Question

what is the minimum sample size required to estimate a population mean with 95% confidence when the desired margin of error is E=1.5? The population standard deviation is known to be 10.75

Answer 1

The minimum sample size is 198

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.

What is the minimum sample size required to estimate a population mean with 95% confidence when the desired margin of error is E=1.5?

This sample size is n.

n is found when M = 1.5.

We have that
`\sigma = 1.5`

`M = z*(\sigma)/(√(n))`

`1.5 = 1.96*(10.75)/(√(n))`

`1.5√(n) = 1.96*10.75`

`√(n) = (1.96*10.75)/(1.5)`

`(√(n))^(2) = ((1.96*10.75)/(1.5))^(2)`

`n = 197.3`

Rounding up

The minimum sample size is 198