SOLUTION: SAT scores are distributed with a mean of 1,500 and a standard deviation of 289. You are interested in estimating the average SAT score of first year students at your college. If y
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Question 1202132: SAT scores are distributed with a mean of 1,500 and a standard deviation of 289. 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 confidence interval to 50 points with 85 percent confidence, how many students should you sample? (Round up to a whole number of students.)
You can put this solution on YOUR website! population mean is 1500.
population standard deviation is 289.
margin of error is less than or equal to plus or minus 50.
z-score formula is z = (x-m) / s
z is the z is the critical z-score.
(x-m) is the margin of error.
s is the standard error.
standard error = standard deviation / sqrt(n)
sd is the standard deviation.
n is the sample size.
z-score formula becomes z = (x-m) / (sd / sqrt(n))
simplify to get z = (x-m) / sd * sqrt(n).
solve for sqrt(n) to get sqrt(n) = z / (x-m) * sd *****
once you have sqrt(n), confirm by solving for (x-m) to get (x-m) = z * sd / sqrt(n) *****
critical z-score at 85% confidence interval is plus or minus z = 1.439531471.
the .15 alpha is divided by 2 to get .075 alpha on each side of the confidence interval.
when z = that and (x-m) = 50 and sd = 289, solve for sqrt(n) to get:
sqrt(n) = 8.320491902.
solve for (x-m) to get (x-m) = 1.439531471 * 289 / 8.320491902. = 50.
this confirms the margin of error is what you want it to be.
n = sqrt(n)^2 = 8.320491902^2 = 69.23058549.
n has to be an integer to set it to the next higher integer = 70.
sqt(n) is now equal to sqrt(70) = 8.366600265.
solve for (x-m) to get (x-m) = 1.439531471 * 289 / 8.366600265 = 49.72444983.
the mrgin of error is less than 50 as desired.
since s = sd / sqrt(n), you get s = 289 / 8.366600265 = 34.54210681.
when m = 1500 and s = 34.54210681 and z = 1.439531471, z-score formula becomes:
1.439531471 = (x - 1500) / 34.54210681.
solve for x to get:
x = 1.439531471 * 34.54210681 + 1500 = 1549.72445.
that's on the high side of the 85% confidence interval.
on the low side, you get -1.439531471 * 34.54210681 + 1500 = 1450.27555.
your margin of error is less than 50, as it should be.
once you found sample size to the next higher integer, you solve for standard error to get s = 289 / sqrt(sample size).
you then use that value of standard error in the z-score formula to find x.
