Answer to Question #339626 in Statistics and Probability for Yashu

Question

 A bulb manufacturer claims that the lives of its bulbs are normally distributed with a mean of 6000 hours and a standard deviation of 400 hours. A random sample of 16 bulbs had an average life of 5850 hours. If the manufacturer’s claim is correct-

a. What is the sampling distribution of the sample mean?

b. what is the probability of finding a sample mean of 5850 or less?

Answer 1

a. If the population is normally distributed, then the sampling distribution of $\bar{X}$ is normally distributed for any sample size $n.$

Let $\bar{X}=$ the sample mean: $\bar{X}\sim N(\mu, \sigma^2/n).$

Given $\mu=600h, \sigma=400h, n=16$

$$\bar{X}\sim N(6000, 400^2/16)$$

b.

$$P(\bar{X}\le5850)=P(Z\le\dfrac{5850-6000}{400/\sqrt{16}})$$

$$=P(Z\le-1.5)\approx0.0668$$