Question

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A particular brand of dishwasher soap is sold in three sizes: 30 az, 45 oz, and 6S oz Tventy pece tof al purchases se et tw_ztox 50% siet..s abex and the ,emaining 30% choose a 65-oz box. Let X1 and X2 denote the package sizes selected by two independently selected purchasers. (a) Determine the sampling distribution of x. 30 37.5 45 47.5 65 p(x) 25 30 09 Calculate E(x E(x) 48 Compare E(X) to μ. E(X) < μ (b) Determine the sampling distribution of the sample variance S2 112.5 200 145 pts?)38 300 Calculate E(S E(s2) 124

Answer 1

Given That

Sizes = 30, 45, 65

30 = 20% = 0.2

45 = 50% = 0.5

65 = 30% = 0.3

x2 x bar = (x1+x2)12 P(x bar)-P(x1)*P(x2) 0.2 0.2 0.04 2*0.2*0.5-0.2 0.5*0.5 0.25 2*0.3*0.2-0.12 2*0.3*0.5-0.3 0.3*0.3-0.09 P(x bar) X bar * P(x bar) var: (XI-X bar)^2 + (x2-х bar)^2 S.no 1.2 7.5 11.25 5.7 16.5 5.85 45 45 30 45 37.5 45 47.5 0.2 112.5 3 4 5 45 65 65 65 0.12 0.3 612 200 SUM 47.5 0.12 x bar P(x bar) E(X) 48 30 0.04 37.5 0.2 65 0.09 0.25 0.3 112.5 0.2 0.2 612.5 0.12 0.12 P(var) P(var) 0.3 0.3 0.04+0.25+0.09 0.38

a) Sampling distribution of $\bar{x}$

Possible ways

(30,30) = 1 possible ways

(30,45) (45,30) = 2

(45,45) = 1

(65,30) (30,65) = 2

(65,45) (45,65) = 2

(65,65) = 1

For 1 Possible ways probability = p(x1)*P(x2)

P(T)

For 2 possible ways probability = 2* p(x1)*P(x2)

P(T)

$\bar{x}$	30	37.5	45	47.5	55	65
P(T)	0.04	0.2	0.25	0.12	0.3	0.09

$E(\bar{x}) = \sum (\bar{x}*P(\bar{x}))$

From above excel table

E(z) = 〉 (a, * P(z)) = 48

$E(\bar{x}) to \mu$

$E(\bar{x}) = \mu$

b)Sampling Distribution of the sample variance

$s^{2}$	0	112.5	200	612.5
$P(s^{2})$	0.38	0.2	0.3	0.12

$E(s^{2})$ = (0*0.38) + (112.5*0.2) + (200*0.3) + (612.5*0.12)

= 0+ 22.5+60+73.5

= 156