Statistics Question DRiving me Crazy?

Question

A calculator manufacturer performs a test on its calculators and finds their working life to be normally distributed, with a mean of 2,150 hours and a standard deviation of 450 hours. What should the manufacturer advertise as the life of the calculators so that 90% of the calculators are covered?

a 2890 hours
b. 2726 hours
c. 1935 hours
d. 1574 hours
e. 1410 hours

Answer 1

In a normal distribution:
68% of the points are within 1 standard deviation
95% of the points are within 2 standard deviations

You'll have to look at your normal distribution tables. You want to find the point where the area is 90%. This happens around z = 1.645 standard deviations.

So they should advertise the life of the calculators as 2,150 +/- 450 * 1.645

That works out to a range of between 1,410 and 2,890 hours, or 2,150 +/- 740 hours.

Answer 2

By "covered" we mean that 90% of the calculators will actually last as long as the adversised life (or longer). So let a be the advertised life span. We want
p(X>=a) = 0.9
p((X-2150)/450 >= (a-2150)/450) = 0.9
p(Z >= (a-2150)/450) = 0.9

You need to look up the value x such that p(Z>=x) in a table. The value is -1.29.

So (a-2150)/450) = -1.29 ---> a = -1.29*450 + 2150 = 1569.5. The z-table I used wasn't very precise (only 2 decimals) so answer (d) is close enough.