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Compute the z-score for a vehicle that obtained 36.3 miles per gallon.

(Give your answer to two decimal places.)

[tex] \[ z = \square \] [/tex]

Answer :

To find the z-score for a value of 36.3 miles per gallon, let's go through the steps to calculate it. A z-score tells us how many standard deviations an element is from the mean. Here's a simple breakdown of the process:

1. Determine the Mean ([tex]\(\mu\)[/tex]) of the Data:
- For z-score calculations, the mean is central to locating the data point in comparison to the rest. Here, it's given as 37.5.

2. Determine the Standard Deviation ([tex]\(\sigma\)[/tex]):
- The standard deviation measures the amount of variation or dispersion in a set of values. Here, it's given as 1.5.

3. Identify the Value ([tex]\(X\)[/tex]) for Which You Want the Z-Score:
- We need to find the z-score for 36.3 miles per gallon.

4. Use the Z-Score Formula:

[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]

Plug the values into the formula:

[tex]\[
z = \frac{36.3 - 37.5}{1.5}
\][/tex]

5. Calculate:

[tex]\[
z = \frac{-1.2}{1.5} \approx -0.8
\][/tex]

So, the z-score for a mileage of 36.3 miles per gallon is approximately [tex]\(-0.8\)[/tex]. This means that 36.3 mpg is 0.8 standard deviations below the mean.

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