Chapter 7, Problem 88SE

The efficiency (in lumens per watt) of light bulbs of a certain type has population mean 9.5 and standard deviation .5, according to production specifications. The specifications for a room in which eight of these bulbs are to be installed call for the average efficiency of the eight bulbs to exceed 10. Find the probability that this specification for the room will be met, assuming that efficiency measurements are normally distributed.

To determine

Compute the probability that this specification for the room will met.

Answer to Problem 88SE

The probability that this specification for the room will met is 0.0023.

Explanation of Solution

Let X be the efficiency of light bulbs of a certain type.

From the given information, X follows normal distribution with mean $\mu =9.5$ and $\sigma =0.5$. The sample size is 8. That is, $n=8$.

Then,

$\begin{array}{c}P\left(\overline{x}>10\right)=P\left(\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}>\frac{10-9.5}{0.5/\sqrt{8}}\right)\\ =P\left(Z>\frac{0.5\sqrt{8}}{0.5}\right)\\ =P\left(Z>2.83\right)\\ =0.0023\left[\text{From}\text{}\text{Appendix}\text{3}\text{table}\text{4}\right]\end{array}$

Thus, the probability that this specification for the room will met is 0.0023.