Drying times for newly painted microwave oven cabinets are normally distributed with a mean of 2.5 minutes and a standard deviation of 0.25 minutes. After painting, each cabinet is mated with its electronic modules and mechanical components. The production manager must decide how much time to allow after painting before these other components are installed. If the time is too short, the paint will smudge and the unit will have to be refinished. If the time is too long, production efficiency will suffer. A consultant has concluded that the time delay should be just enough to allow 99.8% of the cabinets to dry completely, with just 0.2% ending up being smudged and sent back for refinishing. Given this information, for what time setting should the production manager set the automatic timer that pauses the production line while each cabinet dries?

where z is the distance from the mean measured in the standard deviation units, x is the value we are interested in, u is the mean distribution, a is the standard deviation of the distribution.

the time delay should be just enough to allow 99.8% of the cabinets to dry completely = 99.8/100 = 0.9980

first we determine an appropriate z value.

Using the standardized normal tables,

value of z for approximately 0.9980 is 2.88

so using our initial equation z = (x - u) / a

we substitute the value

z = (x - u) / a

2.88 = ( x - 2.5) / 0.25

2.88 * 0.25 = x - 2.5

0.72 = x - 2.5

x = 0.72 + 2.5

x = 3.22