Question

The lifetime of a certain kind of bulb has a normal distribution with mean of 500 hours and standard deviation of 45 hours

Find the percentage of bulbs with a lifetime of at least 570 hrs

The percentage of bulbs with lifetime in between 485 and 515 hrs

The minimum lifetime of the best 5% of the bulbs

Answer 1

Let $X=$ lifetime of a certain kind of bulb: $X\sim N(\mu, \sigma^2).$

Given $\mu=500h, \sigma=45h.$

i)

$$P(X\ge570)=1-P(X<570)$$

$$=1-P(Z<\dfrac{570-500}{45})$$

$$\approx1-P(Z<1.555556)\approx0.0599$$

$5.99\%$

ii)

$$P(485<X<515)=P(X<515)-P(X\le485)$$

$$=P(Z<\dfrac{515-500}{45})-P(Z\le\dfrac{485-500}{45})$$

$$\approx P(Z<0.333333)-P(Z<-0.333333)\approx0.2611$$

$26.11\%$

iii)

$$P(X\ge x)=1-P(X<x)$$

$$=1-P(Z<\dfrac{x-500}{45})=0.05$$

$$\dfrac{x-500}{45}\approx1.6449$$ $$x=45(1.6449)+500$$

$x=574\ hours$