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

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$

$x=574\ hours$