Use the accompanying data set to complete the following actions. a. Find the quartiles. b. Find the interquartile range. c. Identify any outliers. \[ \begin{array}{lllllllllllllllllllll}42 & 53 & 37 & 45 & 40 & 37 & 41 & 46 & 43 & 39 & 35 & 56 & 44 & 34 & 15 & 53 & 38 & 51 & 29 & 29\end{array} \] a. Find the quartiles.

To find the quartiles, we first need to sort the data set in ascending order. The given data set is: \[ 42, 53, 37, 45, 40, 37, 41, 46, 43, 39, 35, 56, 44, 34, 15, 53, 38, 51, 29, 29 \] ### Step 1: Sort the Data Sorting the data in ascending order gives us: \[ 15, 29, 29, 34, 35, 37, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 51, 53, 53, 56 \] ### Step 2: Find the Quartiles To find the quartiles, we will use the following definitions: - **Q1 (First Quartile)**: The median of the first half of the data. - **Q2 (Second Quartile)**: The median of the entire data set. - **Q3 (Third Quartile)**: The median of the second half of the data. #### Finding Q2 (Median) The median (Q2) is the average of the two middle numbers in the sorted list. Since there are 20 data points (even number), the median is the average of the 10th and 11th values. 10th value: 40 11th value: 41 \[ Q2 = \frac{40 + 41}{2} = \frac{81}{2} = 40.5 \] #### Finding Q1 To find Q1, we take the first half of the data (the first 10 values): \[ 15, 29, 29, 34, 35, 37, 37, 38, 39, 40 \] The median of this half (Q1) is the average of the 5th and 6th values. 5th value: 35 6th value: 37 \[ Q1 = \frac{35 + 37}{2} = \frac{72}{2} = 36 \] #### Finding Q3 To find Q3, we take the second half of the data (the last 10 values): \[ 41, 42, 43, 44, 45, 46, 51, 53, 53, 56 \] The median of this half (Q3) is the average of the 5th and 6th values. 5th value: 45 6th value: 46 \[ Q3 = \frac{45 + 46}{2} = \frac{91}{2} = 45.5 \] ### Summary of Quartiles - \( Q1 = 36 \) - \( Q2 = 40.5 \) - \( Q3 = 45.5 \) ### Step 3: Find the Interquartile Range (IQR) The interquartile range (IQR) is calculated as: \[ IQR = Q3 - Q1 = 45.5 - 36 = 9.5 \] ### Step 4: Identify Outliers Outliers can be identified using the following criteria: - Any data point below \( Q1 - 1.5 \times IQR \) - Any data point above \( Q3 + 1.5 \times IQR \) Calculating the lower and upper bounds: \[ \text{Lower Bound} = Q1 - 1.5 \times IQR = 36 - 1.5 \times 9.5 = 36 - 14.25 = 21.75 \] \[ \text{Upper Bound} = Q3 + 1.5 \times IQR = 45.5 + 1.5 \times 9.5 = 45.5 + 14.25 = 59.75 \] ### Outlier Identification Now we check for any data points below 21.75 or above 59.75: - The minimum value is 15 (which is below 21.75). - The maximum value is 56 (which is below 59.75). Thus, the only outlier is: - **Outlier**: 15 ### Final Results - **Quartiles**: \( Q1 = 36 \), \( Q2 = 40.5 \), \( Q3 = 45.5 \) - **Interquartile Range (IQR)**: 9.5 - **Outliers**: 15