Answer:
The probability of receiving reimbursement within 42 days of filing is 0.8849 = 88.49%
Step-by-step explanation:
Let [tex]P(a \leq x \leq b)[/tex] the probability of being reimbursed in an interval of a to b days, we want [tex]P(0 \leq x \leq 42)[/tex]
First, standardize x by doing the change
[tex]z=\frac{x-\mu}{\sigma}[/tex]
where [tex]\mu[/tex] is the mean of our original distribution and [tex]\sigma[/tex] the standard deviation. So
[tex]z=\frac{x-36}{5}[/tex]
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,
[tex] 0 \leq x \leq 42\Rightarrow -36 \leq x -36 \leq 6\Rightarrow \frac{-36}{5} \leq \frac{x -36}{5} \leq \frac{6}{5} [/tex]
we then have
[tex] -7.2 \leq z \leq 1.2[/tex]
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%
