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Let's go through each part of the question step-by-step.

### Given Data
- Sample size (\( n \)) = 480
- Sample proportion (\( \hat{p} \)) = 0.5
- Confidence level = 90%

### Step-by-Step Solution

#### Part a: Is it safe to assume that \( n \leq 0.05 \) of all subjects in the population?

This question is asking whether the sample size is at most 5% of the population size. Generally, this assumption is valid when \( n \) is a small portion of the total population. If the population is much larger than 480 (e.g., at least 9,600), then we can select "Yes."

#### Part b: Verify \( n\hat{p}(1 - \hat{p}) \geq 10 \).

Let's calculate:
\[
n\hat{p}(1 - \hat{p}) = 480 \times 0.5 \times (1 - 0.5) = 480 \times 0.5 \times 0.5 = 120
\]

Since 120 is greater than 10, this condition is satisfied. So, we enter **120.0** in the box.

#### Confidence Interval: Constructing the 90% Confidence Interval

The formula for a confidence interval for a population proportion is:
\[
\hat{p} \pm Z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

1. **Determine \( Z \) for a 90% Confidence Level:**
   For a 90% confidence level, the critical \( Z \)-value is approximately 1.645.

2. **Calculate the Margin of Error (ME):**
   \[
   ME = 1.645 \cdot \sqrt{\frac{0.5 \times (1 - 0.5)}{480}} = 1.645 \cdot \sqrt{\frac{0.25}{480}}
   \]
   \[
   ME = 1.645 \cdot \sqrt{0.00052083} \approx 1.645 \cdot 0.0228 \approx 0.0375
   \]

3. **Calculate the Confidence Interval:**
   \[
   \hat{p} - ME = 0.5 - 0.0375 = 0.4625
   \]
   \[
   \hat{p} + ME = 0.5 + 0.0375 = 0.5375
   \]

So, the 90% confidence interval for the population proportion is:
\[
0.463 < p < 0.538
\]

### Summary of Answers
- **a.** Yes
- **b.** \( n\hat{p}(1 - \hat{p}) = 120.0 \)
- **Confidence Interval:** \( 0.463 < p < 0.538 \)

Would you like a detailed explanation of any specific part?

### Related Questions
1. What is the purpose of checking the condition \( n\hat{p}(1 - \hat{p}) \geq 10 \)?
2. How is the Z-value determined for different confidence levels?
3. Why is it important to have a large enough sample size for confidence intervals?
4. What does the margin of error represent in a confidence interval?
5. How does increasing the confidence level affect the confidence interval width?

### Tip
Always check conditions like sample size relative to population and \( n\hat{p}(1 - \hat{p}) \geq 10 \) to ensure the validity of a confidence interval for proportions.

If n = 480 and p̂ = 0.5, construct a 90% confidence interval about the population proportion. Round your answers to three decimal places.

Learn to construct a 90% confidence interval for the population proportion with a sample size of 480 and a sample proportion of 0.5. This tutorial covers important statistical concepts, including confidence intervals, margin of error, and sample size validation. Suitable for college-level statistics students.

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