Question

question 14 points 2

the class took a math quiz, and the scores were \\{9, 8, 7, 9, 6, 9, 10, 8, 5, 9, 7, 6, 9, 9, 2, 8, 10, 7\\}. which score(s) were outliers?

\\(\bigcirc\\) 2

\\(\bigcirc\\) 10

Explanation:

Order the data set

Using the Outlier Effects knowledge point
First, we arrange the given data set of 18 scores in ascending order:

$$ \{2, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10\} $$

Find the median and quartiles

We calculate the median, first quartile (\(Q_1\)), and third quartile (\(Q_3\)) for the \(N = 18\) data points:

• Median (\(Q_2\)) is the average of the 9th and 10th values:

$$ \text{Median} = \frac{8 + 8}{2} = 8 $$

• \(Q_1\) is the median of the lower 9 values (1st to 9th):

$$ Q_1 = \text{5th value} = 7 $$

• \(Q_3\) is the median of the upper 9 values (10th to 18th):

$$ Q_3 = \text{14th value} = 9 $$

Calculate the Interquartile Range

We find the interquartile range (\(IQR\)):

$$ IQR = Q_3 - Q_1 = 9 - 7 = 2 $$

Determine the outlier boundaries

We establish the lower and upper boundaries for outliers using the \(1.5 \times IQR\) rule:

• Lower Boundary:

$$ Q_1 - 1.5 \times IQR = 7 - 1.5(2) = 7 - 3 = 4 $$

• Upper Boundary:

$$ Q_3 + 1.5 \times IQR = 9 + 1.5(2) = 9 + 3 = 12 $$

Identify the outlier scores

Any score strictly less than 4 or strictly greater than 12 is an outlier:

• The score \(2\) is less than \(4\), so it is an outlier.
• The score \(10\) is less than \(12\), so it is not an outlier.
• Therefore, the only outlier is \(2\).

Answer:

• 2 (Correct answer)
• 10