Question

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question

Most Expensive Colleges Listed below are the annual costs (dollars) of tuition and fees at the 10 most expensive colleges in the United States for a recent year (based on data from U.S. News & World Report). The colleges listed in order are Columbia, Vassar, Trinity, George Washington, Carnegie Mellon, Wesleyan, Tulane, Bucknell, Oberlin, and Union. What does this “top 10” list tell us about those costs for the population of all U.S. college tuitions?

49,138 47,890 47,510 47,343 46,962 46,944 46,930 46,902 46,870 46,785

Short answer

(a) The mean is $47327.4.

(b) The median is $46,953.0.

(c) There is no mode.

(d) The midrange is $47,961.5.

The top 10 list can be used to infer that the cost at any U.S college would be less than the observed costs.

Step-by-step solution

Given information

The annual costs of tuition and fees of the most expensive colleges in the U.S. are listed as follows.

49,138
47,890
47,510
47,343
46,962
46,944
46,930
46,902
46,870
46,785

Compute mean

(a)

The mean of the sample of n observations is computed as:

$\overline{x}=\frac{\sum x}{n}$, where xrepresents the observations and nis the count of the observations.

Substitute the values in the formula.

$$\begin{gathered} \overline{x}=\frac{49138+47890+...+46785}{10} \\ =\frac{473274}{10} \\ \approx 47327.4 \end{gathered}$$

Thus, the mean value is approximately $47327.4.

Compute median

(b)

The median of n observations is computed as:

• If n is odd, the middlemost observation is the median.
• If n is even, the average of the two middle observations is the median.

The number of observations is10.

Arrange the observations in ascending order.

46,785
46,870
46,902
46,930
46,944
46,962
47,343
47,510
47,890
49,138

The middlemost observations are 46,944, and 46,962.

The median is given as:

$$\begin{gathered} M=\frac{46944+46962}{2} \\ =\frac{93906}{2} \\ =46953 \end{gathered}$$

Thus, the median is $46,953.0.

Compute mode

(c)

The mode is(are) the observation(s) with the maximum frequency value(s).

None of the values has the highest frequency.

Thus, there is no mode in the data.

Compute midrange

(d)

The midrange of the set of n observations is computed using the minimum and maximum values as shown below:

$\text{Midrange}=\frac{\text{Minimum}\text{value}+\text{Maximum}\text{value}}{2}$

Substitute the values in the formula.

$$\begin{gathered} \text{Midrange}=\frac{46785+49138}{2} \\ =\frac{95923}{2} \\ =47961.5 \end{gathered}$$

Thus, the midrange is $47,961.5.

Explain the cost of the population of all college fees as per the statistics

The sampled list of colleges comprises the 10 most expensive colleges in the United States. So, the sample includes only a certain number of samples, which are the most expensive in terms of fees. Thus, the sample is not a true representative of the population of costs in all U.S.colleges.

Therefore, the top 10 list of costs can only be used to infer that the costs at any other United States college in that year would be less than the sampled observations.