Question

A random sample of 16 adult male wolves from the Canadian Northwest Territories gave an average weight x1 = 97 lb with estimated sample standard deviation s1 = 6.7 lb. Another sample of 22 adult male wolves from Alaska gave an average weight x2 = 88 lb with estimated sample standard deviation s2 = 7.1 lb. (a) Let 1 represent the population mean weight of adult male wolves from the Northwest Territories, and let 2 represent the population mean weight of adult male wolves from Alaska. Find a 75% confidence interval for 1 – 2. (Round your answers to one decimal place.) lower limit upper limit (b) Examine the confidence interval and explain what it means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the 75% level of confidence, does it appear that the average weight of adult male wolves from the Northwest Territories is greater than that of the Alaska wolves? State the null and alternate hypotheses. H0: 1 = 2; H1: 1 > 2H0: 1 = 2; H1: 1 < 2 H0: 1 ≠ 2; H1: 1 = 2H0: 1 = 2; H1: 1 ≠ 2 (ii) What sampling distribution will you use? What assumptions are you making? The standard normal. We assume that both population distributions are approximately normal with known standard deviations.The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations. The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.The Student's t. We assume that both population distributions are approximately normal with known standard deviations. What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate. (Test the difference 1 − 2. Round your answer to three decimal places.)

Answer 1

At the 75% level of confidence, there is strong evidence to suggest that the average weight of adult male wolves from the Northwest Territories is greater than that of the Alaska wolves.

Part (a): Confidence Interval for 1 - 2

We don't know the population variances, so we need to estimate a pooled standard deviation:

s_pooled = sqrt(((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2))

s_pooled = sqrt(((16 - 1) * 6.7^2 + (22 - 1) * 7.1^2) / (16 + 22 - 2))

s_pooled ≈ 6.9 lb

SE_diff = s_pooled * sqrt(1 / n1 + 1 / n2)

SE_diff ≈ 6.9 * sqrt(1 / 16 + 1 / 22)

SE_diff ≈ 1.7 lb

For a 75% confidence interval, the z-score is 1.44 (obtained from the inverse CDF of the standard normal distribution).

Margin of error = z * SE_diff

Margin of error ≈ 1.44 * 1.7 ≈ 2.4 lb

Lower limit = (x1 - x2) - margin of error

Lower limit = 97 - 88 - 2.4

Lower limit = 6.6 lb

Upper limit = (x1 - x2) + margin of error

Upper limit = 97 - 88 + 2.4

Upper limit = 10.4 lb

Therefore, the 75% confidence interval for 1 - 2 is (6.6, 10.4) lb.

Part (b): Interpretation and Hypothesis Testing

Interpretation:

The confidence interval indicates that, with 75% certainty, the true difference between the average weights of adult male wolves from the Northwest Territories and Alaska lies between 6.6 lb and 10.4 lb.

The interval does not contain zero, suggesting that the difference between the means is likely not zero.

However, the interval includes both positive and negative values, indicating that we cannot definitively conclude whether the wolves from the Northwest Territories are heavier or lighter than those from Alaska.

Null hypothesis (H0): The average weights of adult male wolves from the Northwest Territories and Alaska are equal (1 = 2).

Alternate hypothesis (H1): The average weights are different (1 ≠ 2).

Since we don't know the population variances, we should use the Student's t-distribution with degrees of freedom (df) = n1 + n2 - 2 = 36.

The test statistic is calculated similarly to the standard error of the difference:

t = (x1 - x2) / SE_diff

t = (97 - 88) / 1.7

t ≈ 5.3 lb

Conclusion:

The calculated t-value (5.3) is much larger than the critical t-value for a one-tailed test at the 75% confidence level (approximately 1.04).

This strongly rejects the null hypothesis, suggesting that the average weights of adult male wolves from the Northwest Territories and Alaska are indeed different.