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	82	55
Finland	66	65
Finland	64	69
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.)

(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?

H₀: μ_d > 0; H₁: μ_d = 0; right-tailedH₀: μ_d = 0; H₁: μ_d > 0; right-tailed    H₀: μ_d = 0; H₁: μ_d ≠ 0; two-tailedH₀: μ_d = 0; H₁: μ_d < 0; left-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 Student's t. We assume that d has an approximately normal distribution.    The standard normal. We assume that d has an approximately uniform distribution.The standard normal. 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 < 0.025P-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 sufficient evidence to claim that the average percentage of male wolves in winter is higher.Fail to 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.Reject the null hypothesis, there is insufficient evidence to claim that the average percentage of male wolves in winter is higher.

Answer 1

(a) The level of significance is 0.05. H:Ha=0.11:>0;right-tailed. The Student's t. We assume that d has an approximately normal distribution. EXCEL (Data Analysis) output: t-Test: Paired Two Sample for Means % Males (Winter) % Males (Summer) 59.625 54.625 151.6964286 70.55357143 Mean Variance Observations Pearson Correlation Hypothesized Mean Difference 0.476229719 df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail 1.271466938 0.122088434 1.894578604 0.244176869 2.364624251

The value of the sample test statistic is t=1.271. (c) 0.050<P-value <0.125.

P-Value=0.1222 test i=1.277 At the a = 0.05 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 that the average percentage of male wolves in winter is higher.