Question

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).

Answer 1

Answer:

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