A particular brand of dishwasher soap is sold in three sizes: 35 oz, 45 oz, and 65 oz. Twenty percent of all purchasers select a 35-oz box, 50% select a 45-oz box, and the remaining 30% choose a 65-oz box. Let x1 and x2 denote the package sizes select by two independently selected purchasers. Determine the sampling distribution of X. Calculate E (X). Determine the sampling distribution of the sample variance S2. Calculate E (S2).

E (X) = 44.5

E (S2) = б^2 = 212.25

Step-by-step explanation:

Given:

- The possible value of x are: 25,40,65 with probabilities 0.2,0.5,0.3, respectively

Solution:

- There could be nine possible cases for the joint probability distribution.

[ x_1; x_2; p (x_1, x_2) = p (x_1) * p (x_2) ]

x_bar = (x_1 + x_2) / 2, s^2 = (x_1 - x_bar) ^2 + (x_2 - x_bar) ^2

- Now for all 9 possible cases we have:

x_1 = 25; x = 25; p (x_1, x_2) = 0.04; x_bar = 25; s^2 = 0

x_1 = 25; x = 40; p (x_1, x_2) = 0.1; x_bar = 32.5; s^2 = 112.5

x_1 = 25; x = 65; p (x_1, x_2) = 0.06; x_bar = 45; s^2 = 800

x_1 = 40; x = 25; p (x_1, x_2) = 0.1; x_bar = 32.5; s^2 = 112.5

x_1 = 40; x = 40; p (x_1, x_2) = 0.25; x_bar = 40; s^2 = 0

x_1 = 40; x = 65; p (x_1, x_2) = 0.15; x_bar = 52.5; s^2 = 312.5

x_1 = 65; x = 25; p (x_1, x_2) = 0.06; x_bar = 45; s^2 = 800

x_1 = 65; x = 40; p (x_1, x_2) = 0.15; x_bar = 52.5; s^2 = 312.5

x_1 = 65; x = 65; p (x_1, x_2) = 0.09; x_bar = 65; s^2 = 0

- The probability distribution of x_bar:

x_bar = 25 32.5 40 45 52.5 65

P (x) = 0.04 0.20 0.25 0.12 0.30 0.09

- Expected value of x_bar:

E (x_bar) = sum (x_bar*p (x))

= 25 * (0.04) + 32.5 * (0.02) + 40 * (0.25) + 45 * (0.12) + 52.5 * (0.3) + 65 * (0.09)

= 1 + 6.5 + 10 + 5.4 + 15.75 + 5.85

= 44.5

- The population mean is given by:

u = E (X) = sum (x * P (x))

= 25*0.2 + 40*0.5 + 65*0.3

= 44.5

- The probability distribution of s ^2:

s^2 = 0 112.5 312.5 800

P (s^2) = 0.38 0.20 0.30 0.12

- Expected value of s^2:

E (s^2) = sum (s^2*p (s^2))

= 0 * (0.38) + 112.5 * (0.02) + 312.5 * (0.30) + 800 * (0.12)

= 212.25

- The population standard deviation is given by:

б^2 = E (X^2) - [E (X) ]^2

= 25^2*0.2 + 40^2*0.5 + 65^2*0.3 - 44.5^2

= 2192.5 - 1980.25

= 212.25

- Hence, E (S2) = б^2 = 212.25