Answer:
a) The probability is the p-value of
, in which X is the gas mileage of the 2020 Honda Civic.
b) Your dad's truck is 1.65 standard deviations below the mean.
Step-by-step explanation:
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Normal with mean 22.6 miles per gallon (mpg) and standard deviation 5.2 mpg.
This means that ![\mu = 22.6, \sigma = 5.2](https://tex.z-dn.net/?f=%5Cmu%20%3D%2022.6%2C%20%5Csigma%20%3D%205.2)
a. What proportion of vehicles have worse gas mileage than the 2020 Honda Civic?
This is the p-value of Z, given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{X - 22.6}{5.2}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%2022.6%7D%7B5.2%7D)
In which X is the gas mileage of the 2020 Honda Civic.
b. My dad has a truck that gets around 14 mpg. How many standard deviations from the mean is my dad's truck?
This is Z when X = 14. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{14 - 22.6}{5.2}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B14%20-%2022.6%7D%7B5.2%7D)
![Z = -1.65](https://tex.z-dn.net/?f=Z%20%3D%20-1.65)
Your dad's truck is 1.65 standard deviations below the mean.