Question

If it is assumed that the heights of men are normally distributed with a standard deviation of 1.5 inches, how large a sample should be taken to be fairly sure (probability 0.95) that the sample mean does not differ from the true mean (population mean) by more than 0.90

Answer 1

We need a sample of at least 8 men.

Explanation:

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

`\alpha = (1-0.9)/(2) = 0.05`

Now, we have to find z in the Ztable as such z has a pvalue of
`1-\alpha`.

So it is z with a pvalue of
`1-0.05 = 0.95`, so
`z = 1.645`

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.

In this problem:

We need to have a sample of at least n.

n is found when
`M = 0.9, \sigma = 1.5`. So

`M = z*(\sigma)/(√(n))`

`0.9 = 1.645*(1.5)/(√(n))`

`0.9√(n) = 1.645*1.5`

`√(n) = (1.645*1.5)/(0.9)`

`(√(n))^(2) = ((1.645*1.5)/(0.9))^(2)`

`n = 7.51`

Rounding up

We need a sample of at least 8 men.