Question

Let X be the lifetime of an electronic device. It is known that the average lifetime of the device is 767 days and the standard deviation is 121 days. Let x¯ be the sample mean of the lifetimes of 157 devices. The distribution of X is unknown, however, the distribution of x¯ should be approximately normal according to the Central Limit Theorem. Calculate the following probabilities using the normal approximation. (a) P(x¯≤754)= Answer (b) P(x¯≥784)= Answer (c) P(749≤x¯≤784)= Answer

Answer 1

for normal distribution z score =(X-μ)/σx
here mean=       μ=	767
std deviation   =σ=	121.0000
sample size       =n=	157
std error=σx̅=σ/√n=	9.6569

a)

probability =

P(X<754)

=

P(Z<-1.35)=

0.0885

b)

probability =

P(X>784)

=

P(Z>1.76)=

1-P(Z<1.76)=

1-0.9608=

0.0392

c)

probability =

P(749<X<784)

=

P(-1.86<Z<1.76)=

0.9608-0.0314=

0.9294