Question

The Insurance Institute for Highway Safety (www.iihs. org, June 11,2009 ) published data on repair costs for cars involved in different types of accidents. In one study, seven different 2009 models of mini- and micro-cars were driven at 6 mph straight into a fixed barrier. The following table gives the cost of repairing damage to the bumper for each of the seven models. \begin{tabular}{|lc|} \hline Model & Repair Cost \\ \hline Smart Fortwo & \(\$ 1,480\) \\ Chevrolet Aveo & \(\$ 1,071\) \\ Mini Cooper & \(\$ 2,291\) \\ Toyota Yaris & \(\$ 1,688\) \\ Honda Fit & \(\$ 1,124\) \\ Hyundai Accent & \(\$ 3,476\) \\ Kia Rio & \(\$ 3,701\) \\ \end{tabular} a. Calculate and interpret the value of the median for this data set. b. Explain why the median is preferable to the mean for describing center in this situation.

Short answer

The median repair cost for the cars is \$1,688. The median is preferable to the mean in this scenario because it provides a better representation of the central tendency in situations where data is skewed by outliers.

Step-by-step solution

Sorting the data

First, sort all car repair costs in ascending order: \$1,071 (Chevrolet Aveo), \$1,124 (Honda Fit), \$1,480 (Smart Fortwo), \$1,688 (Toyota Yaris), \$2,291 (Mini Cooper), \$3,476 (Hyundai Accent), \$3,701 (Kia Rio)

Calculating the median

Find the median by identifying the middle number in the sorted list. The number of elements is seven, so the fourth (the middle) element onwards is the median. Therefore, the median repair cost is \$1,688, associated with the Toyota Yaris.

Interpret the median

The median is the middle term in a sorted dataset and acts as the central tendency in a distribution. It means half of the car models have repair costs less than or equal to \$1,688, and half have repair costs greater than or equal to \$1,688.

Discussing why the median is preferable to the mean in this data set

The mean can be skewed by outliers, meaning it could provide a misleading representation of the central tendency if there are significantly larger or smaller values. In this case, the Hyundai Accent and Kia Rio have notably higher repair costs that would pull the mean upward. The median, however, is not affected by the magnitude of extreme values hence it provides a better representation of the central tendency.