Question

A researcher is interested in estimating the mean blood alcohol content of people arrested for driving under the influence. Based on past data, the researcher assumes a population standard deviation of 0.065. What sample size is needed to estimate the true mean blood alcohol content within .005 units at the 95% confidence level?

Answer 1

We need a sample size of at least 650

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.

In this problem:

We need a sample of size at least n

n is found when
`M = 0.005, \sigma = 0.065`

So

`M = z*(\sigma)/(√(n))`

`0.005 = 1.96*(0.065)/(√(n))`

`0.005√(n) = 1.96*0.065`

`√(n) = (1.96*0.065)/(0.005)`

`(√(n))^(2) = ((1.96*0.065)/(0.005))^(2)`

`n = 649.23`

Rounding up

We need a sample size of at least 650