Answer to Question #199932 in Statistics and Probability for John

Question

A LED company claims that the average life of the LED light bulbs it

manufactures is 1, 500 hours with a standard deviation of 500 hours. If a random sample of 40 bulbs is chosen, what is the probability that the sample mean will be:

a. greater than 1, 400 hours?

b. less than 1, 400 hours?

Answer 1

Let $X=$ the average life of the LED light bulb: $X\sim N(\mu, \sigma ^2/n).$

Given $\mu=1500\ h, \sigma=500\ h, n=40$

a)

$$P(X>1400)=1-P(\leq1400)$$

$$=1-P(\leq\dfrac{1400-1500}{500/\sqrt{40}})\approx1-P(Z\leq-1.264911)$$

$$\approx0.897048$$

b)

$$P(X<1400)=P(Z<\dfrac{1400-1500}{500/\sqrt{40}})$$

$$\approx P(Z<-1.264911)\approx0.102952$$