Mathematics
The Sampling Distribution of the Sample Mean

### Question Description

A population has mean 1,542 and standard deviation 246.

Find the probability that the mean of a sample of size 100 will be within 100 units of the population mean, that is, between 1,442 and 1,642

The sample mean is the population mean: 1542.

The standard deviation of the sample mean (called the "standard error") is

population standard deviation / sqrt(sample size)

= 246 / sqrt(100)

= 24.6

So the probability that the sample mean will be outside 100 units is given by the normal distribution cdf at 1442, multiplied by two (since we can be in either the lower or upper tail of the distribution):

= 2x Norm.cdf( x, mean, sd )

= 2 x Norm.cdf( 1442, 1542, 24.6 )

4.80241 x10^-05

So the probability this sample mean will be within 100 units is:

1 - 4.80241 x10^-05

0.999951976

iainharlow (1041)
Boston College
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