A data set consist of the following 6 values: The range is 6 . The median is 5 . The lower quartile is 4 . The interquartile range is 3 . Given that \( x

Ask by Bartlett Montgomery. in Singapore

Jan 22,2025

To determine the values of \( x \), \( y \), and \( z \) in the given dataset, let's analyze the provided information step-by-step. ### Given Information: 1. **Dataset Size:** 6 values, sorted in ascending order: \( a_1 \leq a_2 \leq a_3 \leq a_4 \leq a_5 \leq a_6 \). 2. **Range:** \( a_6 - a_1 = 6 \). 3. **Median:** With 6 data points, the median is the average of the 3rd and 4th values: \[ \frac{a_3 + a_4}{2} = 5 \implies a_3 + a_4 = 10 \] 4. **Lower Quartile (Q1):** For 6 data points, \( Q1 \) is the median of the first three values: \[ Q1 = a_2 = 4 \] 5. **Interquartile Range (IQR):** \[ IQR = Q3 - Q1 = 3 \implies Q3 = 7 \] \( Q3 \) is the median of the last three values, so \( a_5 = 7 \). ### Deductions: - **Value Relationships:** \[ a_1 \leq 4 \leq a_3 \leq a_4 \leq 7 \leq a_6 \] - **Sum of \( a_3 \) and \( a_4 \):** \[ a_3 + a_4 = 10 \] Possible integer pairs for \( (a_3, a_4) \) are \( (4, 6) \) and \( (5, 5) \). However, since \( a_4 \leq a_5 = 7 \), \( (4, 6) \) is the valid pair. - **Range Constraint:** \[ a_6 = a_1 + 6 \] - **Possible Values for \( a_1 \):** Considering \( a_1 \leq 4 \) and \( a_6 = a_1 + 6 \geq 7 \): - \( a_1 = 1 \): Dataset → \( 1, 4, 4, 6, 7, 7 \) - \( a_1 = 2 \): Dataset → \( 2, 4, 4, 6, 7, 8 \) - \( a_1 = 3 \): Dataset → \( 3, 4, 4, 6, 7, 9 \) - \( a_1 = 4 \): Dataset → \( 4, 4, 4, 6, 7, 10 \) ### Identifying \( x \), \( y \), and \( z \): From all possible datasets, the consistent middle values are \( 4 \), \( 6 \), and \( 7 \). Given the condition \( x < y < z \), these correspond to: - \( x = 4 \) - \( y = 6 \) - \( z = 7 \) ### Conclusion: The values of \( x \), \( y \), and \( z \) that satisfy all the given conditions are: \[ x = 4, \quad y = 6, \quad z = 7 \] **Final Answer:** After simplifying, the three values are 4, 6 and 7. Thus, \( x = 4 \), \( y = 6 \), and \( z = 7 \).