Question

A particular brand of dishwasher soap is sold in three sizes: 30 oz, 45 oz, and 70 oz. Twenty percent of all purchasers select a 30-oz box, 50% select a 45-oz box, and the remaining 30% choose a 70-oz box. Let X₁ and X₂ denote the package sizes selected by two independently selected purchasers.

(a) Determine the sampling distribution of

X.

x	30	37.5	45	50	57.5	70
p(x)

Calculate

E(X).

E(X) =

  oz

Compare

E(X)

to μ.

E(X) > μ

E(X) = μ

    

E(X) < μ

(b) Determine the sampling distribution of the sample variance S².

s2	0	112.5	312.5	800
p(s2)

Calculate E(S²).
E(S²) =  

Compare E(S²) to σ².

E(S²) > σ²

E(S²) = σ²

    

E(S²) < σ²

Answer 1

µ = E(X) = 0.2 * 30 + 0.5 * 45 + 0.3 * 70 = 49.5

E(X²) = 0.2 * 30² + 0.5 * 45² + 0.3 * 70² = 2662.5

Var(X) = E(X²) - (E(X))² = 2662.5 - 49.5² = 212.25

a)

X              P(X)

30            P(30,30) = 0.2² = 0.04

37.5         P(30,45) = 0.2 * 0.5 * 2 = 0.2

45            P(45,45) = 0.5² = 0.25

50            P(30,70) = 0.2 * 0.3 * 2 = 0.12

57.5         P(45,70) = 0.5 * 0.3 * 2 = 0.3

70            P(70,70) = 0.3² = 0.09

E(X) = 30 * 0.04 + 37.5 * 0.2 + 45 * 0.25 + 50 * 0.12 + 57.5 * 0.3 + 70 * 0.09 = 49.5

Option-B) E(X) = µ

b)

s²              P(s²)

0              P(30,30) + P(45,45) + P(70,70) = 0.2² + 0.5² + 0.3² = 0.38

112.5       P(30,45) = 0.2 * 0.5 * 2 = 0.2

312.5       P(45,70) = 0.5 * 0.3 * 2 = 0.3

800          P(30,70) = 0.2 * 0.3 * 2 = 0.12

E(s²) = 0 * 0.38 + 112.5 * 0.2 + 312.5 * 0.3 + 800 * 0.12 = 212.25

Option-B) E(S²) = σ²