Question

An artide 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) LetW be the sample average lifetime of 114 Duracell and be the sample averege lifetime of 144 Evereatly Energizer batteries. What is the mean value of - (i..., where is the distribution of - centered)? Does your answer depend on the specified sample sizes? The answer decreases as the sample size increases. The answer decreases as the sample size decreases. The answer increases as the sample size decreases The answer is the same Irrespective of the sample sizes. The answer increases as the sample size increases. - 7, and what is its standard deviation? (Round (b) Suppose the population standard deviations of lifetime are 1.6 hours for Durecell batteries and 3.0 hours for Eveready Energizer batteries. With the sample sizes given in pert (a), what is the variance of the statistic your answers to four decimal places.) variance = standard deviation - (c) For the sample sizes given in part (a), drew a picture of the approximate distribution curve of - (include a measurement scale on the horizontal axis), -1.0 -0.5 0.0 0.5 -0.2 0.0 0.2 0.4 0.6 0.8 10 -0.5 0.0 0.5 1.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 Would the shape of the curve necessarily be the same for sample sizes of 10 batteries of each type? Explain. 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. The shape is not necessarily that of a normal curve when the samples are of ze 10, because the distributions of X and Y are unknown and the Central Lirnit Theorem cannot be invoked.

Answer 1

sol: Given data is Population average lifetime of Duracell Mee 4.1 hours "Population average lifetime of Eveready battery thy 2 4.5 houry ca) No. of Duracell battergies Nx = 144 : No. oi Eveready batteries My = 144 then, Find ď and ý are sample average Duracell and Eveready batteries lifetime of 144 respectively. Now , ECX-7) - EC EU) - Hz - Mi = = 4.1-4.5 -0.4 Hence قعطعطم The mean value of B- ý does not depend on the Obition cdj is correct. d. The answer is same irrespective of the sample - size. CB) Standard deviation of lifetime of Duracell battery isy 1.6 Standard Slaviation of lifetime of Eveready battery - - Ox: 3.0 - The Variance of X- t is, - -YJ = Vitt Vio

(3.0) 144 144 = 0.0.178 +0.0625 = 0.0803 Standard deviation = Variance 2 10.0803 _ 26.9834 Variamce 20.0803 Standard deviation = 10.9834] observation lie is the Io we know that gg. 73 % - interwal =(ut 30) - H : -0.4 30. 28.34. = (-0.4 I 3* 0.2834) = (-0.4 + 0.8502) ; =(-1.2502 , 0.4502 This region is respresented by the 1st curve - Hences The option ist is correct (d) option Col is correct Chel The shape is not necessarily that of a normal curve when the sample are of size 10, berause the distribution of a andy are unknown and the central limit Thearem cornmat be mioked.