Question

the tuition and fees (in thousands of dollars) for the top 14 universities in a recent year are listed below. find the mean, median, and mode of the data, if possible. if any of these measures cannot be found or a measure does not represent the center of the data, explain why.

47 44 35 44 47 39 47
44 42 44 47 42 46 44

a. the median cost is 44.
(round to one decimal place as needed.)
b. there is no median cost.

does the median represent the center of the data?

a. the median represents the center.
b. the median does not represent the center because it is not a data value.
c. the median does not represent the center because it is the largest data value.
d. the median does not represent the center because it is the smallest data value.
e. there is no median cost.

find the mode of the costs. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the mode(s) of the costs is (are) 44.
(round to one decimal place as needed. use a comma to separate answers as needed.)
b. there is no mode.

does (do) the mode(s) represent the center of the data?

a. the mode(s) represent(s) the center.
b. the mode(s) does (do) not represent the center because it (one) is the largest data value.
c. there is no mode.
d. the mode(s) does (do) not represent the center because it (they) is (are) not a data value.
e. the mode(s) does (do) not represent the center because it (one) is the smallest data value.

Explanation:

Sort the data set

Using the Median Calculation knowledge point
The given dataset consists of the tuition and fees (in thousands of dollars) for 14 universities:

$$ 47, 44, 35, 44, 47, 39, 47, 44, 42, 44, 47, 42, 46, 44 $$

Sorting these \(N = 14\) values in ascending order:

$$ 35, 39, 42, 42, 44, 44, 44, 44, 44, 46, 47, 47, 47, 47 $$

Calculate the median

Using the Median Calculation knowledge point
Since \(N = 14\) is even, the median is the mean of the two middle values at positions \(7\) and \(8\):

$$ \text{Median} = \frac{44 + 44}{2} = 44 $$

Thus, the median cost is \(44\) thousand dollars.

Interpret the median

Using the Measures of Central Tendency Interpretation knowledge point
The median of \(44\) lies in the middle of the sorted data set, with half of the values below or equal to \(44\) and half above or equal to \(44\). It is not an outlier and is representative of the center of the distribution. Therefore, the median represents the center.

Find the mode

Using the Mode of Data knowledge point
We count the frequency of each value in the sorted dataset:

• \(35\): 1 time
• \(39\): 1 time
• \(42\): 2 times
• \(44\): 5 times
• \(46\): 1 time
• \(47\): 4 times

The value with the highest frequency is \(44\) (occurring 5 times). Thus, the mode is \(44\).

Interpret the mode

Using the Measures of Central Tendency Interpretation and Mode of Data knowledge points
The mode is \(44\), which is also the median and lies near the center of the data set. Since it is the most typical value and positioned centrally, the mode represents the center of the data.

Answer:

Question 1

• A. The median cost is 44. (Correct answer)
• B. There is no median cost.

Question 2

• A. The median represents the center. (Correct answer)
• B. The median does not represent the center because it is not a data value.
• C. The median does not represent the center because it is the largest data value.
• D. The median does not represent the center because it is the smallest data value.
• E. There is no median cost.

Question 3

• A. The mode(s) of the costs is (are) 44. (Correct answer)
• B. There is no mode.

Question 4

• A. The mode(s) represent(s) the center. (Correct answer)
• B. The mode(s) does (do) not represent the center because it (one) is the largest data value.
• C. There is no mode.
• D. The mode(s) does (do) not represent the center because it (they) is (are) not a data value.
• E. The mode(s) does (do) not represent the center because it (one) is the smallest data value.