Answer:
Step-by-step explanation:
Given that:
There are 3 sizes of dishwasher soaps.
Twenty percent of all purchasers select a 25-oz box, 50% select a 40-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.
Therefore, [tex]3 \times 3 = 9[/tex] random samples of size 2 are taken from the population.
The information is well represented in the table below.
[tex]x_1[/tex] [tex]x_2[/tex] [tex]p(x_1 \ x_2) = p(x_1)p(x_2)[/tex] [tex]\bar x = \dfrac{x_1+x_2}{2}[/tex] [tex]s^2= (x_1-\bar x)^2 +(x_2-\bar x)^2[/tex]
25 25 0.20×0.20=0.04 25 0
25 40 0.20×0.50=0.10 32.5 112.5
25 65 0.20×0.30= 0.06 45 800
40 25 0.50×0.20=0.10 32.5 112.5
40 40 0.50×0.50=0.25 40 0
40 65 0.50×0.30=0.15 52.5 312.5
65 25 0.30×0.20=0.06 45 800
65 40 0.30×0.50=0.15 52.5 312.5
65 65 0.30×0.30=0.09 65 0
The sample distribution of [tex]\bar x[/tex] is shown below as ;
[tex]\bar x[/tex] 25 32.5 40 45 52.5 65
[tex]p(\bar x)[/tex] 0.04 0.10+0.10 0.25 0.06+0.06 0.15+0.15 0.09
= 0.20 =0.12 =0.30
The value of [tex]E(\bar x) = \sum \bar x \ P(\bar x)[/tex]
= 25(0.04) + 32.5(0.20) + 40(0.25) + 45(0.12) + 52.5(0.30) + 65(0.09)
= 44.5 oz
From the same table above ;
the sampling distribution of [tex]{s^2}[/tex] is be shown below;
[tex]{s^2}[/tex] 0 112.5 312.5 800
0.04+0.25 0.10+0.10 0.15+0.15 0.06+0.06
+0.09
[tex]p(s)^2[/tex] = 0.38 =0.20 = 0.30 0.12
The value of [tex]E(s)^2 =\sum s^2 .p(s)^2[/tex]
= (0×0.38)+(112.5×0.20)+(312.5×0.30)+(800×0.12)
=0+112+93.75+96
=301.75
