Let's go step by step to find the sample proportion and margin of error (MOE) for each level of confidence:

Information given:

• Total number of people surveyed ($n$) = 600
• Number of people who responded "It is important" = 450

Sample proportion ($\hat{p}$):

The sample proportion is the fraction of people who responded "It is important."

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

So, the sample proportion is 75%.

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Step 1: Margin of Error (MOE) Formula

The margin of error (MOE) is calculated using the formula:

$MOE = z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

Where:

• $z$ is the z-score corresponding to the level of confidence.
• $\hat{p}$ is the sample proportion.
• $n$ is the sample size.

Step 2: Calculate the MOE for each confidence level.

(a) 90% level of confidence

• The z-score for 90% confidence is approximately 1.645.

$MOE = 1.645 \times \sqrt{\frac{0.75(1 - 0.75)}{600}}$ Let’s calculate this.

(b) 95% level of confidence

• The z-score for 95% confidence is approximately 1.96.

$MOE = 1.96 \times \sqrt{\frac{0.75(1 - 0.75)}{600}}$ Let’s calculate this as well.

I will now compute both margins of error.Here are the results:

(a) 90% level of confidence:

• Sample proportion: 75%
• Margin of Error (MOE): 2.9%

(b) 95% level of confidence:

• Sample proportion: 75%
• Margin of Error (MOE): 3.5%

Would you like more details or clarification on any of the steps?

Here are five related questions for further exploration:

1. How does increasing the sample size affect the margin of error?
2. What is the z-score for a 99% confidence level, and how does it affect the MOE?
3. How do confidence intervals relate to the margin of error?
4. Can you explain the relationship between the sample proportion and the width of the confidence interval?
5. What if the sample proportion had been 60% instead of 75%—how would the MOE change?

Tip: The margin of error decreases with a larger sample size or a lower confidence level, which means you can gain more precision either by surveying more people or accepting a slightly lower confidence level.