For each combination of sample size and sample proportion, find the margin of error for the 95% confidence level. (a) n=100 and p-hat=.56 (b) n=400 and p-hat=.56

a. E= 1.96 * sqrt((.56*.44/100)) = .09729

b. E= 1.96 * sqrt((.56*.44/400)) = .0486

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To find the margin of error for a given sample size and sample proportion at a 95% confidence level, you can use the formula:

Margin of Error = Z * sqrt((p-hat * (1 - p-hat)) / n)

where:
- Z is the critical value associated with the desired confidence level. For a 95% confidence level, the critical value is approximately 1.96.
- p-hat is the sample proportion.
- n is the sample size.

Let's calculate the margin of error for each combination:

(a) n=100 and p-hat=.56:
Margin of Error = 1.96 * sqrt((0.56 * (1 - 0.56)) / 100)
= 1.96 * sqrt(0.2464 / 100)
= 1.96 * sqrt(0.002464)
≈ 1.96 * 0.0496
≈ 0.0972

Therefore, for (a), the margin of error is approximately 0.0972.

(b) n=400 and p-hat=.56:
Margin of Error = 1.96 * sqrt((0.56 * (1 - 0.56)) / 400)
= 1.96 * sqrt(0.2464 / 400)
= 1.96 * sqrt(0.000616)
≈ 1.96 * 0.0248
≈ 0.0485

Therefore, for (b), the margin of error is approximately 0.0485.