To solve this, we need to construct the 98% confidence interval for the population proportion $p$. Here's the process:

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

$\hat{p} = \frac{\text{Number of successes (students planning graduate degrees)}}{\text{Total number of students surveyed}} = \frac{658}{1010} \approx 0.651$

Step 2: Determine the critical value $z^*$ for a 98% confidence level

For a 98% confidence level, the critical value $z^*$ corresponds to the 2.5% in each tail of the standard normal distribution. From the standard normal table: $z^* \approx 2.33$

Step 3: Calculate the standard error (SE)

$SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.651(1 - 0.651)}{1010}} \approx \sqrt{\frac{0.651 \cdot 0.349}{1010}} \approx \sqrt{\frac{0.227199}{1010}} \approx 0.01499$

Step 4: Compute the margin of error (ME)

$ME = z^* \cdot SE = 2.33 \cdot 0.01499 \approx 0.0349$

Step 5: Construct the confidence interval

$\text{Confidence interval} = \hat{p} \pm ME = 0.651 \pm 0.0349$ $\text{Lower bound} = 0.651 - 0.0349 \approx 0.616$ $\text{Upper bound} = 0.651 + 0.0349 \approx 0.686$

Final Confidence Interval:

$(0.616, 0.686)$

Interpretation:

We are 98% confident that the proportion of college seniors planning to pursue a graduate degree is between 0.616 and 0.686.

Correct Answer:

C) (0.616, 0.686); we are 98% confident that the proportion of college seniors who are planning to pursue a graduate degree is between 0.616 and 0.686.

Would you like me to provide further clarifications or calculations?

---

Here are 5 related questions for exploration:

1. How does increasing the confidence level to 99% affect the width of the interval?
2. What would happen to the confidence interval if the sample size was increased to 1500 students?
3. How is the critical value $z^*$ determined for different confidence levels?
4. Why is it necessary to assume the sample is random when constructing a confidence interval?
5. How does the margin of error relate to the sample size and confidence level?

Tip: The larger your sample size, the smaller the margin of