Question

4. (15 points) Suppose that a population follows a normal distribution with unknown mean μ and known standard deviation σ=2, and we want to test the null hypothesis that H 0​ :μ=10 versus the alternative that H a​ :μ>10 at significance level .05. (a - 5 pts) Based on a sample of size 25 , for what values of xˉ will H 0​ be rejected? (That is, what is the rejection region for this test?) (b - 5 pts) Suppose that the true value of the population mean is μ=10 and we collect a random sample of size 25 from the population. What is the probability that we will reject H 0​ if we perform a significance test of H 0​:μ=10 versus H a​ :μ>10 ? (c - 5pts ) Suppose that the true value of the population mean is μ=11 and we collect a random sample of size 25 from the population. What is the power of the significance test of H 0​ :μ=10 versus H a​ :μ>10 under this alternative? That is, what is the probability that we will reject H 0 under this alternative?)

Answer 1

For the given hypothesis test and significance level, the rejection region is X> 11.658. If the true mean is 10, the probability of a Type I error is 0.05. The power of the test when the true mean is 11 is approximately 0.9938.

Step-by-step explanation:

To determine the rejection region for a hypothesis test where the population follows a normal distribution with a known standard deviation (σ = 2) and unknown mean (μ), you calculate using the significance level of the test and the distribution of the test statistic under the null hypothesis.

(a) For a sample size of 25 and a significance level of .05, we will reject the null hypothesis H0: μ=10 if the sample meanX lies in the right tail of the distribution beyond the critical value that corresponds to this significance level in a standard normal distribution. Using a Z-table or statistical software, we can find that the critical Z-value is 1.645.

Since the standard error (SE) is σ/√n = 2/5, we multiply the critical Z-value by the SE and add it to the hypothesized mean to find the critical sample mean: 10 + 1.645*(2/5). This gives us a critical sample mean of 11.658. Therefore, the rejection region is X> 11.658.

(b) If the true value of μ is 10, the probability of rejecting H0 when it is true (a Type I error) is equal to the significance level of the test, which is 0.05.

(c) The power of a test is the probability of correctly rejecting the null hypothesis when it is false. If the true mean is μ = 11, we first find the Z-score of the sample mean from this distribution: (11 - 10) / (2/5) = 5 / 2 = 2.5.

The power is then the probability that a standard normal random variable will exceed this Z-score, which we can find using statistical tables or software as 1 - P(Z < 2.5), which is approximately 0.9938. Therefore, the power of the test under the alternative μ = 11 is roughly 0.9938.