Question

 

Automobile repair costs continue to rise with the average cost now at $367 per repair.† Assume that the cost for an automobile repair is normally distributed with a standard deviation of $88. Answer the following questions about the cost of automobile repairs.

(a)

What is the probability that the cost will be more than $490? (Round your answer to four decimal places.)

(b)

What is the probability that the cost will be less than $260? (Round your answer to four decimal places.)

(c)

What is the probability that the cost will be between $260 and $490? (Round your answer to four decimal places.)

(d)

If the cost for your car repair is in the lower 5% of automobile repair charges, what is your maximum possible cost in dollars? (Round your answer to the nearest cent.)

Answer 1

Answer:

Given that ,

mean = \mu = $367

standard deviation = \sigma = $88

a)

P(x > 490) = 1 - P(x <490 )

= 1 - P((x - \mu ) / \sigma< ( 490-367) / 88)

= 1 - P(z <1.39)

= 1 - 0.9177 (Using standard normal table)

= 0.0823

Probability = 0.0823

P(x >490 ) = 0.0823

b)

P(x <260 ) = P((x - \mu ) / \sigma < (367-260 ) / 88)

= P(z < -1.21)

= 0.1131 Using standard normal table,

P(x < 260) =0.1131

Probability = 0.1131

c)

P(260 < x <490 ) = P((260-367 / 88) < (x - \mu ) / \sigma < (490-367 ) / 88) )

P(260 < x <490 ) = P( -1.21< z <1.39)

P(260 < x <490 ) = P(z < 01.39) - P(z <-

1.21 )

P(260 < x <490 ) = 0.9177 - 0.1131

P(290 < x <490 ) = 0.8046

Probability = 0.8046

d)

Using standard normal table,

P(Z < z) = 5%

P(Z < z) = 0.05

P(Z < -1.645 ) = 0.05

z = -1.645

Using z-score formula,

x = z * \sigma + \mu

x = -1.645 * 88 + 367

x = 222.24

Maximum cost = $ 222