The efficiency (in lumens per watt) of light bulbs of a certain type has populat

Question

The efficiency (in lumens per watt) of light bulbs of a certain type has population mean 9.5 and standard deviation 0.5 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.

Explanation / Answer

Given that the efficiency of each bulb is N(9.5, 0.5^2) and that the efficiencies of the eight bulbs are independent we know from probability theory that the sample mean of the 8 efficiencies Xbar will also have a normal distribution with mean 9.5 and variance (0.5^2)/8 =0.03125. We need P(Xbar > 10). In order to use tables we now standardise Xbar to make it have a standard normal distribution with mean 0 and variance 1. Let Z = (Xbar -9.5)/sqrt(0.03125). Then Z is N(0,1) [standard result] Then P(Xbar >10) = P[(Xbar -9.5)/sqrt(0.03125) > (10-9.5)/sqrt(0.03125)] = P(Z > 2.8284) = 1 - Phi(2.8284) = 1 - 0.9977 = 0.0023.

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