Automobile Repair Costs. Automobile repair costs continue to...

Automobile Repair Costs. Automobile repair costs continue to rise with the average cost now at $\$ 367$ per repair (U.S. News \& World Report website). 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 $\$ 450 ?$ 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?

Automobile Repair Costs. Automobile repair costs continue to rise with the average cost now at $\$ 367$ per repair (U.S. News \& World Report website). 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 $\$ 450 ?$
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?

Key Concepts

Percentiles

Percentiles indicate the value below which a given percentage of observations in a distribution falls. They are used to determine thresholds or cut-off points in a dataset, such as identifying the cost that separates the lowest 5% from the rest of the data.

Probability Calculations

Probability calculations in the context of the normal distribution involve finding the area under the curve corresponding to a given range of values. This can be used to determine the likelihood of certain outcomes, such as values above, below, or between specific points.

Z-Score

The z-score is a measure that describes a value's position relative to the mean of a group of values, expressed in terms of standard deviations. It enables the standardization of different datasets for comparison and facilitates the use of the standard normal table for probability calculations.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. This concept is central in statistics as many natural phenomena are approximately normally distributed.

Transcript

00:01 So it says that automobile repair costs are normally distributed.

00:05 You have your mean of $367 and your standard deviation of $88.

00:08 So we're going to use the z -score formula here, which is x minus mean over standard deviation.

00:16 And we're going to use $367 for the mean and $88 for the standard deviation.

00:22 So the first one asks for more than $450.

00:25 So we're talking about to the right side of the normal curve.

00:28 And so we're going to take that $450 and put it in the formula right there.

00:32 $450 minus $367 divided by $88 gives you a z -score of .94.

00:42 So it's basically one deviation above average.

00:46 To find that probability from .94 to the right, you can use your normal cdf button on your calculator, where you'd put in .94 comma infinity, or a big number.

00:58 Or you could go to your table and look up .94, and then you'd have to take one minus that because we want the right side.

01:05 In the end, the answer ends up being .17, 17 % chance that it's more than $450.

01:16 The next one's asking for less than $250.

01:19 So now we're talking about the left side of the curve.

01:21 So i'm going to take a $250 and put in that into the formula.

01:27 $250 minus $367 divided by $188 is a negative 1 .33.

01:37 So it's a little more than one standard deviation below the mean.

01:40 So again, we're going to use the table.

01:41 This time, you can just look up the table...