Average SAT scores of first-year students?

Question

SAT scores are normally distributed with a mean of 1,500 and a standard deviation of 300. An administrator at a college is interested in estimating the average SAT score of first-year students. If the administrator would like to limit the margin of error of the 96% confidence interval to 15 points, how many students should the administrator sample? Make sure to give a whole number answer.

Answer 1

Hi Bailey,

We have normally distributed data and known standard deviation, so we can use z. We are looking for the sample size n, and the formula is:

n=[(z_a/2)*sigma²]/E²

Let’s break this down.

n=sample size we are looking for

z_a/2=z-critical value

sigma=population standard deviation

E=Acceptable Error

Now, the trickiest part may be getting correct z-critical value—96% confidence intervals are not common, so memorization is impractical. To get z_a/2, we first must get alpha, the significance level. Formula for that:

a=1-c

a=alpha

c=confidence level

Here, c=0.96, so:

a=1-0.96

a=0.04

Now, the formula specifies a/2, so we do:

0.04/2=0.02

Now, we need to remember that this is two-sided, which means we take:

1-0.02=0.98

Look for that value on interior of z-table. Closest value I have is 0.9798, which corresponds to:

z=2.05

So,

z_a/2=2.05

Now, we have all we need to compute n, the minimum sample size:

n=[z_a/2*sigma²]/E²

z_a/2=2.05

sigma=300

E=15

n=[2.05*300²]/15²

n=820

I hope this helps.