Questions: An insurance company crashed four cars of the same model at 5 miles per hour. The costs of repair for each of the four crashes were 438, 420, 460, and 220. Compute the mean, median, and mode cost of repair.

Solution Steps

The mean cost of repair is calculated using the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} \]

where \( N \) is the number of data points and \( x_i \) are the individual costs. For our data:

\[ \mu = \frac{438 + 420 + 460 + 220}{4} = \frac{1538}{4} = 384.5 \]

Thus, the mean cost of repair is \( 384.5 \).

To find the median, we first sort the data:

\[ \text{Sorted data} = [220, 420, 438, 460] \]

The formula for the rank of the median \( Q \) is given by:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (4 + 1) = 2.5 \]

Since the rank is not an integer, we take the average of the values at ranks 2 and 3:

\[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{420 + 438}{2} = 429.0 \]

Thus, the median cost of repair is \( 429.0 \).

The mode is the value that appears most frequently in the dataset. In this case, all values are unique, so we list all the costs:

\[ \text{Mode} = [438, 420, 460, 220] \]

Since there is no single mode, we can state that all values are modes.

Final Answer