Answer to Question #105183 in Statistics and Probability for Roland A. Suico

Question

A manufacturer of light bulbs produces bulbs that last a mean of 950 hrs. with a standard deviation of 120 hrs.What is the probability that the mean lifetime of random sample of 10 of these bulbs is less than 900 hours?

Answer 1

Let $X$ be the random variable which represents the length of the life of a light bulb, in hours: $X\sim N(\mu, \sigma^2/n)$

Then

$$Z={X-\mu \over \sigma/\sqrt{n}}\sim N(0,1)$$

Given that $\mu=950\ hrs, \sigma=120\ hrs, n=10.$

The probability that the mean lifetime of random sample of 10 of these bulbs is less than 900 hours is

$$P(X<900)=P(Z<{900-950 \over 120/\sqrt{10}})\approx P(Z<-1.3176)\approx$$

$$\approx0.0938$$