Question

Crash tests on new cars.The National Highway Traffic Safety Administration (NHTSA) crash-tests new car models to determine how well they protect the driver and front-seat passenger in a head-on collision. The NHTSA has developed a “star” scoring system for the frontal crash test, with results ranging from one star (*) to five stars (*****). The more stars in the rating, the better the level of crash protection in a head-on collision. The NHTSA crashtest results for 98 cars (in a recent model year) are stored in the accompanying data file.

a. The driver-side star ratings for the 98 cars are summarized in the Minitab printout shown below. Use the information in the printout to form a pie chart. Interpret the graph.

Tally for Discrete Variables: DRIVSTAR
DRIVSTAR	Count	Percent
2 3 4 5 N =	4 17 59 18 98	4.08 17.35 60.20 18.37

b. One quantitative variable recorded by the NHTSA is the driver’s severity of head injury (measured on a scale from 0 to 1,500). The mean and standard deviation for the 98 driver head-injury ratings are displayed in the Minitab printout below. Give a practical interpretation of the mean.

Descriptive Statistics: DRIVHEAD
Variable	N	Mean	StDev	Minimum	Q1	Median	Q3	Maximum
DRIVHEAD	98	603.7	185.4	216.0	475.0	605.0	724.3	1240.0

C. Use the mean and standard deviation to make a statement about where most of the head-injury ratings fall.

d..Find the z-score for a driver head-injury rating of 408. Interpret the result.

Short answer

a.

b. Mean = 603.7

c. 47.5 and 1,159.9

d. -1.05

Step-by-step solution

 Step 1: Constructing a pie chart and interpreting it

Based on the percentages, we can calculate the angles for the chart.

$\mathrm{Angle}=\frac{\mathrm{Percent}}{100}\times {360}^{\mathrm{o}}$

Based on the pie chart majority of the 98 cars have received a 4-star rating.

Giving a practical interpretation of mean

603.7 is the mean head injury score. Means on average, the severity of a driver’s head injury is 603.7 while measured on a scale of 1 to 1500.

Interpreting the location of most of the head injury ratings

Mean = 603.7 and standard deviation = 185.4

As we do not know the shape of the distribution, we will use the Chebyshev rule to interpret the location. Therefore according to the rule, 89% of the ratings should lie between $\overline{\mathrm{x}}\pm 3\mathrm{s}$.

Using this we will calculate the interval.

$\begin{array}{c}\stackrel{-}{\chi }-3\text{s}=603.7-3\left(185.4\right)\\ =603.7-556.2\\ =47.5\end{array}$

$\begin{array}{c}\stackrel{-}{\chi }+3\text{s}=603.7+3\left(185.4\right)\\ =603.7+556.2\\ =1159.9\end{array}$

Therefore, most of the head injury ratings should fall between 47.5 and 1,159.9.

Finding the z-score for a driver's head-injury rating of 408 and interpret the result

$\begin{array}{c}\text{z-score=}\frac{\text{Value}-\text{Mean}}{\text{Standarddeviation}}\\ \text{=}\frac{\text{408}-\text{603.7}}{\text{185.4}}\\ \text{=}\frac{-\text{195.7}}{\text{185.4}}\\ \text{=}-\text{1.05}\end{array}$

As the z-score lies below -3, it is not an outlier.