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Let's analyze the problem step-by-step.

### Part (a): Describe the Sampling Distribution
The question states:
- **37%** of adults do not own a credit card, so **\( p = 0.37 \)**.
- A **sample of 500 adults** is taken, so **\( n = 500 \)**.

We need to determine whether the sampling distribution of \( \hat{p} \) (the sample proportion) is approximately normal.

To verify if the sampling distribution of \( \hat{p} \) is approximately normal, we check two conditions:
1. **Large sample condition**: \( n \geq 0.05N \) (where \( N \) is the population size, typically large).
2. **Success-failure condition**: Both \( np \geq 10 \) and \( n(1 - p) \geq 10 \).

Calculating:
- **\( np = 500 \times 0.37 = 185 \)** (which is greater than 10).
- **\( n(1 - p) = 500 \times (1 - 0.37) = 500 \times 0.63 = 315 \)** (also greater than 10).

Both conditions are satisfied, so **the sampling distribution of \( \hat{p} \) is approximately normal**.

Thus, **Option A** is correct: "Approximately normal because \( n \geq 0.05N \) and \( np(1 - p) \geq 10 \)."

---

### Part (b): Determine the Mean of the Sampling Distribution of \( \hat{p} \)
The mean of the sampling distribution of \( \hat{p} \) is given by:
\[
\mu_{\hat{p}} = p
\]

Since \( p = 0.37 \), we have:
\[
\mu_{\hat{p}} = 0.37
\]

So, **the mean of the sampling distribution of \( \hat{p} \) is 0.37**.

---

Would you like further explanation on these calculations or details on how sampling distributions work?

### Related Questions
1. How is the standard deviation of the sampling distribution of \( \hat{p} \) calculated?
2. What is the significance of the success-failure condition in determining normality?
3. How does sample size affect the shape of the sampling distribution?
4. Can the sampling distribution of \( \hat{p} \) ever be exactly normal?
5. How would the results change if \( p \) were closer to 0 or 1?

### Tip
When verifying the normality of a sampling distribution of a proportion, always check both the sample size and the success-failure conditions to ensure accurate results.

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. Describe the sampling distribution of p̂, the sample proportion of adults who do not own a credit card, and determine the mean of the sampling distribution of p̂.

Explore how to determine the sampling distribution and mean for the proportion of adults without credit cards in a sample of 500, given that 37% of adults in the population do not own one. This example applies normal approximation principles and uses the Central Limit Theorem.

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