The image contains a statistical problem that involves evaluating assumptions, naming procedures, and calculating intervals for a proportion. Here's a step-by-step breakdown of the task:

1. Evaluate Assumptions

The assumptions to check include:

• $n \hat{p} \geq 10$
• $n (1 - \hat{p}) \geq 10$
• $N \geq 10 \cdot n$

From the problem:

• $n = 3350$, $\hat{p} = \frac{463}{3350} \approx 0.1382$
• Calculate $n \hat{p}$ and $n (1 - \hat{p})$:
  • $n \hat{p} = 3350 \cdot 0.1382 \approx 463$ (passes the condition since $\geq 10$).
  • $n (1 - \hat{p}) = 3350 \cdot (1 - 0.1382) \approx 3350 \cdot 0.8618 \approx 2887.03$ (passes the condition since $\geq 10$).

The third assumption:

• $N \geq 10 \cdot n$:
  • $N = 1,000,000$, $10 \cdot 3350 = 33,500$.
  • $N \geq 33,500$ (passes the condition).

2. Name the Procedure

Based on the conditions:

• Use the 1-Proportion Z-Interval, as the conditions for the normal approximation are met.

3. Point Estimate

The point estimate $\hat{p}$ is: $\hat{p} = \frac{x}{n} = \frac{463}{3350} \approx 0.1382.$

4. Confidence Interval

To calculate the confidence interval: $\hat{p} \pm z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}},$ where $z$ depends on the confidence level.

• For a 95% confidence level, $z = 1.96$.
• Plug values: = 1.96 \cdot \sqrt{\frac{0.1382 \cdot 0.8618}{3350}} = 1.96 \cdot \sqrt{\frac{0.11903}{3350}} = 1.96 \cdot \sqrt{0.00003554} = 1.96 \cdot 0.00596 \approx 0.0117.$$
• Confidence interval: $0.1382 \pm 0.0117 \implies (0.1265, 0.1499).$

Summary of Results

• Point estimate: $\hat{p} = \frac{463}{3350} \approx 0.1382$.
• Confidence interval: $(0.126, 0.150)$.

Would you like further details or calculations?

Related Questions:

1. What are the specific interpretations of the assumptions $n \hat{p}$ and $n (1 - \hat{p})$?
2. How is the $z$-value determined for different confidence levels?
3. What is the significance of a confidence interval in hypothesis testing?
4. How does the sample size $n$ affect the width of the confidence interval?
5. What would happen if the assumptions for the normal approximation weren’t satisfied?

Tip: Always verify your assumptions before applying statistical methods; failing to do so may lead to invalid conclusions.