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 66 59 Finland 68 67 Finland 62 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: μd = 0; H1: μd > 0; right-tailed H0: μd > 0; H1: μd = 0; right-tailed H0: μd = 0; H1: μd ≠ 0; two-tailed H0: μd = 0; H1: μd < 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 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 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 > 0.250 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 fail to reject the null hypothesis and conclude the data are not 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 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 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. 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.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
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
66
59
Finland
68
67
Finland
62
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: μd = 0; H1: μd > 0; right-tailed
H0: μd > 0; H1: μd = 0; right-tailed
H0: μd = 0; H1: μd ≠ 0; two-tailed
H0: μd = 0; H1: μd < 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
The standard normal. We assume that d has an approximately uniform distribution.
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.
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 fail to reject the null hypothesis and conclude the data are not 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 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 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.
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.
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