To find the critical value for this hypothesis test, we need to use the following details:

1. Significance level $\alpha = 0.01$.
2. Hypotheses:
   • Null hypothesis: $H_0: \mu = 58$
   • Alternative hypothesis: $H_1: \mu < 58$
3. This is a left-tailed test because we are testing if the mean is less than the nationwide average.

For a left-tailed test with $\alpha = 0.01$, we will use the z-distribution because the population standard deviation $\sigma = 13$ is known.

Step-by-Step Solution

1. Determine the critical z-value:
   • For a left-tailed test at a $0.01$ significance level, we find the z-score that has a cumulative probability of $0.01$.
   • Looking up the z-score for $0.01$ in a z-table or using statistical software, we get:

$z_{\alpha} = -2.326$

Answer

The critical value for this test is: $-2.326$

Would you like to continue with further steps or have any questions?

5 Relative Questions to Consider:

1. How do we find the test statistic for this hypothesis test?
2. How do we interpret the results if the test statistic falls below the critical value?
3. How does changing the significance level $\alpha$ affect the critical value?
4. Why is the z-distribution appropriate in this scenario instead of the t-distribution?
5. How would we approach this test if the sample size were much smaller?

Tip: Always check if the sample size is large or the population standard deviation is known, as these factors determine whether to use a z-test or t-test.