Question

Let \( 1 < a < 7 \) and \( b > 7 \). If both mean and median of the data set \( \{1, 7, -7, a, b\} \) are equal to 4, then the value of \( b \) is _____.

• A. 20
• B. 15
• C. -1
• D. 5

Hint:

When dealing with both mean and median, always use the median to fix the middle value of your sorted list first. This usually determines one variable immediately, making the mean calculation much simpler.

Answer 1

The Correct Option is B

Concept:
The mean is the sum of observations divided by the number of observations. The median is the middle value when the data is arranged in ascending order. For a set with 5 elements, the median is the 3rd element.

Step 1: Arrange the data and find \( a \).
Given conditions: \( -7 < 1 < a < 7 < b \). The sorted data set is: \( \{-7, 1, a, 7, b\} \). Since the median is 4, and the 3rd term in our sorted list is \( a \), we have: \[ \text{Median} = a = 4 \] This satisfies the given condition \( 1 < a < 7 \) as \( 1 < 4 < 7 \).

Step 2: Use the mean to find \( b \).
The mean of the 5 numbers is also 4. \[ \text{Mean} = \frac{-7 + 1 + a + 7 + b}{5} = 4 \] Substitute \( a = 4 \): \[ \frac{-7 + 1 + 4 + 7 + b}{5} = 4 \] \[ \frac{5 + b}{5} = 4 \]

Step 3: Solve the algebraic equation.
\[ 5 + b = 20 \] \[ b = 20 - 5 = 15 \] This satisfies the condition \( b > 7 \) as \( 15 > 7 \).