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.