Consider two brands of batteries for calculators Xmax and Yc

Consider two brands of batteries for calculators, Xmax and Y-cell, manufactured independently by two companies. Assume that the lifetimes of each battery has a Normal distribution. The Xmax batteries has a mean of 1000 hours with a standard deviation of 100 hours, and Y-cell batteries has a mean of 1200 hours with a standard deviation of 200 hours. For each of the following questions, you must state the random variable you are using and the distribution assumption you make.

(a.) What is the probability of an Xmax battery will last at most 800 hours? What is this probability for a Y-cell battery?

(b.) What is the probability that the average lifetime of 400 Xmax batteries is less than 1012 hours?

(c.) Assume, two batteries are randomly selected, one Xmax and one Y-cell. What can you say about the difference D in their lifetimes? That is, find the distribution of D, it

Solution

(a.) What is the probability of an Xmax battery will last at most 800 hours? What is this probability for a Y-cell battery?

P(X<800) = P((X-mean)/s <(800-1000)/100)

=P(Z<-2) =0.0228 (From standard normal table)

P(Y<800) = P(Z<(800-1200)/200) = P(Z<-2) = 0.0228 (From standard normal table)

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(b.) What is the probability that the average lifetime of 400 Xmax batteries is less than 1012 hours?

P(xbar<1012) = P((xbar-mean)/(s/vn) <(1012-1000)/(100/sqrt(400)))

=P(Z<2.4) =0.9918 (From standard normal table)

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(c.) Assume, two batteries are randomly selected, one Xmax and one Y-cell. What can you say about the difference D in their lifetimes? That is, find the distribution of D, it