A car manufacturer is concerned about poor customer satisfaction at one of its d

Question

A car manufacturer is concerned about poor customer satisfaction at one of its dealerships. The management decides to evaluate the satisfaction surveys of its next 60 customers. The dealer will be fined if the number of customers who report favorably is between 42 and 49. The dealership will be dissolved if fewer than 42 customers report favorably. It is known that 72% of the dealer’s customers report favorably on satisfaction surveys. Use Table 1.

a. What is the probability that the dealer will be fined? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

Probability

b. What is the probability that the dealership will be dissolved? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

sad: piscecurwaneprobablSes,thats theins nor 4901 0A761 unu one probabl ses, th 09 199 ail86 ù8212 da238 t.iio4 asag B315 34> ai365 083as IL9C32 0.500.9066 0.0052 t.3099 09113 0.9131147 0.5102 0.3177 1.512,0332 0.5313 0.0357 a9370 2.3382 0.9301 00406 118 0.3420 age11 rs772 ana 09793 gs750 t3793 0979a 09923 .9ge asE12 09617 27 |L95G3 as9ic 09967 aose c.3500 ason 00071 .9972 a3573 a9071 3.3 |19595 0.5995 09995 a9996 t.33% 09996 09996 .9996 a39% 0.5997 3.5lL9see ase a 0999B 099 AF99 a3000 09999 a0999 t.3599 a%es 00099 ·0099 ase99 aeo9 M11 aot 1p00D 10000 100:o ince lasco 10030 10:0 1:00p 1,0coo

Explanation / Answer

Answer to the question)

given:

.

Part a) if 42 to 49 report favorably , then the dealership will be fined

x1 = 42 , x2 = 49

.

This is a question of binomial probability , but the question requires us to solve it use normal distribution

Hence we make use of normal approximation

Mean M = n*p = 60*0.72 = 43.2

Standard deviation s = sqrt(npq) = sqrt(60 * 0.72 *0.28) = 3.4779

.

The correction factor would be : x1' = 42-0.5 = 41.5

x2' = 49+0.5 = 49.5

.

P(41.5 < x < 49.5) = P(x < 49.5) - P(x < 41.5)

P(x < 41.5) = P( z < 41.5 - 43.2 / 3.48) = P(z < -0.49) = 0.3121

P(x < 49.5) = P(z < 49.5-43.2/3.48) = P(z < 1.81) = 0.9649

P(41.5 < x < 49.5) = 0.9649-0.3121 = 0.6528

.

Part b)

Fewer than 42 will dissolve

So we need to find P(x < 42)

conitnuity correction is x' = 42+0.5 = 42.5

P(x < 42.5) = P(z < 42.5-43.2/3.48) = P(z < -0.20) = 0.4207

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