Question

Suppose the heights of seasonal pine saplings are normally distributed. If the population standard deviation is 14 millimeters, what minimum sample size is needed to be 95% confident that the sample mean is within 4 millimeters of the true population mean?

Answer 1

The minimum sample size needed to be 95% confident that the sample mean is within 4 millimeters of the true population mean is 47.

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 Ztable as such z has a pvalue of
`1-\alpha`.

So it 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.

If the population standard deviation is 14 millimeters, what minimum sample size is needed to be 95% confident that the sample mean is within 4 millimeters of the true population mean?

This is n when
`\sigma = 14, M = 4`. So

`M = z*(\sigma)/(√(n))`

`4 = 1.96*(14)/(√(n))`

`4√(n) = 1.96*14`

`√(n) = (1.96*14)/(4)`

`(√(n))^(2) = ((1.96*14)/(4))^(2)`

`n = 47`

The minimum sample size needed to be 95% confident that the sample mean is within 4 millimeters of the true population mean is 47.