Question

(a) Indicate whether each of the following is a two-tailed test, a left-tailed test, or a right-tailed test.

Is the hypothesis test H0: μ = 12, H1: μ < 12 a two-tailed test, a left-tailed test, or a right-tailed test?

Is the hypothesis test H0: μ ≤ 12, H1: μ > 12 a two-tailed test, a left-tailed test, or a right-tailed test?

Is the hypothesis test H0: μ = 12, H1: μ ≠ 12 a two-tailed test, a left-tailed test, or a right-tailed test?

(b) Indicate the structure of the rejection region for each of the following hypothesis tests.

For the hypothesis test H0: μ = 12, H1: μ < 12, what is the structure of the rejection region?

For the hypothesis test H0: μ ≤ 12, H1: μ > 12, what is the structure of the rejection region?

For the hypothesis test H0: μ = 12, H1: μ ≠ 12, what is the structure of the rejection region?

Answer 1

The hypothesis test H0: μ = 12, H1: μ < 12 is a left-tailed test, the test H0: μ ≤ 12, H1: μ > 12 is a right-tailed test, and the test H0: μ = 12, H1: μ ≠ 12 is a two-tailed test. The respective rejection regions for these tests are in the left tail, the right tail, and split evenly between both tails of the distribution.

Step-by-step explanation:

The type of hypothesis test (two-tailed, left-tailed, or right-tailed) depends on the alternative hypothesis (H1 or Ha).

• The hypothesis test H0: μ = 12, H1: μ < 12 is a left-tailed test because we are looking for values of μ that are less than 12, indicating a decrease.
• The hypothesis test H0: μ ≤ 12, H1: μ > 12 is a right-tailed test because we are looking for values of μ that are greater than 12, indicating an increase.
• The hypothesis test H0: μ = 12, H1: μ ≠ 12 is a two-tailed test since we are looking for values of μ that are either less than or greater than 12.

The rejection region for each of these hypothesis tests is based on where we would expect to find extreme values that would lead us to reject the null hypothesis, H0:

• For H0: μ = 12, H1: μ < 12, the rejection region is in the left tail of the distribution since we are concerned with values significantly lower than 12.
• For H0: μ ≤ 12, H1: μ > 12, the rejection region is in the right tail of the distribution because we are looking for values significantly higher than 12.
• For H0: μ = 12, H1: μ ≠ 12, the rejection region is split evenly between the lower and upper tails of the distribution because values significantly different from 12 in either direction would cause us to reject H0.