Step 1: Understanding the Concept:
This problem combines two major concepts of statistics: Mean and Median.
The "Mean" is the arithmetic average, calculated by dividing the sum of all values by the total count.
The "Median" is the middle value of a data set when it is arranged in order.
The challenge here is that some values are dependent on an unknown variable \( p \).
We must first solve for \( p \) using the mean, then reconstruct the data set to find the median.
Step 2: Key Formula or Approach:
Mean = \( \frac{\text{Sum of observations}}{\text{Number of observations}} \)
For a set of \( N \) observations where \( N \) is odd, Median = \( \left(\frac{N+1}{2}\right)^{\text{th}} \) value in a sorted list.
Step 3: Detailed Explanation:
The list of numbers is: \( \{3, (3p+3), 8, 14, 18, 5, (p-2)\} \).
The total number of terms is \( N = 7 \).
The mean is given as 9.
Applying the mean formula:
\[ 9 = \frac{3 + (3p + 3) + 8 + 14 + 18 + 5 + (p - 2)}{7} \]
Multiply both sides by 7:
\[ 63 = 3 + 3p + 3 + 8 + 14 + 18 + 5 + p - 2 \]
Combine the \( p \) terms: \( 3p + p = 4p \).
Combine the constants: \( 3 + 3 + 8 + 14 + 18 + 5 - 2 = 49 \).
The equation is now:
\[ 63 = 4p + 49 \]
\[ 14 = 4p \implies p = \frac{14}{4} = 3.5 \]
Now substitute \( p = 3.5 \) into the variable terms:
Second term: \( 3p + 3 = 3(3.5) + 3 = 10.5 + 3 = 13.5 \).
Seventh term: \( p - 2 = 3.5 - 2 = 1.5 \).
The actual data set is: \( \{3, 13.5, 8, 14, 18, 5, 1.5\} \).
To find the median, we must sort the data in ascending order:
Sorted List: \( 1.5, 3, 5, 8, 13.5, 14, 18 \).
Since there are 7 terms, the median is the \( \frac{7+1}{2} = 4^{\text{th}} \) term.
The \( 4^{\text{th}} \) term in the sorted list is 8.
Step 4: Final Answer:
The median of the given set of numbers is 8.