To determine the median number of musical instruments played, we need to follow a series of steps:
1. List All Data Points:
Start by creating a list of all the individual responses based on the given frequencies. The table provides the frequency for each number of instruments played:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Number of instruments} & \text{Frequency} \\
\hline
0 & 2 \\
\hline
1 & 3 \\
\hline
2 & 2 \\
\hline
3 & 4 \\
\hline
\end{array}
\][/tex]
- For the number 0: It appears 2 times.
- For the number 1: It appears 3 times.
- For the number 2: It appears 2 times.
- For the number 3: It appears 4 times.
Putting these together, we get the following list of data points:
[tex]\[ [0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3] \][/tex]
2. Sort the Data Points:
The data points are already presented in ascending order:
[tex]\[ [0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3] \][/tex]
3. Determine the Position of the Median:
To find the median, we need to determine the middle value of the sorted list. First, find the total number of data points ([tex]\(n\)[/tex]):
[tex]\[
n = 11
\][/tex]
Since [tex]\(n\)[/tex] is odd, the median will be the value at position [tex]\(\left(\frac{n+1}{2}\right)\)[/tex]-th position in the list. Calculating this position:
[tex]\[
\text{Median position} = \frac{11+1}{2} = 6
\][/tex]
4. Find the Median Value:
The median is the value at the 6th position in the sorted list, which is:
[tex]\[ [0, 0, 1, 1, 1, \textbf{2}, 2, 3, 3, 3, 3] \][/tex]
Therefore, the median number of musical instruments played is:
[tex]\[
\boxed{2}
\][/tex]
