SAT scores are distributed with a mean of 1,500 and a standard deviation of 300. You are interested in estimating the average SAT score of first year students at your college. If you would like to limit the margin of error of your 95% confidence interval to 25 points, how many students should you sample?
The error=+/-z*sigma/sqrt (n)
25=1.96*300/sqrt(n)
25 sqrt(n)=588
divide both sides by 25 and square
n=553.19 or 554
To estimate the number of students you need to sample, you can use the formula for sample size calculation for estimating the population mean.
The formula is:
n = (Z * σ / ME)²
Where:
- n is the required sample size
- Z is the z-score corresponding to the desired level of confidence (in this case, for a 95% confidence level, Z = 1.96)
- σ is the standard deviation of the population (300 in this case)
- ME is the maximum margin of error (25 in this case)
Plugging in these values into the formula, we get:
n = (1.96 * 300 / 25)²
n = (588 / 25)²
n = 23.52²
n ≈ 554.86
Since you cannot have a fraction of a student, you would round up the sample size to the nearest whole number. Therefore, you should sample approximately 555 students to estimate the average SAT score of first-year students at your college, with a maximum margin of error of 25 points and a 95% confidence level.