Chapter 9.1, Problem 1E

An article in the November 1983 Consumer Reports compared various types of batteries. The average lifetimes of Duracell Alkaline AA batteries and Eveready Energizer Alkaline AA batteries were given as 4.1 hours and 4.5 hours, respectively. Suppose these are the population average lifetimes.

 a. Let $\bar{X}$ be the sample average lifetime of 100 Duracell batteries and $\bar{Y}$ be the sample average lifetime of 100 Eveready batteries. What is the mean value of $\bar{X}-\bar{Y}$(i.e., where is the distribution of $\bar{X}-\bar{Y}$ centered)? How does your answer depend on the specified sample sizes? b. Suppose the population standard deviations of lifetime are 1.8 hours for Duracell batteries and 2.0 hours for Eveready batteries. With the sample sizes given in part (a), what is the variance of the statistic $\bar{X}-\bar{Y}$ , and what is its standard deviation?

 c. For the sample sizes given in part (a), draw a picture of the approximate distribution curve of $\bar{X}-\bar{Y}$  (include a measurement scale on the horizontal axis). Would the shape of the curve necessarily be the same for sample sizes of 10 batteries of each type? Explain.