Questionnaires designed to assess initial quality of new car models are sent to

Question

Questionnaires designed to assess initial quality of new car models are sent to randomly selected samples of buyers and overall quality scores are derived from responses to this questionnaire. Suppose that scores among cars in the luxury sedan category had approximately a normal distribution with mean 92 and s.d. 6.
(a) What proportion of scores in this category were greater than 100?
(b) What proportion were between 80 and 90?
(c) What is the upper 20% of initial quality scores in this category?
(d) Suppose the scores of cars in the economy sedan category were approximately normally distributed with mean 84 and s.d. 8. If the initial quality score of a particular luxury sedan model was 94 and the score of a particular economy sedan model was 89, which model was relatively higher within its category? Explain.

Explanation / Answer

Ans:

Given that

mean=92

stdandard deviation=6

a)

z=(100-92)/6=1.33

P(z>1.33)=0.0912

b)

z(80)=(80-92)/6=-2

z(90)=(90-92)/6=-0.33

P(-2<z<-0.33)=P(z<-0.33)-P(z<-2)=0.3694-0.0228=0.3467

c)P(Z>=z)=0.2

P(Z<=z)=0.8

z=0.8416

x=92+0.8416*6=97.05

Greater than 97.05 are in upper 20%

d)For economy:

mean=84

std. dev=8

z(89)=(89-84)/8=0.625

For luxury:

z(94)=(94-92)/6=0.33

As,z score is higher for economy,so economy model is relatively higher within its category.

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