A particular brand of dishwasher soap is sold in three sizes: 30 oz, 35 oz, and

Question

A particular brand of dishwasher soap is sold in three sizes: 30 oz, 35 oz, and 65 oz. Twenty percent of all purchasers select a 30-oz box, 50% select a 35-oz box, and the remaining 30% choose a 65-oz box. Let X1 and X2 denote the package sizes selected by two Independently selected purchasers. (Give answers accurate to 2 decimal places.) (a) Determine the sampling distribution of pR) Calculate E(X) and compare to u Calculate the variance of X. (b) Determine the sampling distribution of the sample variance S2. (Hint: Note that S2-(X1 -X2)2/2) s2 p(s2) Calculate E(S2) and compare to 2.

Explanation / Answer

E(XBar) = Sum of all { Xbar*[P(Xbar)] } = 1.2 + 6.5 + 5.7 + 8.75 + 15 + 5.85 = 43

Mean = 0.2*30 + 0.5*35 + 0.3* = 43

THEY ARE EQUAL!

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Var(Xbar) = Sum of all [ { (Xbar - Mean(Xbar))^2 } * P (X = Xbar) ] ..... for each Xbar

= [(30-43)^2]*(0.04) + ... + [(65-43)^2]*(0.09) = 105.5

S^2 = [ (X1-X2)^2 ] / 2

=> E(s^2) = 0*(0.38) + (12.5)*(0.20) + (450)*(0.30) + (612.5)*(0.12) = 211

THEY ARE NOT EQUAL; E(s^2) = 2 times the variance (sigma^2)

Xbar 30 32.5 47.5 35 50 65 P(Xbar) 0.04 0.20 0.12 0.25 0.30 0.09

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