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 0.15 .

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 0.03 ? Round your answer up to the next integer.
Answer
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Step 1 ：The problem is asking for the sample size needed to estimate a population proportion with a certain level of confidence and margin of error. This can be solved using the formula for the sample size of a proportion: \(n = \frac{{Z^2 * p * (1-p)}}{{E^2}}\) where:

Step 2 ：\(n\) is the sample size

Step 3 ：\(Z\) is the Z-score, which corresponds to the desired confidence level

Step 4 ：\(p\) is the estimated population proportion

Step 5 ：\(E\) is the desired margin of error

Step 6 ：In this case, we have: \(Z = 1.96\) (for a 85% confidence level), \(p = 0.15\) (from the earlier study), \(E = 0.03\) (the desired margin of error)

Step 7 ：We can plug these values into the formula to find the required sample size. However, since we can't have a fraction of a student, we'll need to round up to the next whole number.

Step 8 ：By substituting the given values into the formula, we get \(n = 545\)

Step 9 ：Final Answer: The required sample size 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 0.03 is \(\boxed{545}\)