Questions: Listed below are the annual tuition amounts of the 10 most expensive colleges in a country for a recent year. What does this "Top 10 " list tell us about the population of all of that country's college tuitions? 50,622, 52,519, 53,602, 50,554, 500,732, 52,519, 53,238, 51,635, 53,153 Find the mean, midrange, median, and mode of the data set. The mean of the data set is (Round to two decimal places as needed.) The midrange of the data set is (Round to two decimal places as needed.) The median of the data set is . (Round to two decimal places as needed.) What is (are) the mode(s) of the data set? Select the correct choice below and, if necessary, fill in the answer box within your choice.

Solution Steps

The mean \( \mu \) of the annual tuition amounts 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 tuition amounts. For the given data:

\[ \mu = \frac{918574}{9} = 102063.78 \]

The midrange is calculated as the average of the minimum and maximum values in the dataset:

\[ \text{Midrange} = \frac{\text{min}(x) + \text{max}(x)}{2} \]

For the tuition amounts, the minimum is \( 50554 \) and the maximum is \( 500732 \):

\[ \text{Midrange} = \frac{50554 + 500732}{2} = 275643.0 \]

To find the median, we first sort the data:

\[ \text{Sorted data} = [50554, 50622, 51635, 52519, 52519, 53153, 53238, 53602, 500732] \]

The median is the value at the position given by the formula:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (9 + 1) = 5.0 \]

Since the rank is \( 5.0 \), the median corresponds to the value at position \( 5 \):

\[ \text{Median} = 52519 \]

Final Answer