Q. 56 - The Practice of Statistics for AP

Question

56. The SAT again High school students who take the SAT Math exam a second time generally score higher than on their first try. Past data suggest that the score increase has a standard deviation of about $50$ points.
How large a sample of high school students would be needed to estimate the mean change in SAT score to within $2$ points with $95\%$ confidence? Show your work.

Step-by-Step Solution

Answer

The mean change in SAT score to within $2$points with $95\%$ confidence is $n\ge 2401$.

Step 1: Given information

Past data suggest that the score increase has a standard deviation of about $50$ points. To estimate the mean change in SAT score to
within $2$ points with $95\%$ confidence.

Step 2: Explanation

Convert the confidence interval $95\%$ into decimal.
$\frac{95}{100}=0.95$
Using table A, calculate the critical value:
$z_{\alpha /2}=z_{(1-0.95)/2}=1.96$

Note that, Margin of error $E$ is $2,$ and, the standard deviation $\sigma$ is $50$
Determine the sample size:

$E\ge z_{\alpha /2}\frac{\sigma }{\sqrt{n}}$

$=2\ge 1.96\frac{50}{\sqrt{n}}$

$=2\times \sqrt{n}\ge 1.96\times 50$

$=\sqrt{n}\ge 49$

Remove the square from both sides:
$=(\sqrt{n}{)}^2\ge {49}^2$
$=n\ge 2401$

Hence, the sample size is $n\ge 2401$.