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:

Answer 1

Let $X=$ the sample mean: $X\sim N(\mu, \sigma^2/n).$ Then $Z=\dfrac{X-\mu}{\sigma/\sqrt{n}}\sim N(0, 1)$

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

a. greater than 1, 400 hours?

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

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

$$\approx1-P(Z\leq-1.264911)\approx0.897048$$

b. less than 1, 400 hours?

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

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

Check

$$0.897048+0.102952=1$$