Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim. Choose the correct option based on the problem information.

Question

A test of sobriety involves measuring the subject's motor skills. Twenty randomly selected sober subjects take the test and produce a mean score of 41.0 with a standard deviation of 3.7. At the 0.01 level of significance, test the claim that the true mean score for all sober subjects is equal to 35.0. Identify the correct null and alternative hypotheses.

A) H_0: μ < 35.0. H_1: μ ≥ 35.0.
B) H_0: μ = 35.0. H_1: μ > 35.0.
C) H_0: μ > 35.0. H_1: μ ≤ 35.0.
D) H_0: μ = 35.0. H_1: μ ≠ 35.0.

Answer 1

To define the correct null and alternative hypotheses based on the claim that the true mean score for all sober subjects is equal to 35.0, we need to reflect the nature of the claim in hypothesis testing.

The null hypothesis (\( H_0 \)) generally represents a statement of no effect or no difference, and it is what we test against. The alternative hypothesis (\( H_1 \)) represents what we are trying to provide evidence for.

In this case:

• The claim states that the true mean score (μ) is equal to 35.0, which we would consider our null hypothesis.
• The alternative hypothesis would encompass all other possibilities for the mean, indicating that it is not equal to 35.0.

So, we can define the hypotheses as follows:

• Null hypothesis (\( H_0 \)): \( μ = 35.0 \) (the mean score is equal to 35.0)
• Alternative hypothesis (\( H_1 \)): \( μ ≠ 35.0 \) (the mean score is not equal to 35.0)

This leads us to option D.

Thus, the correct choice is:

D) \( H_0: μ = 35.0 \). \( H_1: μ ≠ 35.0 \).