To solve this hypothesis testing problem, we need to follow several steps. Let's go through them one by one.

1. Determine the critical value

Since the population standard deviation is unknown and the sample size $n = 293$ is large, we will use the t-distribution to find the critical value.

Given:

• Significance level $\alpha = 0.01$
• The test is two-tailed.

For a two-tailed test at a significance level of $\alpha = 0.01$, we need to find the critical value $t_{\alpha/2}$ using the degrees of freedom $df = n - 1 = 293 - 1 = 292$.

Using a t-distribution table or statistical software, the critical value for $df = 292$ at $\alpha/2 = 0.005$ is approximately:

$\text{critical value} = \pm 2.580$

2. Calculate the test statistic

The test statistic for a one-sample t-test is calculated using the formula:

$t = \frac{M - \mu}{\frac{SD}{\sqrt{n}}}$

Where:

• $M = 59.6$ (sample mean)
• $\mu = 60.7$ (population mean under null hypothesis)
• $SD = 5.3$ (sample standard deviation)
• $n = 293$ (sample size)

Plugging in the values:

$t = \frac{59.6 - 60.7}{\frac{5.3}{\sqrt{293}}}$ $t = \frac{-1.1}{\frac{5.3}{17.1175}}$ $t = \frac{-1.1}{0.3097} \approx -3.552$

3. Compare the test statistic with the critical value

Now, we compare the calculated test statistic with the critical value:

• The test statistic $t = -3.552$
• The critical value $\pm 2.580$

Since $-3.552$ is less than $-2.580$, the test statistic falls in the critical region.

4. Decision

Because the test statistic is in the critical region, we reject the null hypothesis.

5. Conclusion

Since we reject the null hypothesis, the conclusion is:

"There is sufficient evidence to warrant rejection of the claim that the population mean is equal to 60.7."

Summary:

• Critical value: $\pm 2.580$
• Test statistic: $-3.552$
• Test statistic is in the critical region: Yes
• Decision: Reject the null hypothesis
• Conclusion: There is sufficient evidence to warrant rejection of the claim that the population mean is equal to 60.7.

Would you like more details, or do you have any questions about this? Here are 5 related questions that might help deepen your understanding:

1. What assumptions are necessary for the t-test to be valid in this context?
2. How would the conclusion change if the sample size were much smaller?
3. What is the difference between a one-tailed and two-tailed test?
4. How would you interpret the p-value associated with this test statistic?
5. What would be the impact on the test result if the significance level was changed to 0.05?

Tip: Always verify if your test statistic calculation matches the context of the hypothesis test (one-tailed vs. two-tailed).