Question

A confidence interval was used to estimate the proportion of math majors that are female. A random sample of 72 math majors generated the following confidence interval: (0.438, 0.642). Using the information above, what size sample would be necessary if we wanted to estimate the true proportion to within 4% using 99% reliability?

Answer 1

To estimate the true proportion of female math majors within a 4% margin of error using 99% confidence, a sample size of 476 is needed. This is calculated using the midpoint of the provided confidence interval and the Z-score for 99% confidence.

Step-by-step explanation:

To determine the necessary sample size to estimate the true proportion of female math majors with a 4% margin of error and 99% reliability, we use the formula for the sample size of a proportion:

n = (Z² * p * (1-p)) / E²

• Z is the Z-score corresponding to the desired confidence level (for 99%, Z-score ≈ 2.576)
• p is the estimated proportion (we can use the midpoint of the given confidence interval: (0.438 + 0.642)/2)
• E is the desired margin of error (in this case, 4% or 0.04).

Step 1: Calculate the midpoint of the confidence interval to estimate p:

p = (0.438 + 0.642) / 2 = 0.540

Step 2: Calculate the sample size using the formula:

n = (2.576² * 0.540 * (1 - 0.540)) / 0.04²

Step 3: Perform the calculations:

n = (6.635776 * 0.540 * 0.460) / 0.0016

n = (1.654 * 0.460) / 0.0016

n = 0.76084 / 0.0016 ≈ 475.525

Since we cannot have a fraction of a person, we round up to the next whole number.

Therefore, a sample size of 476 would be necessary to estimate the true proportion of female math majors within a 4% margin of error at 99% confidence.