Question

Automobile repair costs continue to rise with the average cost now at $367 per repair (U.S. News & World Report website, January 5, 2015). 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 $450 (to four decimals)? b. What is the probability that the cost will be less than $250 (to four decimals)? c. What is the probability that the cost will be between $250 and $450 (to four decimals)? d. If the cost for your car repair is in the lower 5% of automobile repair charges, what is your cost (to two decimals)?

Answer 1

Solution :

Given that,

mean = $\mu$ = 367

standard deviation = $\sigma$ = 88

A ) P (x > 450)

= 1 - P (x < 450 )

= 1 - P ( x -  $\mu$/ $\sigma$ ) < ( 450 - 367 / 88)

= 1 - P ( z < 83 / 88 )

= 1 - P ( z < 0.94 )

Using z table

= 1 - 0.8264

= 0.1736

Probability = 0.1736

B) P( x <250)

P ( x - $\mu$ / $\sigma$ ) < ( 250 - 367 / 88)

P ( z < - 117 / 88 )

P ( z < -1.33 )

= 0.0918

Probability =0.0918

C ) P (250 < x < 450 )

P ( 250 - 367 / 88) < ( x -  $\mu$/ $\sigma$ ) < ( 331 - 367 / 88)

P ( - 117 / 88 < z < 83 / 88 )

P (-1.33 < z < 0.9)

P ( z < - 0.259 ) - P ( z < -.1.33 )

Using z table

= 0.8264 -0.0918

= 0.7346

Probability = 0.7346

a ) P( Z < z) = 5%  

P(Z < z) = 0.05

z = 0.41

Using z-score formula,

x = z * $\sigma$ + $\mu$

x = -1.64 * 88 + 367

x = 222.68