Question

Select the true statement(s).

A. In general, you reject the null hypothesis when the population standard deviation σ is known, and the test statistic has a large magnitude.
B. In a hypothesis test for a population mean with a known population standard deviation σ, the probabilities of type I error (α) and type II error (β) always sum to 1, i.e., α + β = 1.
C. The probability of a type II error in a hypothesis test for a population mean, where the population standard deviation σ is known, does not depend on the true value of the population mean.
D. There are only three possible alternative hypotheses in a hypothesis test for a population mean when the population standard deviation is known.
E. The hypothesis test for a population mean with a known population standard deviation can be used exclusively when the underlying population is normal.
F. The conclusion in a hypothesis test for a population mean with a known population standard deviation depends on whether the test statistic lies in the rejection region.

Answer 1

In hypothesis testing with known population standard deviation, you reject the null hypothesis based on the test statistic's critical region. The sum of type I and type II error probabilities does not equal 1, and the probability of a type II error does depend on the actual population mean. The hypothesis test can be used even when the population is not normal if the sample size is sufficiently large.

Step-by-step explanation:

When evaluating the true statements regarding hypothesis tests for a population mean with known population standard deviation \(\sigma\), we need to consider the principles of statistical hypothesis testing. First, consider the following response to the student's query:

• A: This statement is typically true. You reject the null hypothesis if the test statistic falls within the critical region, which usually corresponds to large magnitudes of the test statistic when the population standard deviation is known.
• B: This statement is false. The sum of the probabilities of type I error (\(\alpha\)) and type II error (\(\beta\)) does not necessarily equal 1. These are separate probabilities for different types of errors.
• C: This statement is false. The probability of a type II error (\(\beta\)) does depend on the true value of the population mean because it is related to the power of the test, which varies with the actual mean.
• D: This statement is true. There are indeed only three alternative hypotheses in a hypothesis test for a population mean: a not equal to hypothesis (two-tailed), a greater than hypothesis (right-tailed), and a less than hypothesis (left-tailed).
• E: This statement is false. While the normal distribution assumption makes the hypothesis test more robust, the Central Limit Theorem assures that for large sample sizes the sampling distribution of the mean will be approximately normal, even if the population is not.
• F: This statement is true. In hypothesis testing, the decision to reject or not reject the null hypothesis is based upon whether the calculated test statistic falls within the critical region (rejection region) or not.

The subject of this question is hypothesis testing under the branch of statistics, which is a topic within the study of Mathematics.