To state the null and alternative hypotheses for the problem, let's carefully break it down:

Problem Interpretation:

• Course 1 is "Mountain Mania" and Course 2 is "Raging River."
• The goal is to test if the mean time for Course 1 (Mountain Mania) is less than the mean time for Course 2 (Raging River).
• Keith wants to test if the mean time for Course 1 is less than Course 2, which means we are looking at a one-tailed hypothesis test (testing if Course 1 is less than Course 2).

Define the difference as $\mu_1 - \mu_2$ (Course 1 minus Course 2):

• $\mu_1$ is the population mean for Course 1.
• $\mu_2$ is the population mean for Course 2.

Hypothesis Setup:

• Null Hypothesis (H₀): There is no difference between the means of Course 1 and Course 2. Mathematically, this means $\mu_1 = \mu_2$.
• Alternative Hypothesis (Hₐ): The mean time to complete Course 1 is less than the mean time to complete Course 2. Mathematically, this means $\mu_1 < \mu_2$.

The correct answer is:

(A) H₀: $\mu_1 = \mu_2$, Hₐ: $\mu_1 < \mu_2$.

Explanation:

• The null hypothesis assumes no difference between the two course times, while the alternative hypothesis suggests that Course 1 takes less time than Course 2.

Would you like further clarification on hypothesis testing or other related concepts?

Here are some follow-up questions:

1. What is the significance of using a one-tailed hypothesis test in this case?
2. How would you calculate the test statistic for this hypothesis test?
3. What is the meaning of a p-value in hypothesis testing?
4. How do you determine the critical value for a hypothesis test?
5. What assumptions need to be checked before conducting this test?

Tip: Make sure to check the assumptions of normality and independence before applying statistical tests like t-tests to ensure the validity of your results.