Question #347244

Suppose that the lifetimes of electric light bulbs follow an Exponential distribution with mean

1000 hours. A

random sample of 40 light bulbs is tested. The probability that the mean

lifetime is at least 900 hours is:


1
Expert's answer
2022-06-06T16:36:51-0400

The Variance will be Var(Xˉ)=λ2n.Var(\bar{X})=\dfrac{\lambda^2}{n}.

μ=1000 h,n=40\mu=1000\ h, n=40


λ=1μ\lambda=\dfrac{1}{\mu}

β2=Var(nX)\beta^2=Var(nX)


β=Var(X)=λ2n\beta=\sqrt{Var(X)}=\sqrt{\dfrac{\lambda^2}{n}}

=140(1000)2=1.58×104=\sqrt{\dfrac{1}{40(1000)^2}}=1.58\times10^{-4}

Exponential distribution


P(X900)=P(X>900)P(X\ge900)=P(X>900)

=eβ(900)=e1.58×104(900)=0.867448=e^{-\beta(900)}=e^{-1.58\times10^{-4}(900)}=0.867448


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