Question

Suppose the number of square feet per house is normally distributed. If the population standard deviation is 155 square feet, what minimum sample size is needed to be 90% confident that the sample mean is within 47 square feet of the true population mean

Answer 1

The minimum sample size to be 90% confident that the sample mean is within 47 square feet of the true population mean is 30.

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.

What minimum sample size is needed to be 90% confident that the sample mean is within 47 square feet of the true population mean

This is n when
`\sigma = 155, M = 47`. So

`M = z*(\sigma)/(√(n))`

`47 = 1.645*(155)/(√(n))`

`47√(n) = 1.645*155`

`√(n) = (1.645*155)/(47)`

`(√(n))^(2) = ((1.645*155)/(47))^(2)`

`n = 29.4`

Rounding up

The minimum sample size to be 90% confident that the sample mean is within 47 square feet of the true population mean is 30.