True or False? In Exercises 7-10, determine whether the statement is true or false. If it is false, rewrite it as a true statement. 7. About one-quarter of a data set falls below \( Q_{1} \). 8. The second quartile is the mean of an ordered data set. 9. An outlier is any number above \( Q_{3} \) or below \( Q_{1} \). 10. It is impossible to have a \( z \)-score of 0 . Using and Interpreting Concepts Finding Quartiles, Interquartile Range, and Outliers In Exercises 11 and 12, (a) find the quartiles, (b) find the interquartile range, and (c) identify any outliers. 11. \( 56 \quad 63 \) 51 \( \begin{array}{llllllllllllll}60 & 57 & 60 & 60 & 54 & 63 & 59 & 80 & 63 & 60 & 62 & 65\end{array} \) \[ \text { 12. } \begin{array}{llllllllll} 22 & 25 & 22 & 24 & 20 & 24 & 19 & 22 & 29 & 21 \\ 21 & 20 & 23 & 25 & 23 & 23 & 21 & 25 & 23 & 22 \end{array} \]

Sure, let's go through each of the statements and exercises step by step. --- ### **True or False** #### **7. About one-quarter of a data set falls below \( Q_{1} \).** **True.** - **Explanation:** \( Q_{1} \) (the first quartile) represents the 25th percentile of a data set. This means that 25% of the data falls below \( Q_{1} \). #### **8. The second quartile is the mean of an ordered data set.** **False.** - **Corrected Statement:** The second quartile is the **median** of an ordered data set. - **Explanation:** The second quartile (\( Q_{2} \)) is indeed the median, which is the middle value of an ordered data set. It is not necessarily the mean unless the data set is perfectly symmetrical. #### **9. An outlier is any number above \( Q_{3} \) or below \( Q_{1} \).** **False.** - **Corrected Statement:** An outlier is any number that is significantly above \( Q_{3} \) or below \( Q_{1} \), typically defined as being more than 1.5 times the interquartile range (IQR) above \( Q_{3} \) or below \( Q_{1} \). - **Explanation:** Simply being above \( Q_{3} \) or below \( Q_{1} \) does not necessarily make a data point an outlier. The standard definition involves being a certain distance beyond these quartiles, often 1.5 times the IQR. #### **10. It is impossible to have a \( z \)-score of 0.** **False.** - **Corrected Statement:** It is possible to have a \( z \)-score of 0, which indicates that the data point is equal to the mean of the data set. - **Explanation:** A \( z \)-score of 0 signifies that the value is exactly at the mean, neither above nor below it. --- ### **Using and Interpreting Concepts** #### **Exercise 11** **Data Set:** 56, 63, 51, 60, 57, 60, 60, 54, 63, 59, 80, 63, 60, 62, 65 **Step 1:** **Order the data set from smallest to largest.** Ordered Data: 51, 54, 56, 57, 59, 60, 60, 60, 62, 63, 63, 63, 65, 80 (Note: There are 15 data points.) **a. Find the Quartiles (\( Q_{1} \), \( Q_{2} \), \( Q_{3} \)):** - **Median (\( Q_{2} \))**: - Position: (15 + 1) / 2 = 8 - \( Q_{2} = 60 \) - **First Quartile (\( Q_{1} \))**: - Lower half: 51, 54, 56, 57, 59, 60, 60 - Median of lower half: 56 (4th value) - \( Q_{1} = 56 \) - **Third Quartile (\( Q_{3} \))**: - Upper half: 60, 62, 63, 63, 63, 65, 80 - Median of upper half: 63 (4th value) - \( Q_{3} = 63 \) **b. Find the Interquartile Range (IQR):** \[ IQR = Q_{3} - Q_{1} = 63 - 56 = 7 \] **c. Identify Any Outliers:** - **Lower Bound:** \( Q_{1} - 1.5 \times IQR = 56 - 10.5 = 45.5 \) - **Upper Bound:** \( Q_{3} + 1.5 \times IQR = 63 + 10.5 = 73.5 \) **Outliers:** Any data points below 45.5 or above 73.5 are considered outliers. - **Outliers in this data set:** 80 --- #### **Exercise 12** **Data Set:** 22, 25, 22, 24, 20, 24, 19, 22, 29, 21, 21, 20, 23, 25, 23, 23, 21, 25, 23, 22 **Step 1:** **Order the data set from smallest to largest.** Ordered Data: 19, 20, 20, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 25, 25, 25, 29 (Note: There are 20 data points.) **a. Find the Quartiles (\( Q_{1} \), \( Q_{2} \), \( Q_{3} \)):** - **Median (\( Q_{2} \))**: - Position: (20 + 1) / 2 = 10.5 - Average of 10th and 11th values: (22 + 23) / 2 = 22.5 - \( Q_{2} = 22.5 \) - **First Quartile (\( Q_{1} \))**: - Lower half (first 10 data points): 19, 20, 20, 21, 21, 21, 22, 22, 22, 22 - Median of lower half: (21 + 21) / 2 = 21 - \( Q_{1} = 21 \) - **Third Quartile (\( Q_{3} \))**: - Upper half (last 10 data points): 23, 23, 23, 23, 24, 24, 25, 25, 25, 29 - Median of upper half: (24 + 24) / 2 = 24 - \( Q_{3} = 24 \) **b. Find the Interquartile Range (IQR):** \[ IQR = Q_{3} - Q_{1} = 24 - 21 = 3 \] **c. Identify Any Outliers:** - **Lower Bound:** \( Q_{1} - 1.5 \times IQR = 21 - 4.5 = 16.5 \) - **Upper Bound:** \( Q_{3} + 1.5 \times IQR = 24 + 4.5 = 28.5 \) **Outliers:** Any data points below 16.5 or above 28.5 are considered outliers. - **Outliers in this data set:** 29 --- ### **Summary** - **Statements 7:** True - **Statement 8:** False – Corrected to state that \( Q_{2} \) is the median. - **Statement 9:** False – Corrected to define outliers based on 1.5×IQR beyond quartiles. - **Statement 10:** False – Corrected to acknowledge that a \( z \)-score of 0 is possible. - **Exercise 11:** - \( Q_{1} = 56 \) - \( Q_{2} = 60 \) - \( Q_{3} = 63 \) - \( IQR = 7 \) - **Outlier:** 80 - **Exercise 12:** - \( Q_{1} = 21 \) - \( Q_{2} = 22.5 \) - \( Q_{3} = 24 \) - \( IQR = 3 \) - **Outlier:** 29 If you need further explanations or assistance with any of these problems, feel free to ask!