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 probability of type II error is approximately
A)0.076
B)0.016
C)0.05
D)0.011
E)0.983

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 probability of type II error is approximately
A)0.076
B)0.016
C)0.05
D)0.011
E)0.983

Quizplus · Accepted Answer

To calculate the probability of a Type II error (β), we first need to find the critical value for the test at α = 0.1 for a one-tailed test. Since the hypothesis is μ > 40, we're looking at the upper tail. The z-score for α = 0.1 in a one-tailed test is approximately 1.28 (from z-tables or calculators, knowing that the upper 10% of the normal distribution corresponds to this z-score).Given μ0 = 40, μ1 = 45 (the true mean under the alternative hypothesis), σ = 7, and n = 25, we calculate the z-score that corresponds to μ0 under the assumption that μ1 is true. The formula for the test statistic under the alternative hypothesis is:$$ z = \frac{\bar{x} - \mu_1}{\sigma/\sqrt{n}} $$We want to find the z-score that corresponds to the critical value at μ0. The critical value at μ0 = 40 is the mean under H0, so we calculate the z-score for x = μ0 using μ1:$$ z = \frac{40 - 45}{7/\sqrt{25}} = \frac{-5}{7/5} = \frac{-5}{1.4} = -3.57 $$This z-score is the distance from the mean under H1 (45) to the critical value under H0 (40) in terms of standard error units. To find β, we look up the probability of observing a z-score less than -3.57 in a standard normal distribution, which corresponds to observing a sample mean that does not lead us to reject H0 when H1 is true.Looking up -3.57 in z-tables or using a calculator gives a very small probability, but since we're interested in the probability of not rejecting H0 (which is the upper tail in this context, given the direction of the test), we need to find the probability to the right of the z-score corresponding to the critical value under H1. The correct answer, based on standard normal distribution calculations, is closest to option D) 0.011, which represents a very small probability of making a Type II error under these specific conditions.

The Lifetime of a Squirrel Follows a Normal Distribution with Mean