Answer to Question #314550 in Quantitative Methods for jack allen

Question

The lifetimes of light bulbs produced by a particular manufacturer have a mean of 1,200 hours and a standard deviation of 400 hours. The population distribution is normal. Suppose that you purchase nine bulbs, which can be regarded as a random sample from the manufacturer’s output

a. What is the mean of the sample mean life? (2 marks)

b. What is the variance of the sample mean? (2 marks)

c. What is the standard error of the sample mean? (2 marks)

d. What is the probability that, on average, those nine light bulbs have lives of fewer than 1,050 hours? (4 marks)

Answer 1

$a:\\\bar{x}=\mu =1200\\b:\\s^2=\frac{\sigma ^2}{n}=\frac{400^2}{9}=17777.8\\c:\\s=\sqrt{s^2}=\frac{400}{3}=133.333\\d:\\P\left( \bar{x}<1050 \right) =P\left( \sqrt{n}\frac{\bar{x}-\mu}{\sigma}<\sqrt{n}\frac{1050-1200}{400} \right) =P\left( Z<\sqrt{9}\frac{1050-1200}{400} \right) =\\=P\left( Z<-1.125 \right) =\varPhi \left( -1.125 \right) =0.1303$