Question
The state education commission wants to estimate the fraction of tenth grade students that have reading skills at or below the eighth grade level. In an earlier study, the population proportion was estimated to be 0.23
.
How large a sample would be required in order to estimate the fraction of tenth graders reading at or below the eighth grade level at the 85%
confidence level with an error of at most 0.02
? Round your answer up to the next integer.
.
How large a sample would be required in order to estimate the fraction of tenth graders reading at or below the eighth grade level at the 85%
confidence level with an error of at most 0.02
? Round your answer up to the next integer.
Answers
GPT 3.5
The formula for calculating the sample size needed for estimating a population proportion is:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-score corresponding to the confidence level (for 85% confidence level, Z ≈ 1.44)
p = estimated population proportion (0.23)
E = maximum error allowed (0.02)
Plugging in the values, we get:
n = (1.44^2 * 0.23 * (1-0.23)) / 0.02^2
n = (2.0736 * 0.23 * 0.77) / 0.0004
n = 0.3650084 / 0.0004
n = 912.52
Since we need to round up to the next integer, the sample size required would be 913.
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-score corresponding to the confidence level (for 85% confidence level, Z ≈ 1.44)
p = estimated population proportion (0.23)
E = maximum error allowed (0.02)
Plugging in the values, we get:
n = (1.44^2 * 0.23 * (1-0.23)) / 0.02^2
n = (2.0736 * 0.23 * 0.77) / 0.0004
n = 0.3650084 / 0.0004
n = 912.52
Since we need to round up to the next integer, the sample size required would be 913.