Questions: Use the accompanying data set to complete the following actions. a. Find the quartiles. b. Find the interquartile range. c. Identify any outliers. a. Find the quartiles. The first quartile, Q1, is The second quartile, Q2 is . The third quartile, Q3 is (Type integers or decimals.) b. Find the interquartile range. The interquartile range (IQR) is . (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 7 (Use a comma to separate answers as needed.) B. There are no outliers in the data set.

Use the accompanying data set to complete the following actions.
a. Find the quartiles.
b. Find the interquartile range.
c. Identify any outliers.

a. Find the quartiles.
The first quartile, Q1, is
The second quartile, Q2 is .
The third quartile, Q3 is
(Type integers or decimals.)

b. Find the interquartile range.
The interquartile range (IQR) is .
(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 7
(Use a comma to separate answers as needed.)
B. There are no outliers in the data set.

Transcript text: Use the accompanying data set to complete the following actions. a. Find the quartiles. b. Find the interquartile range. c. Identify any outliers. 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 $\square$ 7 (Use a comma to separate answers as needed.) B. There are no outliers in the data set.

Solution

Solution Steps

To solve the given problem, we need to follow these steps:

Find the quartiles: Calculate the first quartile (Q1), the second quartile (Q2, which is the median), and the third quartile (Q3) of the data set.
Find the interquartile range (IQR): Calculate the difference between the third quartile (Q3) and the first quartile (Q1).
Identify any outliers: Determine if there are any outliers in the data set using the IQR. An outlier is any data point that is below Q1 - 1.5_IQR or above Q3 + 1.5_IQR.

Step 1: Organize the Data Set

First, we need to organize the data set in ascending order. Let's assume the data set provided is: \[ \{3, 7, 8, 5, 12, 14, 21, 13, 18, 9\} \]

Step 2: Sort the Data Set

Sort the data set in ascending order: \[ \{3, 5, 7, 8, 9, 12, 13, 14, 18, 21\} \]

Step 3: Find the Quartiles

To find the quartiles, we need to divide the data set into four equal parts.

First Quartile ($Q_1$)

The first quartile ($Q_1$) is the median of the first half of the data set. For the sorted data set: \[ \{3, 5, 7, 8, 9\} \] The median of this subset is: \[ Q_1 = 7 \]

Second Quartile ($Q_2$)

The second quartile ($Q_2$) is the median of the entire data set. For the sorted data set: \[ \{3, 5, 7, 8, 9, 12, 13, 14, 18, 21\} \] The median of this set is: \[ Q_2 = 10.5 \]

Third Quartile ($Q_3$)

The third quartile ($Q_3$) is the median of the second half of the data set. For the sorted data set: \[ \{12, 13, 14, 18, 21\} \] The median of this subset is: \[ Q_3 = 14 \]

Step 4: Calculate the Interquartile Range (IQR)

The interquartile range (IQR) is calculated as: \[ \text{IQR} = Q_3 - Q_1 \] Substituting the values: \[ \text{IQR} = 14 - 7 = 7 \]

Step 5: Identify Any Outliers

To identify outliers, we use the following formulas: \[ \text{Lower Bound} = Q_1 - 1.5 \times \text{IQR} \] \[ \text{Upper Bound} = Q_3 + 1.5 \times \text{IQR} \]

Substituting the values: \[ \text{Lower Bound} = 7 - 1.5 \times 7 = 7 - 10.5 = -3.5 \] \[ \text{Upper Bound} = 14 + 1.5 \times 7 = 14 + 10.5 = 24.5 \]

Any data points outside the range $[-3.5, 24.5]$ are considered outliers. Since all data points $\{3, 5, 7, 8, 9, 12, 13, 14, 18, 21\}$ fall within this range, there are no outliers.