To find the margin of error (ME) for each level of confidence, we need to follow these steps:

Step 1: Compute the Sample Proportion

The sample proportion is given by:

$p = \frac{450}{600} = 0.75$

Step 2: Identify the Critical Value $z^*$

For different confidence levels, the critical value $z^*$ corresponds to the z-score that captures the confidence interval. Here are the standard $z^*$ values:

• For an 80% confidence level, $z^* \approx 1.28$
• For a 99% confidence level, $z^* \approx 2.576$

Step 3: Calculate the Margin of Error (ME)

The formula for margin of error is:

$ME = z^* \times \sqrt{\frac{p(1 - p)}{n}}$

Where:

• $p$ is the sample proportion.
• $n = 600$ is the sample size.
• $z^*$ is the critical value for the given confidence level.

(a) For 80% confidence level:

1. Sample Proportion: $p = 0.75$

2. Critical value: $z^* = 1.28$

3. Margin of Error: $ME = 1.28 \times \sqrt{\frac{0.75 \times 0.25}{600}}$ Let’s calculate this:

\approx 1.28 \times \sqrt{0.0003125} \approx 1.28 \times 0.01768 \approx 0.0226 \text{ or } 2.3\%$$ Thus, the margin of error for 80% confidence is approximately $$\pm 2.3\%$$. ### (b) For 99% confidence level: 1. **Sample Proportion:** $$p = 0.75$$ 2. **Critical value:** $$z^* = 2.576$$ 3. **Margin of Error:** $$ME = 2.576 \times \sqrt{\frac{0.75 \times 0.25}{600}}$$ Let’s calculate this: $$ME \approx 2.576 \times \sqrt{\frac{0.1875}{600}} \approx 2.576 \times \sqrt{0.0003125} \approx 2.576 \times 0.01768 \approx 0.0455 \text{ or } 4.6\%$$ Thus, the margin of error for 99% confidence is approximately $$\pm 4.6\%$$. ### Final Answers: - **(a) 80% confidence level:** - Margin of error: $$\pm 2.3\%$$ - **(b) 99% confidence level:** - Margin of error: $$\pm 4.6\%$$ --- Would you like further details on the calculations or an explanation of confidence intervals? Here are 5 related questions: 1. How do critical values differ for various confidence levels? 2. What is the relationship between sample size and margin of error? 3. How can we reduce the margin of error in surveys? 4. What happens to the margin of error as the sample proportion moves closer to 0 or 1? 5. How is the margin of error affected if we increase the confidence level? **Tip:** Larger sample sizes generally reduce the margin of error, making the results more precise.