Estimate the percent of salespeople that earn more than $885, but less than $1095.

Question

An automobile dealership pays its salespoeple a salary plus a commision on sales.  The mean biweekly commision is $990, the median $950, amd standard deviation $70.

Answer 1

The mean and median are fairly close so a normal distribution probably applies.

Z₁=(885-990)/70=-1.5, Z₂=(1095-990)/70=1.5.

Z₁ is 1.5 standard deviations below the mean and Z₂ is 1.5 standard deviations above the mean.

The normal distribution probability is given by the function N(Z) and is the area beneath the distribution curve to the left, that is, the percentage less than the value of Z. So let's use Z₂ first. N(Z₂)=93.32%.

Since Z₁ is -1.5, the area above this point is the same as the area below Z₂, since both values are the same number of SDs from the mean, and the distribution is symmetrical. 100%-93.32%=6.68% represents those earning less than $885 and 93.32% represents those earning more than $885. The area between is 93.32-6.68=86.64%, that is, the percentage of those earning more than $885 but less than $1095.