We guess, based on historical data, that 30% of graduating high-school seniors in a large city will have completed a first-year calculus course. What's the minimum sample size needed to construct a 95% confidence interval for a proportion with a margin of error of 2.5%?
1537
1291
1286
1533
1223
Formula to find sample size:
n = [(z-value)^2 * p * q]/E^2
... where n = sample size, z-value is found using a z-table for 95% confidence (which will be 1.96), p = .30, q = 1 - p, ^2 means squared, * means to multiply, and E = .025 (2.5%)
Convert all fractions to decimals, then plug values into the formula and calculate n. Round your answer to the next highest whole number.
1533
To calculate the minimum sample size needed to construct a confidence interval for a proportion with a given margin of error, we can use the formula:
n = (Z^2 * p * (1 - p)) / E^2
Where:
- n is the sample size
- Z is the Z-score corresponding to the desired confidence level (in this case, 95% confidence level can be associated with a Z-score of 1.96)
- p is the estimated proportion (30% in this case)
- E is the margin of error as a proportion (2.5% = 0.025)
Plugging in the values:
n = (1.96^2 * 0.30 * (1 - 0.30)) / (0.025^2)
n = (3.8416 * 0.30 * 0.70) / 0.000625
n = (0.8073648) / 0.000625
n = 1292.5832
We round up the sample size to the nearest whole number, so the minimum sample size needed is 1293.
Therefore, the correct answer is 1293.