Question

A manufacturer of light bulbs produces bulbs that last a mean of 950 hours with a standard variation of 120 hours. What is the probability that the mean life-time of a 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. We suppose that $X\sim N(\mu, \sigma^2).$ Then $\bar{X}\sim N(\mu, \sigma^2/n)$

Given $\mu=950, \sigma=120, n=10$

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

$$\approx P(Z<1.317616)\approx0.906184$$

The probability that the mean life-time of a random sample of 10 of these bulbs is less than 900 hours is $0.906184.$