Question

Assume that we want to estimate the mean IQ score for the population of statistics students. How many statistics students must be randomly selected for IQ tests if we want 95% confidence that the sample mean is within 3 IQ points of the population mean? The Standard Deviation is 15.

Answer 1

97 students must be randomly selected for IQ tests.

Explanation:

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1 - 0.95)/(2) = 0.025`

Now, we have to find z in the Z-table as such z has a p-value of
`1 - \alpha`.

That is z with a pvalue of
`1 - 0.025 = 0.975`, so Z = 1.96.

Now, find the margin of error M as such

`M = z(\sigma)/(√(n))`

In which
`\sigma` is the standard deviation of the population and n is the size of the sample.

The Standard Deviation is 15.

This means that
`\sigma = 15`

How many statistics students must be randomly selected for IQ tests if we want 95% confidence that the sample mean is within 3 IQ points of the population mean?

This is n for which M = 3. So

`M = z(\sigma)/(√(n))`

`3 = 1.96(15)/(√(n))`

`3√(n) = 1.96*15`

Simplifying both sides by 3

`√(n) = 1.96*5`

`(√(n))^2 = (1.96*5)^2`

`n = 96.04`

Rounding up(as with a sample size of 96 the margin of error will be slightly above 3):

97 students must be randomly selected for IQ tests.