Question

use the accompanying data set to complete the following actions.
a. find the quartiles.
b. find the interquartile range.
c. identify any outliers.

40 51 35 44 41 37 40 46 44 38 34 55 42 35 15 53 37 50 30 29

a. find the quartiles.
the first quartile, \\(q_1\\), is \square.
the second quartile, \\(q_2\\), is \square.
the third quartile, \\(q_3\\), is \square.
(type integers or decimals.)

b. find the interquartile range.
the interquartile range (iqr) is \square.
(type an integer or a decimal.)

c. identify any outliers. choose the correct choice below.
a. there exists at least one outlier in the data set at \square.
(use a comma to separate answers as needed.)
b. there are no outliers in the data set.

Explanation:

Sort the data set and determine the sample size

The given data set of size \(n = 20\) sorted in ascending order:

$$ 15, 29, 30, 34, 35, 35, 37, 37, 38, 40, 40, 41, 42, 44, 44, 46, 50, 51, 53, 55 $$

Calculate the quartiles

Using standard textbook/software methods for finding quartiles (splitting the sorted data set into halves):

• Lower half (first 10 values): \(15, 29, 30, 34, 35, 35, 37, 37, 38, 40\)
• \(Q_1\) is the median of the lower half:

$$ Q_1 = \frac{35 + 35}{2} = 35 $$

• Median (\(Q_2\)) of the entire data set (average of the 10th and 11th values):

$$ Q_2 = \frac{40 + 40}{2} = 40 $$

• Upper half (last 10 values): \(40, 41, 42, 44, 44, 46, 50, 51, 53, 55\)
• \(Q_3\) is the median of the upper half:

$$ Q_3 = \frac{44 + 46}{2} = 45 $$

Calculate the IQR and identify outliers

• Interquartile Range (IQR):

$$ \text{IQR} = Q_3 - Q_1 = 45 - 35 = 10 $$

• Outlier boundaries:

$$ \text{Lower Bound} = Q_1 - 1.5 \times \text{IQR} = 35 - 1.5(10) = 20 $$

$$ \text{Upper Bound} = Q_3 + 1.5 \times \text{IQR} = 45 + 1.5(10) = 60 $$

• Identify data points outside \([20, 60]\):
• The value \(15\) is less than \(20\), so it is an outlier.

Answer:

Question a

The first quartile, \(Q_1\), is <blank>35</blank>.
The second quartile, \(Q_2\), is <blank>40</blank>.
The third quartile, \(Q_3\), is <blank>45</blank>.

Question b

The interquartile range (IQR) is <blank>10</blank>.

Question c

• A. There exists at least one outlier in the data set at 15. (Correct answer)
• B. There are no outliers in the data set.