Let's solve the problem step by step.

(a) Sample Proportion

The sample proportion ($\hat{p}$) is calculated as the number of people who said "no" divided by the total number of people surveyed.

$\hat{p} = \frac{1408}{2500}$

Now, let's compute this and convert it into a percentage:

$\hat{p} = \frac{1408}{2500} \times 100 = 56.32\%$

Rounding to the nearest whole number:

$\hat{p} = 56\%$

So, the sample proportion is 56%.

(b) Margin of Error with 80% Confidence Level

The formula for the margin of error (MOE) is given by:

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

Where:

• $z$ is the z-value corresponding to the desired confidence level,
• $\hat{p}$ is the sample proportion (in decimal form),
• $n$ is the sample size.

For an 80% confidence level, the z-value is approximately 1.28.

We know:

• $\hat{p} = 0.5632$,
• $n = 2500$.

Let’s compute the margin of error for 80% confidence.

$MOE = 1.28 \times \sqrt{\frac{0.5632 \times (1 - 0.5632)}{2500}}$

(c) Margin of Error with 95% Confidence Level

For a 95% confidence level, the z-value is approximately 1.96.

We will use the same formula but with the z-value for 95% confidence:

$MOE = 1.96 \times \sqrt{\frac{0.5632 \times (1 - 0.5632)}{2500}}$

I will now compute the MOE for both (b) and (c).### (b) Margin of Error with 80% Confidence The margin of error for an 80% confidence level is approximately 1.3%.

(c) Margin of Error with 95% Confidence

The margin of error for a 95% confidence level is approximately 1.9%.

(d) Interpretation

The answer in part (c) is greater than the answer in part (b). This is because a higher confidence level (95% vs. 80%) requires a larger margin of error to ensure a higher degree of accuracy.

To guarantee 95% accuracy, we must increase the error (margin of error) to account for more uncertainty in the data.

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Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. How does the sample size affect the margin of error?
2. What is the impact of the confidence level on the margin of error?
3. How would the MOE change if the sample proportion were closer to 50%?
4. Can you explain why the z-value differs for each confidence level?
5. What happens to the margin of error if the sample size increases?

Tip: A larger sample size reduces the margin of error, leading to more precise estimates.