Answer to Question #155911 in Statistics and Probability for bert

Question

Suppose and LED light manufacturer want to set a lifespan guarantee on its new LED light. life test revealed that the mean of 50,000 life hours and the standard deviation of the normal distribution of 2,140 hours, The manufacturer wants to set the guaranteed life hours so that no more than 5% of the LED lights will have to be replaced. What guaranteed life hours should the manufacturer announce?

Answer 1

Let $X=$ the number of life hours: $X\sim N(\mu,\sigma^2).$ Then $Z=\dfrac{X-\mu}{\sigma}\sim N(0,1).$

Given $\mu=50000\ hours, \sigma=2140\ hours$

$$P(X<x)=P(Z<\dfrac{x-50000}{2140})\leq0.05$$

$$\dfrac{x-50000}{2140}\leq-1.6449$$

$$x\leq46480$$

The guaranteed life hours should be 46480 hours.