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Ms. Jamieson compared the math test scores of her students from her morning and afternoon classes. Which statement is true?




\multicolumn{3}{|c|}{Math Test Scores}

Leaf (Morning Class)		StemLeaf (Afternoon Class)

67

577		7022689

0245666799		8013334789

1335899		90026





AThe midrange of the afternoon test scores is greater than the midrange of the morning test scores.
BThe median of the morning scores is less than the median of the afternoon scores.
CThe range of the morning test scores is less than the range of the afternoon test scores.
DThe mode of the morning test scores is less than the mode of the afternoon test scores.

Verified Answer :

  • Calculate the statistical metrics for both classes:
  • Morning: Range = \(24\), Midrange = \(87.0\), Median = \(86.5\), Mode = \(86\).
  • Afternoon: Range = \(29\), Midrange = \(81.5\), Median = \(83.0\), Mode = \(83\).
  • Compare the metrics:
  • Range: \(24 < 29\) (True).
  • Midrange: \(81.5 < 87.0\) (False).
  • Median: \(83.0 < 86.5\) (False).
  • Mode: \(83 < 86\) (False).
  • The true statement is: \(\boxed{\text{The range of the morning test scores is less than the range of the afternoon test scores.}}\)

Explanation

  1. Extract the test scores

First, we extract the scores for each class from the provided stem-and-leaf plot.
For the Morning class, the scores are: \(75, 77, 77, 80, 82, 84, 85, 86, 86, 86, 87, 89, 89, 91, 93, 93, 95, 98, 99, 99\).
For the Afternoon class, the scores are: \(67, 70, 72, 72, 76, 78, 79, 80, 81, 83, 83, 83, 84, 87, 88, 89, 90, 90, 92, 96\).
Both datasets contain \(20\) scores.

  1. Calculate statistical measures

Next, we calculate the key statistical measures:

  • Range: The difference between the maximum and minimum score.
  • Morning: \(99 - 75 = 24\).
  • Afternoon: \(96 - 67 = 29\).
  • Midrange: The average of the maximum and minimum score.
  • Morning: \(\frac{75 + 99}{2} = 87.0\).
  • Afternoon: \(\frac{67 + 96}{2} = 81.5\).
  • Median: The average of the \(10^{th}\) and \(11^{th}\) values in the sorted list.
  • Morning: \(\frac{86 + 87}{2} = 86.5\).
  • Afternoon: \(\frac{83 + 83}{2} = 83.0\).
  • Mode: The most frequent score.
  • Morning: \(86\) (appears \(3\) times).
  • Afternoon: \(83\) (appears \(3\) times).
  1. Evaluate the statements and determine the true one

Now, we evaluate each of the given statements:

  • Midrange of the afternoon class is greater than the morning class: \(81.5 > 87.0\) is False.
  • Median of the morning class is less than the afternoon class: \(86.5 < 83.0\) is False.
  • Range of the morning class is less than the range of the afternoon class: \(24 < 29\) is True.
  • Mode of the morning class is less than the mode of the afternoon class: \(86 < 83\) is False.

Since the range of the morning test scores is \(24\) and the range of the afternoon scores is \(29\), the statement "The range of the morning test scores is less than the range of the afternoon test scores" is the only true statement.

Examples

In real life, understanding these statistics helps teachers or analysts compare the performance of different groups. For instance, if you're assessing student progress or product quality across shifts, the range helps you identify which group has more variability in its performance. Calculating the range is simple: \(\text{Range} = \text{Max} - \text{Min}\). In this case, \(\text{Morning Range} = 99 - 75 = 24\) and \(\text{Afternoon Range} = 96 - 67 = 29\). By comparing these values, one can quickly see that the afternoon class had a wider spread of test scores.

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