Answer to Question #153454 in Statistics and Probability for samantha

Question

An Urban council has installed 2000 lamps with mercury bulbs in the   streets of town area. The lifetimes of these bulbs are normally distributed with a mean of 1200 burning hours and having a standard deviation of 200 hours. 

(1) After what number of burning hours would you expect that 10 % of the bulbs would fail?

(2) After what number of burning hours would you expect that 150 bulbs are still in good condition?

I need the answers with simple functions and little explanations.

Answer 1

Let $X=$ the lifetime of the mercury bulb: $X\sim N(\mu, \sigma^2)$

Given $\mu=1200\ h, \sigma=200\ h, n=2000$

(1)

$$P(X<x)=P(Z<\dfrac{x-\mu}{\sigma})=0.1$$

$$\dfrac{x-\mu}{\sigma}\approx-1.281552$$

$$x\approx1200+200(-1.281552)=943.69(h)$$

After about 944 hours, 10% of the bulbs would fail.

(2)

$$P(X>x)=1-P(X \leq x)=1-P(Z \leq\dfrac{x-\mu}{\sigma} )$$

$$=\dfrac{150}{2000}=0.075$$

$$\dfrac{x-\mu}{\sigma}\approx1.439531$$

$$x\approx1200+200(1.439531)=1487.91(h)$$

After 1488 hours, we expect that 150 bulbs are still in good condition.