Question

Dave has two jobs. For daytime work at a jewelry store he is paid $15,000 per month, plus a commission. His monthly commission is normally distributed with a mean of $10,000 and a standard deviation of $2,000. At night he works occasionally as a waiter, for which his monthly income is normally distributed with a mean of $1,000 and a standard deviation of $300. Dave's income levels from these two sources are independent of each other. Dave's commission from the jewelry store will be between what two values, symmetrically distributed around the population mean 80% of the time?

Answer 1

Let X be the Dave's commission.  

X ~ N( = $10,000 , = $2,000)

For symmetrically distributed around the population mean 80% of the time, the critical limits of the percentile are,

(1 - 0.8)/2 and 1 - (1 - 0.8)/2

= 0,1 and 0.9

Z score for p = 0.1 and 0.9 are -1.28 and +1.28 respectively.

The two values, symmetrically distributed around the population mean 80% of the time are,

(10000 - 1.28 * 2000 ,  10000 + 1.28 * 2000)

($7440 , $12560)