https://file.aiquickdraw.com/mathbot/file/17302965607220l3ylrz5.jpg

To solve this problem, let’s go through each part step-by-step.

### Given Data:
- Population proportion (\( p \)) of students who took out federal loans = 68% or 0.68.
- Sample size (\( n \)) = 50.

### Part (a): Mean of the Distribution of Sample Proportions
The mean of the sampling distribution of sample proportions is equal to the population proportion \( p \). Therefore:

\[
p = 0.68
\]

### Part (b): Shape of the Distribution of Sample Proportions
To determine the shape of the distribution of sample proportions, we use the Central Limit Theorem (CLT) for proportions. According to the CLT, if the sample size \( n \) is large enough, the sampling distribution of the sample proportion will be approximately normal.

For a sample proportion to be normally distributed, the following conditions should be met:
1. \( n \cdot p \geq 10 \)
2. \( n \cdot (1 - p) \geq 10 \)

Let’s check these conditions:

\[
n \cdot p = 50 \cdot 0.68 = 34 \quad (\text{which is greater than 10})
\]
\[
n \cdot (1 - p) = 50 \cdot (1 - 0.68) = 50 \cdot 0.32 = 16 \quad (\text{which is also greater than 10})
\]

Since both conditions are satisfied, the distribution of sample proportions can be approximated as normal.

### Answers:
- **(a)** The mean of the distribution of sample proportions \( p = 0.68 \).
- **(b)** The sampling distribution is **normally distributed**.

Would you like further details or have questions?

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Here are five related questions for further exploration:
1. How does the sample size affect the accuracy of the sampling distribution?
2. What happens if the sample size is less than 30 in such problems?
3. How would this change if the population proportion were 0.5?
4. What is the standard error of this sampling distribution?
5. How does the Central Limit Theorem apply to sampling distributions?

**Tip:** For any sample proportion, the closer the sample size gets to the population, the more accurate the sampling distribution becomes!

According to the National Center for Education Statistics conducted by the U.S. Department of Education in 2020, among 2015-16 bachelor's degree recipients, 68% took out federal loans. If we randomly sample 50 students from this population, at a time:

(a) What will be the mean of the distribution of sample proportions?

(b) What is the shape of the distribution of sample proportions?

Learn how to calculate the mean and determine the shape of the sampling distribution of sample proportions, based on federal loan statistics for 2015-16 bachelor’s degree recipients. Understand how the Central Limit Theorem applies to sample proportions with a population proportion of 68% and a sample size of 50.

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