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. What is the probability that the cost will be more than $450? What is the probability that the cost will be less than $250? What is the probability that the cost will be between $250 and $450? If the cost for your car repair is in the lower 5% of automobile repair charges, what is your cost?

Answer 1

Answer:

1) 0.943 2)0.092 3) 0.081 4) $222.24

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = $367

Standard Deviation, σ = $88

We are given that the cost for an automobile repair is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

a) P(cost is more than $450)

P(x > 450)

$P( x > 450) = P( z > \displaystyle\frac{450 - 367}{88}) = P(z > 0.943)$

$= 1 - P(z \leq 0.943)$

Calculation the value from standard normal z table, we have,  $P(x > 450) = 1 - 0.827= 0.173= 17.3\%$

b) P(cost is less than $250)

$P(x < 250) = P(z < \displaystyle\frac{250-367}{88}) = P(z < -1.329)$

Calculating the value from the standard normal table we have,

$P(z < -1.329) = 0.092 = 9.2\%\\P( x \leq 250) = 9.2\%$

c) P(cost between $250 and $450)

$P(250 \leq x \leq 450) = P(-1.329 \leq z \leq 0.943)\\\\= P(z \leq 0.943) - P(z < -1.329)\\= 0.173 - 0.092 = 0.081 = 8.1\%$

$P(250 \leq x \leq 450) = 8.1\%$

d) P(x<a) = 0.05

Calculating the value from the standard normal table we have

$P(z < -1.645) = 0.05\\\displaystyle\frac{a-367}{88} = -1.645\\\\a = (-1.645\times 88) + 367\\\\x = 222.24$

If the cost for your car repair is in the lower 5% of automobile repair charges, then the cost is less than $222.24.

Answer 2

The probability that the cost will be more than $450 is  0.173 and the probability that the cost will be less than $250 is 0.092 and  the probability that the cost will be between $250 and $450 is 0.081 If the cost for car repair is in the lower 5% of automobile repair charges, then cost is $222.24

What is probability ?

Probability is chances of occurring of an event.

Given that

Mean, μ = $367

Standard Deviation, σ = $88

Now  probability that the cost will be more than $450 can be calculated as

$\mathrm{P}(\mathrm{x} > 450) \\\\=P\left(z > \frac{450-367}{88}\right)\\\\=P(z > 0.943) \\\\&=1-P(z \leq 0.943)$

By using  standard normal table

$P(x > 450)=1-0.827=\\\\0.173$

Similarly probability that the cost will be less than $250 can be calculated as

$P(x < 250)=P\left(z < \frac{250-367}{88}\=P(z < -1.329) \\\\=0.092$

Similarly probability that the cost will be between $250 and $450 can be calculated as

$P(250 \leq x \leq 450)=P(-1.329 \leq z \leq 0.943) \\\\=0.173-0.092=0.081$

If the cost for car repair is in the lower 5% of automobile repair charges are

$\begin{aligned}&P(z < -1.645)=0.05 \\&\frac{a-367}{88}=-1.645 \\&a=(-1.645 \times 88)+367\end{aligned}$

a=222.24

The probability that the cost will be more than $450 is  0.173 and the probability that the cost will be less than $250 is 0.092 and  the probability that the cost will be between $250 and $450 is 0.081 If the cost for car repair is in the lower 5% of automobile repair charges, then cost is $222.24

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