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.