Let's go through the solution step-by-step.

Part 1: State the Hypotheses

The problem states that an economist claims that less than 13% of U.S. adults have a great deal of confidence in banks. This means we are testing whether the population proportion $p$ is less than 0.13.

• Null Hypothesis ($H_0$): $p = 0.13$
• Alternative Hypothesis ($H_1$): $p < 0.13$

This is a left-tailed test since we are testing if the proportion is less than the hypothesized value.

Part 2: Find the Critical Values

For a left-tailed test, we need to find the critical $z$-values for both significance levels, $\alpha = 0.10$ and $\alpha = 0.01$.

1. For $\alpha = 0.10$, the critical value $z_{\alpha}$ corresponds to the 10th percentile of the standard normal distribution.
2. For $\alpha = 0.01$, the critical value $z_{\alpha}$ corresponds to the 1st percentile of the standard normal distribution.

Using the standard normal table:

• For $\alpha = 0.10$: $z_{0.10} \approx -1.282$
• For $\alpha = 0.01$: $z_{0.01} \approx -2.326$

Part 3: Compute the Test Statistic

1. Sample Proportion $\hat{p}$: $\hat{p} = \frac{149}{1358} \approx 0.1097$

2. Standard Error $SE$: $SE = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.13 \times (1 - 0.13)}{1358}} \approx 0.0094$

3. Test Statistic $z$: $z = \frac{\hat{p} - p}{SE} = \frac{0.1097 - 0.13}{0.0094} \approx -2.16$

Now that we have the test statistic $z \approx -2.16$, we can compare it to our critical values.

Would you like me to proceed with the conclusions for both significance levels?

Follow-up Questions

1. How do you interpret the results of a hypothesis test?
2. What does a left-tailed test signify in this context?
3. Why is it essential to specify the significance level in hypothesis testing?
4. How do we interpret the critical value in a hypothesis test?
5. What steps would we take if the test were two-tailed?

Tip

Always check if the assumptions of the hypothesis test are met, such as sample size and randomness.