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 ques- tions about the cost of automobile repairs. a. what is the probability that the cost will be more than $450? b. what is the probability that the cost will be less than $250? c. what is the probability that the cost will be between $250 and $450? d. if the cost for your car repair is in the lower 5% of automobile repair charges, what is your cost?

Using the normal distribution, we have that:

a) There is a 0.8264 - 82.64% probability that the cost will be more than $450.

b) There is a 0.0918 = 9.18% probability that the cost will be less than $250.

c) There is a 0.7346 = 73.46% probability that the cost will be between $250 and $450.

d) The cost will be of at most $222.

In a normal distribution with mean and standard deviation , the z-score of a measure X is given by:

It measures how many standard deviations the measure is from the mean. After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.  

In this problem:

Item a:

This probability is 1 subtracted by the p-value of Z when X = 450, so:

has a p-value of 0.8264.

0.8264 - 82.64% probability that the cost will be more than $450.

Item b:

This probability is the p-value of Z when X = 250, so:

has a p-value of 0.0918.

0.0918 = 9.18% probability that the cost will be less than $250.

Item c:

This probability is the p-value of Z when X = 450 subtracted by the p-value of Z when X = 250, so:

0.8264 - 0.0918 = 0.7346.

0.7346 = 73.46% probability that the cost will be between $250 and $450.

Item d:

The cost is of at most the 5th percentile, which is X when Z has a p-value of 0.05, so X when Z = -1.645.

The cost will be of at most $222.

A similar problem is given at link

a. What is the probability that the cost will be more than $450?

We are supposed to find P(x > 450)

Formula :

Refer the z table :

P(z<0.9431)=0.8264

P(z> 0.9431)=1-P(z<0.9431)=1- 0.8264= 0.1736

Hence the probability that the cost will be more than $450 is 0.1736

b)What is the probability that the cost will be less than $250?

We are supposed to find P(x<250)

Formula :

Refer the z table :

P(z<-1.3295)=0.0934

Hence the probability that the cost will be less than $250 is 0.0934

c)What is the probability that the cost will be between $250 and $450?

P(250<z<450)=P(z<450)-P(z<250) = 0.8264 - 0.0934 =0.733

Hence the probability that the cost will be between $250 and $450 is 0.733

d)  If the cost for your car repair is in the lower 5% of automobile repair charges, what is your cost?

p = 0.05

refer the z table

z = -1.65

Formula :

Hence If the cost for your car repair is in the lower 5% of automobile repair charges, so, cost is $221.8

step-by-step explanation:

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