Mastery Check Solutions

1. If 70.2% of customers bring in their Model G every 8,435 miles, what is the standard deviation for oil changes to the nearest whole number?

$$\begin{gathered} \mathrm{P}\left(X<8435\right)=0.702 \\ z=0.53 \\ \begin{array}{c}0.53=\frac{8435–8255}{\sigma }\\ 0.53\sigma =180\\ \sigma =339.62\\ \sigma =340\end{array} \end{gathered}$$

2. What is the likelihood that a car is serviced at the manufacturer’s recommendation of every 7,500 miles?

$$\begin{gathered} \begin{array}{c}z=\frac{7500–8255}{340}\\ z=–2.22\\ \end{array} \\ \mathrm{P}\left(X<7500\right)=0.0132 \end{gathered}$$

There is a 1.32% chance a customer will get an oil change at 7,500 miles.

3. The same Model G cars need an alignment between 6,000 and 12,000 miles. On average, a customer has their car aligned every 10,500 miles with a standard deviation of 1,982 miles. What percentage of cars are in the given range?

$$\begin{gathered} z=\frac{6000–10500}{1982} \\ z=–2.27 \\ \mathrm{P}(X<6000)=0.0116 \end{gathered}$$

$$\begin{gathered} z=\frac{12000–10500}{1982} \\ z=0.756=0.76 \\ \mathrm{P}(X<12000)=0.7764 \\ 0.7764–0.0116=0.7648 \end{gathered}$$

Approximately 76% of Model G cars get an alignment between 6,000 and 12,000 miles.