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 0.18
.
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 90%
confidence level with an error of at most 0.02
? Round your answer up to the next integer.
Answers
GPT 3.5
To determine the sample size needed, we can use the formula for sample size calculation for estimating a population proportion:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = z-score corresponding to the desired confidence level (for 90% confidence level, Z = 1.645)
p = estimated population proportion (0.18)
E = maximum error (0.02)
Plugging the values into the formula:
n = (1.645^2 * 0.18 * (1-0.18)) / 0.02^2
n = (2.701025 * 0.18 * 0.82) / 0.0004
n = 0.39174405 / 0.0004
n = 979.360125
Rounding up to the next integer, the sample size required would be 980.
Therefore, a sample size of 980 tenth graders would be needed to estimate the fraction of students with reading skills at or below the eighth grade level with a confidence level of 90% and a maximum error of 0.02.
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = z-score corresponding to the desired confidence level (for 90% confidence level, Z = 1.645)
p = estimated population proportion (0.18)
E = maximum error (0.02)
Plugging the values into the formula:
n = (1.645^2 * 0.18 * (1-0.18)) / 0.02^2
n = (2.701025 * 0.18 * 0.82) / 0.0004
n = 0.39174405 / 0.0004
n = 979.360125
Rounding up to the next integer, the sample size required would be 980.
Therefore, a sample size of 980 tenth graders would be needed to estimate the fraction of students with reading skills at or below the eighth grade level with a confidence level of 90% and a maximum error of 0.02.