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Let's go through the problem step-by-step.

### Problem Breakdown

1. **Given Information:**
   - Population proportion \( p = 0.37 \).
   - Sample size \( n = 500 \).

2. **Objective:**
   - To describe the sampling distribution of \( \hat{p} \) (the sample proportion).
   - To find the mean and standard deviation of the sampling distribution of \( \hat{p} \).

### Part (a) - Describe the Sampling Distribution of \( \hat{p} \)

For the sampling distribution of \( \hat{p} \) to be approximately normal, two conditions need to be met:

- **Condition 1:** The sample size \( n \) should be less than or equal to 5% of the population size, which is represented by \( n \leq 0.05N \).
- **Condition 2:** Both \( np \) and \( n(1 - p) \) must be at least 10.

#### Verification:
1. **Condition 1**: Since this is a survey of adults, the population \( N \) (number of adults in the country) is likely large enough that \( n = 500 \leq 0.05N \) holds true.

2. **Condition 2**: 
   - Calculate \( np \): 
     \[
     np = 500 \times 0.37 = 185
     \]
   - Calculate \( n(1 - p) \): 
     \[
     n(1 - p) = 500 \times (1 - 0.37) = 500 \times 0.63 = 315
     \]

   Since both values (185 and 315) are greater than 10, this condition is also satisfied.

Therefore, the sampling distribution of \( \hat{p} \) is approximately normal, which matches option **A**.

### Part (b) - Mean of the Sampling Distribution of \( \hat{p} \)

The mean of the sampling distribution of \( \hat{p} \), denoted by \( \mu_{\hat{p}} \), is equal to the population proportion \( p \):
\[
\mu_{\hat{p}} = p = 0.37
\]

This answer is already provided as **0.37**.

### Part (c) - Standard Deviation of the Sampling Distribution of \( \hat{p} \)

The standard deviation of the sampling distribution of \( \hat{p} \), denoted by \( \sigma_{\hat{p}} \), is calculated by the formula:
\[
\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}
\]

Substitute the values:
\[
\sigma_{\hat{p}} = \sqrt{\frac{0.37 \times 0.63}{500}} = \sqrt{\frac{0.2331}{500}}
\]

Calculate this value:
\[
\sigma_{\hat{p}} \approx \sqrt{0.0004662} \approx 0.0216
\]

So, the standard deviation of \( \hat{p} \) is approximately **0.022** when rounded to three decimal places.

---

### Summary of Answers

- **(a)** Option A: Approximately normal because \( n \leq 0.05N \) and \( np \geq 10 \) and \( n(1 - p) \geq 10 \).
- **Mean (b)**: \( 0.37 \)
- **Standard Deviation (c)**: \( 0.022 \)

---

Would you like more details or have any questions?

### Related Questions:

1. How would the sampling distribution change if the sample size were smaller?
2. Why is the condition \( n \leq 0.05N \) important for the sampling distribution?
3. What happens if \( np \) or \( n(1 - p) \) are less than 10?
4. How does increasing the sample size affect the standard deviation of \( \hat{p} \)?
5. What would the mean and standard deviation be if \( p \) changed to 0.5?

### Tip:
Always check if the sample size and population proportion meet normality conditions before assuming the sampling distribution is approximately normal.

According to a survey in a country, 37% of adults do not own a credit card. Suppose a simple random sample of 500 adults is obtained. Complete parts (a) through (d) below.

Explore the sampling distribution of the sample proportion of adults without credit cards based on a survey where 37% do not own one. This solution includes verification of normality conditions, calculations of the mean, and the standard deviation of the sample proportion. Suitable for students in Grades 10-12.

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