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Math  /  Data & Statistics

Question37. A particular brand of dishwasher soap is sold in three sizes: 25oz,40oz25 \mathrm{oz}, 40 \mathrm{oz}, and 65 oz . Twenty percent of all purchasers select a 25 oz box, fifty percent select a 40 oz box, and the remaining thirty percent choose a 65 oz box. Let X1X_{1} and X2X_{2} denote the package sizes selected by two independently selected purchasers. a. Determine the sampling distribution of Xˉ\bar{X}, calculate E(Xˉ)E(\bar{X}), and compare to μ\mu. b. Determine the sampling distribution of the sample variance S2S^{2}, calculate E(S2)E\left(S^{2}\right), and compare to σ2\sigma^{2}.

Studdy Solution

STEP 1

1. The random variable X X represents the size of the dishwasher soap package selected by a purchaser.
2. The probabilities for each package size are: P(X=25)=0.2 P(X = 25) = 0.2 , P(X=40)=0.5 P(X = 40) = 0.5 , P(X=65)=0.3 P(X = 65) = 0.3 .
3. X1 X_1 and X2 X_2 are independent and identically distributed (i.i.d.) random variables.
4. The sample mean Xˉ\bar{X} is given by Xˉ=X1+X22\bar{X} = \frac{X_1 + X_2}{2}.
5. The sample variance S2S^2 is given by S2=(X1−Xˉ)2+(X2−Xˉ)21S^2 = \frac{(X_1 - \bar{X})^2 + (X_2 - \bar{X})^2}{1}.

STEP 2

1. Determine the probability distribution of XX.
2. Calculate the expected value E(X)E(X) and variance σ2\sigma^2 of XX.
3. Determine the sampling distribution of Xˉ\bar{X} and calculate E(Xˉ)E(\bar{X}).
4. Compare E(Xˉ)E(\bar{X}) to μ\mu.
5. Determine the sampling distribution of S2S^2 and calculate E(S2)E(S^2).
6. Compare E(S2)E(S^2) to σ2\sigma^2.

STEP 3

Determine the probability distribution of XX:
P(X=25)=0.2,P(X=40)=0.5,P(X=65)=0.3P(X = 25) = 0.2, \quad P(X = 40) = 0.5, \quad P(X = 65) = 0.3

STEP 4

Calculate the expected value E(X)E(X):
E(X)=25×0.2+40×0.5+65×0.3=5+20+19.5=44.5E(X) = 25 \times 0.2 + 40 \times 0.5 + 65 \times 0.3 = 5 + 20 + 19.5 = 44.5
Calculate the variance σ2\sigma^2:
E(X2)=252×0.2+402×0.5+652×0.3=125+800+1267.5=2192.5E(X^2) = 25^2 \times 0.2 + 40^2 \times 0.5 + 65^2 \times 0.3 = 125 + 800 + 1267.5 = 2192.5
σ2=E(X2)−(E(X))2=2192.5−44.52=2192.5−1980.25=212.25\sigma^2 = E(X^2) - (E(X))^2 = 2192.5 - 44.5^2 = 2192.5 - 1980.25 = 212.25

STEP 5

Determine the sampling distribution of Xˉ\bar{X}:
Since X1X_1 and X2X_2 are i.i.d., Xˉ=X1+X22\bar{X} = \frac{X_1 + X_2}{2}.
Calculate E(Xˉ)E(\bar{X}):
E(Xˉ)=E(X1+X22)=E(X1)+E(X2)2=44.5+44.52=44.5E(\bar{X}) = E\left(\frac{X_1 + X_2}{2}\right) = \frac{E(X_1) + E(X_2)}{2} = \frac{44.5 + 44.5}{2} = 44.5

STEP 6

Compare E(Xˉ)E(\bar{X}) to μ\mu:
Since E(Xˉ)=44.5E(\bar{X}) = 44.5 and μ=44.5\mu = 44.5, they are equal.

STEP 7

Determine the sampling distribution of S2S^2:
S2=(X1−Xˉ)2+(X2−Xˉ)21S^2 = \frac{(X_1 - \bar{X})^2 + (X_2 - \bar{X})^2}{1}
Calculate E(S2)E(S^2):
Since X1X_1 and X2X_2 are i.i.d., E(S2)=σ2=212.25E(S^2) = \sigma^2 = 212.25.

STEP 8

Compare E(S2)E(S^2) to σ2\sigma^2:
Since E(S2)=212.25E(S^2) = 212.25 and σ2=212.25\sigma^2 = 212.25, they are equal.
The expected value of the sample mean Xˉ\bar{X} is equal to the population mean μ\mu, and the expected value of the sample variance S2S^2 is equal to the population variance σ2\sigma^2.

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