: μd = 0; H1: μd ≠ 0; two-tailed

H0: μd > 0; H1: μd = 0; right-tailed

H0: μd = 0; H1: μd > 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.

 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)

H0: μd = 0; H1: μd > 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.

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