Question

According to records kept by Mariposa County, the average (mean) amount of time that it takes for employees to be reimbursed for professional expenses incurred in service to the county is 36 days, with a standard deviation of 5 days. The distribution is normal. About 2 months ago, Latisha MacNeil attended a training conference for her job in the County Recreation Department. She filed for reimbursement of expenses that she incurred at the conference 42 days ago, but she has not received payment. What is the probability of receiving reimbursement within 42 days of filing?

Answer 1

The probability of receiving reimbursement within 42 days of filing is 0.8849 = 88.49%

Explanation:

Let
`P(a \leq x \leq b)` the probability of being reimbursed in an interval of a to b days, we want
`P(0 \leq x \leq 42)`

First, standardize x by doing the change

`z=(x-\mu)/(\sigma)`

where
`\mu` is the mean of our original distribution and
`\sigma` the standard deviation. So

`z=(x-36)/(5)`

This change transforms the original distribution in a normal distribution N(0,1) with mean 0 and standard deviation 1.

This is done in order to easily compute the area under the normal curve.

Since,

`0 \leq x \leq 42\Rightarrow -36 \leq x -36 \leq 6\Rightarrow (-36)/(5) \leq (x -36)/(5) \leq (6)/(5)`

we then have

`-7.2 \leq z \leq 1.2`

And we want the area under the normal curve between -7.2 and 1.2. This can be done either by using a table or a computer, and we find this area is 0.8849,

Hence, the probability of receiving reimbursement within 42 days of filing is 0.8849 = 88.49%