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

Answer:

We need a sample size of at least 650

Step-by-step explanation:

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

$\alpha = \frac{1-0.95}{2} = 0.025$

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

$M = z*\frac{\sigma}{\sqrt{n}}$

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

$M = z*\frac{\sigma}{\sqrt{n}}$

$0.005 = 1.96*\frac{0.065}{\sqrt{n}}$

$0.005\sqrt{n} = 1.96*0.065$

$\sqrt{n} = \frac{1.96*0.065}{0.005}$

$(\sqrt{n})^{2} = (\frac{1.96*0.065}{0.005})^{2}$

$n = 649.23$

Rounding up

We need a sample size of at least 650