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 0.19
.
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 98%
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 98%
confidence level with an error of at most 0.02
? Round your answer up to the next integer.
Answers
GPT 3.5
To calculate the sample size required, we can use the formula:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-score corresponding to 98% confidence level, which is approximately 2.33
p = estimated population proportion (0.19)
E = maximum error (0.02)
Plugging in the values:
n = (2.33^2 * 0.19 * (1-0.19)) / 0.02^2
n = (5.4289 * 0.19 * 0.81) / 0.0004
n = 0.8249 / 0.0004
n = 2062.25
Since the sample size has to be a whole number, we round up to the next integer:
n = 2063
Therefore, a sample size of 2063 tenth grade students would be required to estimate the fraction of students reading at or below the eighth grade level at the 98% confidence level with an error of at most 0.02.
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size
Z = Z-score corresponding to 98% confidence level, which is approximately 2.33
p = estimated population proportion (0.19)
E = maximum error (0.02)
Plugging in the values:
n = (2.33^2 * 0.19 * (1-0.19)) / 0.02^2
n = (5.4289 * 0.19 * 0.81) / 0.0004
n = 0.8249 / 0.0004
n = 2062.25
Since the sample size has to be a whole number, we round up to the next integer:
n = 2063
Therefore, a sample size of 2063 tenth grade students would be required to estimate the fraction of students reading at or below the eighth grade level at the 98% confidence level with an error of at most 0.02.