Question

Suppose the finishing times for cyclists in a race are normally distributed. If the population standard deviation is 16 minutes, what minimum sample size is needed to be 95% confident that the sample mean is within 5 minutes of the true population mean

Answer 1

Answer:

The minimum sample size is needed to be 95% confident that the sample mean is within 5 minutes of the true population mean is 40.

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.

If the population standard deviation is 16 minutes, what minimum sample size is needed to be 95% confident that the sample mean is within 5 minutes of the true population mean

This is n when $M = 5, \sigma = 16$. So

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

$5 = 1.96*\frac{16}{\sqrt{n}}$

$5\sqrt{n} = 1.96*16$

$\sqrt{n} = \frac{1.96*16}{5}$

$(\sqrt{n})^{2} = (\frac{1.96*16}{5})^{2}$

$n = 39.3$

The minimum sample size is needed to be 95% confident that the sample mean is within 5 minutes of the true population mean is 40.