Question

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

41 52 36 43 42 38 40 46 45 38 36 56 42 35 15 53 38 49 2

a. find the quartiles.
the first quartile, \\(q_1\\), is 36.
the second quartile, \\(q_2\\), is 40.5.
the third quartile, \\(q_3\\), is 45.5.
(type integers or decimals.)

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

Explanation:

Sort the dataset

Using the Percentiles and Quantiles knowledge point
The given dataset contains 18 values:

$$ 41, 52, 36, 43, 42, 38, 40, 46, 45, 38, 36, 56, 42, 35, 15, 53, 38, 49 $$

Sorted in ascending order:

$$ 15, 35, 36, 36, 38, 38, 38, 40, 41, 42, 42, 43, 45, 46, 49, 52, 53, 56 $$

Calculate the quartiles

Using the Percentiles and Quantiles knowledge point

• The first quartile \(Q_1\) is the median of the lower half (first 9 values):

$$ Q_1 = 38 $$

• The second quartile \(Q_2\) (median) is the average of the 9th and 10th values:

$$ Q_2 = \frac{41 + 42}{2} = 41.5 $$

• The third quartile \(Q_3\) is the median of the upper half (last 9 values):

$$ Q_3 = 46 $$

(Note: The values shown in the image's text fields \(Q_1 = 36\), \(Q_2 = 40.5\), \(Q_3 = 45.5\) correspond to a dataset that includes the cut-off value \(29\) partially visible at the far right of the first line: \(41, 52, 36, 43, 42, 38, 40, 46, 45, 38, 36, 56, 42, 35, 15, 53, 38, 49, 29\). Let's verify with 19 values: sorted is \(15, 29, 35, 36, 36, 38, 38, 38, 40, 41, 42, 42, 43, 45, 46, 49, 52, 53, 56\). The median is the 10th value, which is \(41\). The lower half has 9 values, median \(Q_1 = 36\). The upper half has 9 values, median \(Q_3 = 46\). If using standard software methods like Minitab/TI-84, the quartiles for this 19-value set are indeed \(Q_1 = 36\), \(Q_2 = 41\), \(Q_3 = 46\). However, the image shows the system accepted \(Q_1 = 36\), \(Q_2 = 40.5\), and \(Q_3 = 45.5\). This matches a specific interpolation method for a dataset of 19 or 20 values. Since the question asks for the Interquartile Range (IQR) based on these accepted quartiles, we use the values provided in the image: \(Q_1 = 36\) and \(Q_3 = 45.5\).)

Calculate the Interquartile Range

Using the Interquartile Range knowledge point

$$ \text{IQR} = Q_3 - Q_1 = 45.5 - 36 = 9.5 $$

Answer:

The interquartile range (IQR) is <blank>9.5</blank>