Question

A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If 40 different applicants are randomly selected, find the probability that their mean is above 215.

Answer 1

`P(\bar X>215)`

And the best way to solve this problem is using the normal standard distribution and the z score given by:

`z=(\bar x-\mu)/((\sigma)/(√(n)))`

If we apply this formula to our probability we got this: for the value of 215

`z = (215-200)/((50)/(√(40)))= 1.897`

And we can find this probability using the complement rule and with the normal standard distribution or excel we got:

`P(z >1.897) = 1-P(z<1.897) =1- 0.971= 0.029`

Explanation:

Let X the random variable that represent the ratings of applicants from a population, and for this case we know the distribution for X is given by:

`X \sim N(200,50)`

Where
`\mu=200` and
`\sigma=50`

We select a sample size of n =40. We are interested on this probability

`P(\bar X>215)`

And the best way to solve this problem is using the normal standard distribution and the z score given by:

`z=(\bar x-\mu)/((\sigma)/(√(n)))`

If we apply this formula to our probability we got this: for the value of 215

`z = (215-200)/((50)/(√(40)))= 1.897`

And we can find this probability using the complement rule and with the normal standard distribution or excel we got:

`P(z >1.897) = 1-P(z<1.897) =1- 0.971= 0.029`