Question

The life time of a particular brand smartphone's battery is known to have Gamma distribution with αα = 4 and ββ = 2. If we take a random sample of 50 these batteries, what is the probability that the average life time of this random sample will be between 7 and 9 months?

Answer 1

mean μ =	αβ =	8
varaince =σ2=	αβ2 =	16.00
standard deviation σ=β*√α=		4.0000

since n>=30, we can use normal approximation from central limit theorem:

for normal distribution z score =(X-μ)/σx
here mean=       μ=	8
std deviation   =σ=	4.000
sample size       =n=	50
std error=σx̅=σ/√n=	0.56569

  probability that the average life time of this random sample will be between 7 and 9 months:

probability =P(7<X<9)=P((7-8)/0.566)<Z<(9-8)/0.566)=P(-1.77<Z<1.77)=0.9616-0.0384=0.9232