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To solve this question, we need to determine the mean of the sampling distribution of \(\hat{p}\), the sample proportion of students who live on campus.

### Solution:
In a sampling distribution of a sample proportion:
- The mean of the sampling distribution of \(\hat{p}\) is simply the population proportion, \( p \).

Given:
- The population proportion \( p = 0.71 \) (71% of all students live on campus).

So, the mean of the sampling distribution of \(\hat{p}\) is:
\[
\text{Mean of } \hat{p} = p = 0.71
\]

**Answer:** The correct choice is **0.71**.

Would you like a detailed explanation of why the sample size (241) does not affect the mean in this case?

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Here are some related questions to expand your understanding:

1. What is the standard deviation of the sampling distribution of \(\hat{p}\)?
2. How would the mean change if the population proportion changed to 0.75?
3. Why does the sample size not affect the mean of the sampling distribution?
4. What would happen to the sampling distribution if we increased the sample size to 500?
5. How can we interpret a sample proportion that is different from the population proportion?

**Tip:** In sampling distributions, the sample size affects the spread (standard deviation) but not the mean.

At a large university, 71% of all students live on campus. Suppose a random sample of 241 students is selected from this university and the sample proportion is defined as the proportion of students in the sample who live on campus. Determine the mean of the sampling distribution of \( \hat{p} \).

Understand how to determine the mean of the sampling distribution of a sample proportion in statistics. This example uses a population proportion of 71% of students living on campus with a random sample size of 241. Learn why the sample size does not affect the mean of the distribution in this case.

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