A high school guidance counselor has a pamphlet that says that 35% of all high school students go to a community college after graduation. In a survey of 275 randomly selected high school seniors, 116 replied that they planned to go to a community college in the fall. Use a 95% confidence interval to test and see if the pamphlet needs updating.
Based on this sample, use your calculator to construct a 95% confidence interval for the proportion of all high school seniors who plan to go to a community college in fall. Round your answers to the nearest tenth of a percent.

To determine if the pamphlet needs updating, we can construct a 95% confidence interval for the proportion of high school seniors planning to attend a community college. Here's the step-by-step process:

1. **Calculate the sample proportion (\(\hat{p}\)):**
   \[
   \hat{p} = \frac{\text{number of students planning to go to community college}}{\text{total number of students surveyed}}
   \]
   \[
   \hat{p} = \frac{116}{275} \approx 0.4218
   \]

2. **Calculate the standard error (SE) for the proportion:**
   \[
   SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
   \]
   \[
   SE = \sqrt{\frac{0.4218 \times (1 - 0.4218)}{275}} \approx 0.0299
   \]

3. **Determine the critical value (z) for a 95% confidence interval:**
   The critical value for a 95% confidence interval is approximately 1.96.

4. **Calculate the margin of error (ME):**
   \[
   ME = z \times SE
   \]
   \[
   ME = 1.96 \times 0.0299 \approx 0.0586
   \]

5. **Construct the confidence interval:**
   \[
   \hat{p} \pm ME
   \]
   \[
   0.4218 \pm 0.0586
   \]
   This gives us:
   \[
   \left(0.4218 - 0.0586, 0.4218 + 0.0586\right) \approx \left(0.3632, 0.4804\right)
   \]

6. **Convert the confidence interval to percentages:**
   \[
   (0.3632 \times 100\%, 0.4804 \times 100\%) \approx (36.3\%, 48.0\%)
   \]

### Conclusion:
The 95% confidence interval for the proportion of high school seniors planning to attend a community college is approximately \(36.3\%\) to \(48.0\%\). Since the pamphlet states that 35% of students go to a community college, and this value is not within the confidence interval, there is evidence to suggest that the pamphlet may need updating.

Would you like more details or have any questions?

Here are 5 questions you might consider next:
1. How do you interpret the results of a confidence interval?
2. What steps can be taken if the confidence interval does not align with the stated proportion?
3. How do you calculate a sample size needed for a desired margin of error?
4. What are the assumptions underlying the use of confidence intervals?
5. How can confidence intervals be used in other fields or scenarios?

**Tip:** Always ensure the sample size is sufficiently large to provide an accurate estimate when constructing confidence intervals.

Learn how to construct a 95% confidence interval for proportions in statistics using sample data. This problem involves calculating sample proportions, standard errors, and margins of error, and interpreting confidence intervals. Suitable for high school seniors studying statistics.

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