Question

Albert Abbasi, VP of Operations at Ingleside International Bank, is evaluating the service level provided to walk-in customers. Accordingly, he plans a sample of waiting times for walk-in customers. If the population of waiting times has a mean of 15 minutes and a standard deviation of 4 minutes, the probability that Albert's sample of 64 will have a mean between 13.5 and 16.5 minutes is ________.

Answer 1

Answer:

The probability that Albert's sample of 64 will have a mean between 13.5 and 16.5 minutes is 0.9973.

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Let X the random variable that represent interest on this case, and for this case we know the distribution for X is given by:

$X \sim N(\mu=15,\sigma=4)$  

And let $\bar X$ represent the sample mean, the distribution for the sample mean is given by:

$\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})$

On this case  $\bar X \sim N(15,\frac{4}{\sqrt{64}})$

Solution to the problem

We are interested on this probability

$P(13.5$  

If we apply the Z score formula to our probability we got this:

$P(13.5$

$=P(\frac{13.5-15}{\frac{4}{\sqrt{64}}}$

And we can find this probability on this way:

$P(-3$

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

$P(-3$

The probability that Albert's sample of 64 will have a mean between 13.5 and 16.5 minutes is 0.9973.