Question

a class of 37 students took a test worth 100 points. the five-number summary of their scores is 55, 67, 73, 81, 98. identify the false statement.
o at least 25% of the students scored a 67 or below.
o at least 18 students scored below a 67.
o at least 9 students scored between 67 and 73.
o at least 25% of the students scored an 81 or above.

Explanation:

Step1: Recall five - number summary

The five - number summary is \( \text{Minimum}, Q_1, \text{Median}(Q_2), Q_3, \text{Maximum} \). Here, the five - number summary is \( 55, 67, 73, 81, 98 \). So, \( \text{Min}=55 \), \( Q_1 = 67 \), \( \text{Median}=73 \), \( Q_3 = 81 \), \( \text{Max}=98 \).

Interpretation of quartiles:

• Approximately \( 25\% \) of the data lies between \( \text{Min} \) and \( Q_1 \), \( 25\% \) between \( Q_1 \) and \( \text{Median} \), \( 25\% \) between \( \text{Median} \) and \( Q_3 \), and \( 25\% \) between \( Q_3 \) and \( \text{Max} \). Also, at least \( 25\% \) of the data is less than or equal to \( Q_1 \), at least \( 25\% \) is greater than or equal to \( Q_3 \), and the inter - quartile range (IQR) is \( Q_3 - Q_1 \), and at least \( 50\% \) of the data lies between \( Q_1 \) and \( Q_3 \).

Step2: Analyze each option

• Option 1: At least 25% of the students scored a 67 or below

Since \( Q_1 = 67 \), by the property of quartiles, at least \( 25\% \) of the data is less than or equal to \( Q_1 \). So this statement is true.

• Option 2: At least 18 students scored below a 67

The total number of students \( n = 37 \). \( 25\% \) of \( 37 \) is \( 0.25\times37=\frac{37}{4} = 9.25 \). The number of students with scores less than or equal to \( Q_1 = 67 \) is at least \( 25\% \) of \( 37 \), which is at least \( 9.25 \) (so at least 10 students). But the statement says at least 18 students scored below 67. \( 18\) is more than half of \( 37\) (\( 37\div2 = 18.5\)), and we know that \( Q_1 \) divides the data such that at most \( 25\% \) is below \( Q_1 \) (in a non - skewed distribution, it's approximately \( 25\% \)). So this statement is false. Let's check the other options for confirmation.

• Option 3: At least 9 students scored between 67 and 73

The proportion of data between \( Q_1 = 67 \) and \( \text{Median}=73 \) is at least \( 25\% \) (since the median divides the data into two halves, and the first half is from \( Q_1 \) to median in a sense). \( 25\% \) of \( 37 \) is \( 0.25\times37 = 9.25 \), so at least 9 students (since we can't have a fraction of a student) scored between 67 and 73. This statement is true.

• Option 4: At least 25% of the students scored an 81 or above

Since \( Q_3 = 81 \), by the property of quartiles, at least \( 25\% \) of the data is greater than or equal to \( Q_3 \). So this statement is true.

Answer:

At least 18 students scored below a 67.