Question

The lifetime X in hours of a certain electrical component has the pdf

If a random sample of 36 is taken from these components,

find P(3$\overline{X}$ + 100 < 111)

Answer 1

Here X has exponential distribution with parameter 1/3 So

$mu=E(X)=3,sigma=3$

Since sample size n=36 is greater than 30 so according to CLT sampling distribution of sample mean will be approximately normal with mean and SD as follows:

$mu_{ar{x}}=mu=3,sigma_{ar{x}}=rac{sigma}{sqrt{n}}=rac{3}{sqrt{36}}=0.5$

Now,

$P(3ar{X}+100<111)=P(3ar{X}<11)=P(ar{X}<11/3)$

The z-score for is

$z=rac{ar{x}-mu_{ar{x}}}{sigma_{ar{x}}}=rac{11/3-3}{0.5}=1.33$

Therefore required probability is

$P(3ar{X}+100<111)=P(z<1.33)=0.9082$

Answer: 0.9082