Question

Suppose you buy a new car whose advertised mileage is 25 miles per gallon (mpg). After driving your car for several months, you find that its mileage is 21.4mpg. You telephone the manufacturer and learn that the standard deviation of gas mileages for all cars of the model you bought is 1.15 mpg.

Part (a): Find the z-score for the gas mileage of your car, assuming the advertised claim is correct.

Part (b): Does it appear that your car is getting unusually low gas mileage?

Short answer

Part (a): The z-score for the gas mileage of your car is $-3.13$.

Part (b): Yes. Assuming the advertise claim is correct, the three standard deviations rule implies that your car's mileage is lower than most other cars of the that model.

Step-by-step solution

Part (a) Step 1. Given information.

Consider the given question,

The mean mileage is 21.4 mpg and standard deviation is 1.15 mpg.

Part (a) Step 2. Find the z-score for the gas mileage of your car.

On assuming the advertised claim correct which is buying a new car whose advertised mileage is 25 miles per gallon (mpg).

The z-score for the gas mileage of your car,

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{21.4-25}{1.15} \\ =-3.1304 \\ \approx -3.13 \end{gathered}$$

Part (b) Step 1. Check if it appears that your car is getting unusually on low gas mileage.

For unusual mileage we have to find the one, two and three standard deviation of the mean.

Unusual observations lie between the ranges $\overline{x}\pm \sigma$,

$$\begin{gathered} =\left(\overline{x}-\sigma ,\overline{x}+\sigma \right) \\ =\left(25-1.15,25+1.15\right) \\ =\left(23.85,26.15\right) \end{gathered}$$

Unusual observations lie between the ranges $\overline{x}\pm 2\sigma$,

$$\begin{gathered} =\left(\overline{x}-2\sigma ,\overline{x}+2\sigma \right) \\ =\left(25-2\times 1.15,25+2\times 1.15\right) \\ =\left(21.55,28.45\right) \end{gathered}$$

By observing the three standard deviation rule implies that your car's mileage is lower than most other cars of the model.

Hence, our assumption the advertised claim is correct.