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 X be the sample average lifetime of 121 Duracell and Y be the sample average lifetime of 121 Eveready Energizer batteries. What is the mean value of X - Y (i.e., where is the distribution of X - Y centered)? Does your answer depend on the specified sample sizes? O The answer is the same irrespective of the sample sizes. O The answer decreases as the sample size increases. O The answer increases as the sample size increases. O The answer increases as the sample size decreases. O The answer decreases as the sample size decreases. (b) Suppose the population standard deviations of lifetime are 1.7 hours for Duracell batteries and 2.0 hours for Eveready Energizer batteries. With the sample sizes given in part (a), what is the variance of the statistic X - Y, and what is its standard deviation? (Round your answers to four decimal places.) variance = standard deviation = (c) For the sample sizes given in part (a), draw a picture of the approximate distribution curve of X - Y (include a measurement scale on the horizontal axis). 0.1 0.0 0.2 0.3 0.4 0.5 0.5 0.6 1.0 0.7 -1.0 -0.7 -0.6 -0.5 0.0 -0.5 -0.4 -0.3 -0.2 -0.1 Would the shape of the curve necessarily be the same for sample sizes of 10 batteries of each type? Explain. O The shape is not necessarily that of a normal curve when the samples are of size 10, because the distributions of X and Y are unknown and the Central Limit Theorem cannot be invoked. O The shape will be that of a normal curve when the samples are of size 10, because even though the distributions of X and Y are unknown, the Central Limit Theorem can be invoked.
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 X be the sample average lifetime of 121 Duracell and Y be the sample average lifetime of 121 Eveready Energizer batteries. What is the mean value of X - Y (i.e., where is the distribution of X - Y centered)? Does your answer depend on the specified sample sizes? O The answer is the same irrespective of the sample sizes. O The answer decreases as the sample size increases. O The answer increases as the sample size increases. O The answer increases as the sample size decreases. O The answer decreases as the sample size decreases. (b) Suppose the population standard deviations of lifetime are 1.7 hours for Duracell batteries and 2.0 hours for Eveready Energizer batteries. With the sample sizes given in part (a), what is the variance of the statistic X - Y, and what is its standard deviation? (Round your answers to four decimal places.) variance = standard deviation = (c) For the sample sizes given in part (a), draw a picture of the approximate distribution curve of X - Y (include a measurement scale on the horizontal axis). 0.1 0.0 0.2 0.3 0.4 0.5 0.5 0.6 1.0 0.7 -1.0 -0.7 -0.6 -0.5 0.0 -0.5 -0.4 -0.3 -0.2 -0.1 Would the shape of the curve necessarily be the same for sample sizes of 10 batteries of each type? Explain. O The shape is not necessarily that of a normal curve when the samples are of size 10, because the distributions of X and Y are unknown and the Central Limit Theorem cannot be invoked. O The shape will be that of a normal curve when the samples are of size 10, because even though the distributions of X and Y are unknown, the Central Limit Theorem can be invoked.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text: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 X be the sample average lifetime of 121 Duracell and Y be the sample average lifetime of 121 Eveready Energizer
batteries. What is the mean value of X - Y (i.e., where is the distribution of X - Y centered)?
Does your answer depend on the specified sample sizes?
The answer is the same irrespective of the sample sizes.
The answer decreases as the sample size increases.
The answer increases as the sample size increases.
O The answer increases as the sample size decreases.
The answer decreases as the sample size decreases.
(b) Suppose the population standard deviations of lifetime are 1.7 hours for Duracell batteries and 2.0 hours for Eveready
Energizer batteries. With the sample sizes given in part (a), what is the variance of the statistic X - Y, and what is its
standard deviation? (Round your answers to four decimal places.)
variance =
standard deviation =
(c) For the sample sizes given in part (a), draw a picture of the approximate distribution curve of X - Y (include a measurement
scale on the horizontal axis).
0.1
0.0
0.2 0.3 0.4
0.5
0.5
1.0
0.6 0.7
-1.0
-0.5
-0.7 -0.6 -0.5 -0.4
0.0
-0.3 -0.2 -0.1
Would the shape of the curve necessarily be the same for sample sizes of 10 batteries of each type? Explain.
The shape is not necessarily that of a normal curve when the samples are of size 10, because the distributions of X
and Y are unknown and the Central Limit Theorem cannot be invoked.
O The shape will be that of a normal curve when the samples are of size 10, because even though the distributions of X
and Y are unknown, the Central Limit Theorem can be invoked.
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