Question

The SAT againHigh 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?

Short answer

Sample size is $2401$.

Step-by-step solution

Given Information

It is given that $\left(\sigma \right)=50$

Confidence Interval$=95\%$

$E=2$

Concept Used

Formula to be used is:

$n={\left(\frac{z_{\frac{\alpha }{2}}\times \sigma }{E}\right)}^2$

Using table, value of $Z_{\frac{a}{2}}=1.96$corresponding to $95\%$confidence interval.

Calculation

Hence, $n={\left(\frac{\frac{3}{2}\times \sigma }{E}\right)}^2$

$={\left(\frac{1.96\times 50}{2}\right)}^2$

$=2401$

Sample size is$2401$