Answer to Question #310181 in Statistics and Probability for nyenye

Question

A manufacturer of light bulbs produces bulbs that last a mean of 950 hours with a standard deviation of 120

hours. What is the probability that the mean lifetime of a random sample of 10 of these bulbs is less than 900

hours?

Answer 1

We have

$X_1\sim N\left( 950,120^2 \right) \Rightarrow \bar{x}\sim N\left( 950,\frac{120^2}{n} \right) =N\left( 950,1440 \right) \\n=10\\P\left( \bar{x}<900 \right) =P\left( \frac{\bar{x}-950}{\sqrt{1440}}<\frac{900-950}{\sqrt{1440}} \right) =P\left( Z<-1.31762 \right) =\\=\varPhi \left( -1.13762 \right) =0.12764\\$