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To solve this hypothesis testing problem, we need to follow the steps for a one-tailed test of the population mean.

### Given Data:
- Population mean (μ) under the null hypothesis (\( H_0 \)): 7.2 hours
- Sample mean (\( \bar{x} \)): 7.9 hours
- Sample standard deviation (s): 1.4 hours
- Sample size (n): 15
- Significance level (α): 0.01

### Step-by-Step Solution:

1. **State the Hypotheses:**
   - Null Hypothesis (\( H_0 \)): \( \mu = 7.2 \)
   - Alternative Hypothesis (\( H_1 \)): \( \mu > 7.2 \) (claim)

2. **Determine the Test Statistic:**
   We use the t-test statistic for the sample mean because the population standard deviation is not known.

   The formula for the t-test statistic is:
   \[
   t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
   \]
   
   Substituting the given values:
   \[
   t = \frac{7.9 - 7.2}{1.4 / \sqrt{15}}
   \]

3. **Calculate the Test Statistic:**
   \[
   t = \frac{0.7}{1.4 / \sqrt{15}} = \frac{0.7}{0.361}
   \]
   \[
   t \approx 1.94
   \]

4. **Find the Critical Value:**
   For a one-tailed test with \( \alpha = 0.01 \) and degrees of freedom \( df = n - 1 = 15 - 1 = 14 \), we look up the critical t-value from the t-distribution table.

   The critical t-value for \( df = 14 \) and \( \alpha = 0.01 \) is approximately 2.624.

5. **Make the Decision:**
   - If the calculated t-value is greater than the critical t-value, we reject the null hypothesis.
   - Here, \( t \approx 1.94 \) is less than 2.624.

   Thus, we do not reject the null hypothesis \( H_0 \).

### Conclusion:
At the 0.01 significance level, there is not enough evidence to conclude that college students sleep more than 7.2 hours on average.

Would you like further details or have any questions?

Here are some more questions you can ask:

1. Can you explain how to find the critical t-value from the t-distribution table?
2. How would the test change if we used a two-tailed test instead?
3. Can you show the steps for calculating the p-value for this test?
4. How do we interpret the p-value in the context of hypothesis testing?
5. What assumptions must be met for the t-test to be valid?

**Tip:** Always check the assumptions of the test you are performing to ensure the validity of your results.

Learn how to solve a one-tailed hypothesis testing problem for the population mean with a detailed step-by-step solution. This problem involves calculating the t-test statistic using sample data, determining the critical t-value, and making a decision based on the significance level. Suitable for undergraduate students studying statistics.

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