SOLUTION: An IQ test is designed so that the mean is 100 and the standard deviation is 12 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of

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Question 1182680: An IQ test is designed so that the mean is 100 and the standard deviation is 12 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 2 IQ points of the true mean. Assume that σ = 12 and determine the required sample size using technology. Then determine if this is a reasonable sample size for a real world calculation.
The required sample size is ______. (Round up to the nearest integer.)
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
critical z-score for 95% confidence level is plus or minus 1.96.
z-score formula is:
z = (x - m) / s
z is the z-score
x is the raw score
m is the mean
s is the standard error.
when m is 100, formula becomes:
plus or minus 1.96 = (x - 100) / s
formula for standard error is:
s = standard deviation divided by square root of sample size.
with a standard deviation of 12, this becomes:
s = 12 / sqrt(sample size)
when x = 102 and s = 12 / sqrt(sample size), the formula becomes:
1.96 = (102 - 100) / (12 / sqrt(sample size).
this is equivalent to:
1.96 = (102 - 100) * sqrt(sample size) / 12
simplify to get:
1.96 = 2 * sqrt(sample size) / 12
multiply both sides of this equation by 12 and divide both sides of this equation by 2 to get to get:
1.96 * 12 / 2 = sqrt(sample size)
solve for sqrt(sample size) to get:
sqrt(sample size) = 1.96 * 12 / 2 = 11.76
solve for s (standard error) to get:
s = standard deviation / sqrt(sample size) = 12 / 11.76 = 1.020408163
z=score formula becomees 1.96 = (x - 100) / 1.020408163
solve for x to get:
x = 1.020408163 * 1.96 + 100 = 102.
this is what you wanted since 102 - 100 = 2.
when z = -1.96, the formula becomes:
-1.96 = (x - 100) / 1.020408163
solve for x to get:
x = -1.96 * 1.020408163 + 100 = 98.
this is what you wanted since 98 - 100 = -2.
your 95% interval is between 98 and 102.
the sample size required for this is 11.76 squared = 138.2976.
if you need an integer answer, then you would round up to 139.
here's what it looks like on a graph.

SD on the graph standard for standard deviation which would be the standad deviation of the population or the standard error of the sample of a given size.
in this case, we are dealing with the standard error of a sample of size of 138.2976 which is equal to the standard deviation of the population divided by the square root of 138.2976 which is equal to 1.020408163.
there i some rounding with this calculator for input and outputs. the results are accurate enough for normal problem requirements.