Question

A quality control manager at a chemical company wants to measure the average amount of cleaning fluid the company is placing in their standard size bottle. From previous studies, they believe their population standard deviation is 0.4 ounces. If the manager would like to be 99% confident that their estimate of the mean is within 0.05 ounces of the true mean, how large of a sample is needed?

Answer 1

Answer:

We need a sample size of at least 425.

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:

$\alpha = \frac{1-0.99}{2} = 0.01$

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 $1-0.01 = 0.99$, so $z = 2.575$

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.

If the manager would like to be 99% confident that their estimate of the mean is within 0.05 ounces of the true mean, how large of a sample is needed?

We need a sample size of at least n.

n is found when $M = 0.05, \sigma = 0.4$. So

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

$0.05 = 2.575*\frac{0.4}{\sqrt{n}}$

$0.05\sqrt{n} = 2.575*0.4$

$\sqrt{n} = \frac{2.575*0.4}{0.05}$

$(\sqrt{n})^{2} = (\frac{2.575*0.4}{0.05})^{2}$

$n = 424.36$

Rounding up

We need a sample size of at least 425.