SOLUTION: An IQ test is designed so that the mean is 100 and the standard deviation is 17 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of

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Question 1162115: An IQ test is designed so that the mean is 100 and the standard deviation is 17 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with 95​% confidence that the sample mean is within 6 IQ points of the true mean. Assume that sigmaequals17 and determine the required sample size using technology. Then determine if this is a reasonable sample size for a real world calculation.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
population mean = m = 100
population standard deviation = sd = 17
sample size = n
standard error = s = sd / sqrt(n)
z-score formula is z = (x - m) / s
at 95% confidence limit, critical z-score is plus or minus 1.959963986.
you want (x - m) to be equal to plus or minus 6.
work with the upper limit first.
(x - m) = (106 - 100) = 6
z-score formula becomes z = 6 / s
solve for s to get s = 6 / z
since s = sd / sqrt(n), you get sd / sqrt(n) = 6 / z
solve for sqrt(n) to get sqrt(n) = z * sd / 6
since z = 1.959963986 and sd = 17, you get sqrt(n) = 1.959963986 * 17 / 6 = 5.553231204
solve for n to get n = sqrt(n)^2 = 30.8383778
that's your sample size required to get (x-m) equal to plus or minus 6 at 95% confidence limit.
to confirm, use s = 17 / sqrt(n) = 3.061280739 in your calculations.
at 95% confidence limits, your lower and upper z-score formulas become:
-1.959963986 = (x - 100) / 3.061280739 for lower z-score.
1.959963986 = (x - 100) / 3.061280739 for upper z-score.
solve for lower x and upper x to get:
lower x = -1.959963986 * 3.061280739 + 100 = 94
upper x = 1.959963986 * 3.061280739 + 100 = 106
your sample size needs to be equal to 30.8383778 is your answer.
since sample size needs to be an integer, then use sample size = 31.
that will ensure that (x - m) will be less than or equal to 6.
visually, this will look like this.