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.

a) Determine the sampling distribution of X.

x(bar)	30	32.5	35	47.5	50	65
P(x)

B) Calculate E(X). / Compare E(X) to μ.

C) Determine the sampling distribution of the sample variance S².

s^2	0	12.5	450	612.5
p(s^2)

D) Calculate E(S²). / Compare E(S²) to σ².

Answer 1

(a)

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

(b) Calculate E(X). / Compare E(X) to μ.

sum of all {x bar * p( x bar)} =( 30*0.04)+(32.5*0.20)+(47.5*0.12)+(35*0.25)+(50*0.30)+(65*0.09)

=1.2+6.5+5.7+8.75+15+5.83 = 43

(c) the sampling distribution of the sample variance S²

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

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

(d) Calculate E(S²). / Compare E(S²) to σ².

variance (s^2) = [ (X1-X2)^2 ] / 2

s^2	0	12.5	450	612.5
P(s^2)	0.38	0.20	0.30	0.12

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

            = 0+2.5+135+73.5

            =211

they are not equal ,E(s^2) = 2 times the variance