To construct the 95% confidence interval for the proportion of students enrolled in college or a trade school within 12 months of graduating high school in 2017, follow these steps:

Step 1: Calculate the sample proportion $\hat{p}$

The sample proportion $\hat{p}$ is given by:

$\hat{p} = \frac{x}{n}$ where:

• $x = 94$ (number of students enrolled in college or trade school),
• $n = 154$ (total number of students in the sample).

$\hat{p} = \frac{94}{154} = 0.610$

Step 2: Determine the standard error (SE)

The standard error for the proportion is calculated using the formula:

$SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

$SE = \sqrt{\frac{0.610(1 - 0.610)}{154}} = \sqrt{\frac{0.610 \times 0.390}{154}} = \sqrt{\frac{0.2379}{154}} = \sqrt{0.001545} = 0.0393$

Step 3: Find the critical value for a 95% confidence level

For a 95% confidence level, the critical value $z_{\alpha/2}$ is 1.96 (from the standard normal distribution table).

Step 4: Calculate the margin of error (ME)

The margin of error is given by:

$ME = z_{\alpha/2} \times SE = 1.96 \times 0.0393 = 0.077$

Step 5: Construct the confidence interval

The confidence interval is calculated as:

$\hat{p} \pm ME = 0.610 \pm 0.077$

Thus, the 95% confidence interval is:

$(0.610 - 0.077, 0.610 + 0.077) = (0.533, 0.687)$

Final Answer:

• The lower limit of the confidence interval is 0.533.
• The upper limit of the confidence interval is 0.687.

Would you like further details or clarifications on the calculation steps?

Here are 5 related questions to consider:

1. How would a larger sample size affect the confidence interval?
2. What would happen to the confidence interval if we used a 99% confidence level?
3. How is the critical value for the confidence level determined?
4. How do you interpret the confidence interval in terms of real-world significance?
5. How does the 2017 sample compare to the 2015 proportion of 63.8%?

Tip: A smaller confidence interval (with a narrower range) indicates more precision, but it requires a larger sample size or lower confidence level.