The lifetime of a squirrel follows a normal distribution with mean μ = 40 months and a standard deviation $\sigma$ = 7 months.To test the hypothesis that μ 40, 25 squirrels are randomly selected.Assuming that μ = 45, and α = 0.1, the power is approximately
A)0.076
B)0.016
C)0.05
D)0.011
E)0.989

Question

The lifetime of a squirrel follows a normal distribution with mean μ = 40 months and a standard deviation $\sigma$ = 7 months.To test the hypothesis that μ > 40, 25 squirrels are randomly selected.Assuming that μ = 45, and α = 0.1, the power is approximately
A)0.076
B)0.016
C)0.05
D)0.011
E)0.989

Quizplus · Accepted Answer

The power of a test is the probability of correctly rejecting the null hypothesis when it is false. In this case, we are testing if the true mean lifetime of squirrels, μ, is greater than 40 months. Given μ = 45 for the alternative hypothesis, we want to find the probability of rejecting the null hypothesis (that μ ≤ 40) when the true mean is indeed 45.To calculate the power, we first determine the z-score corresponding to the significance level α = 0.1 for a one-tailed test. The z-score for α = 0.1 (90% confidence) is approximately 1.28 (since we are looking at the upper tail).Next, we calculate the z-score for the sample mean under the assumption that the true mean is 45. The formula for this z-score is:$$ z = \frac{\bar{x} - \mu_0}{(\sigma/\sqrt{n})} $$where $\bar{x}$ is the assumed true mean (45), $\mu_0$ is the mean under the null hypothesis (40), $\sigma$ is the standard deviation (7), and $n$ is the sample size (25).$$ z = \frac{45 - 40}{(7/\sqrt{25})} = \frac{5}{(7/5)} = \frac{5}{1.4} = 3.57 $$This z-score corresponds to a very high percentile in the normal distribution, indicating a high probability of rejecting the null hypothesis when the true mean is 45. Looking up this z-score in the standard normal distribution table or using a calculator, we find that the area to the left of z = 3.57 is very close to 1, meaning the power of the test is very high. The closest answer choice to this high probability is 0.989, indicating a very high power to correctly reject the null hypothesis when the true mean is indeed greater than 40.

The Lifetime of a Squirrel Follows a Normal Distribution with Mean