To determine the mean of the sums from Violet's game, we need to find the weighted average of the sums, taking into account their frequencies. Here’s a step-by-step breakdown of the process:
1. List of Sums and their Frequencies:
The possible sums and their corresponding frequencies are given in the table. We summarize these:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Sum} & \text{Frequency} \\
\hline
5 & 1 \\
\hline
7 & 2 \\
\hline
9 & 3 \\
\hline
11 & 4 \\
\hline
13 & 3 \\
\hline
15 & 2 \\
\hline
17 & 1 \\
\hline
\end{array}
\][/tex]
2. Calculate the Total of the Products of Sums and Frequencies:
Multiply each sum by its frequency and then sum these products:
[tex]\[
\begin{align*}
5 \times 1 & = 5 \\
7 \times 2 & = 14 \\
9 \times 3 & = 27 \\
11 \times 4 & = 44 \\
13 \times 3 & = 39 \\
15 \times 2 & = 30 \\
17 \times 1 & = 17 \\
\end{align*}
\][/tex]
Adding these products together gives:
[tex]\[
5 + 14 + 27 + 44 + 39 + 30 + 17 = 176
\][/tex]
3. Calculate the Total Frequency:
Add up all the frequencies:
[tex]\[
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
\][/tex]
4. Determine the Mean:
The mean (average) is calculated by dividing the total sum of the products by the total frequency:
[tex]\[
\text{Mean} = \frac{176}{16} = 11.0
\][/tex]
Now, let’s consider the statements provided for the mean:
- The mean is 12: The mean we calculated is 11.0, not 12, so this statement is false.
- The mean is 16: Again, the mean we calculated is 11.0, not 16, so this statement is false.
- The mean is the same as the median:
To find the median, we first need to list the sums in ascending order and find the middle value. By expanding the frequencies into a list of sums, we get:
[tex]\[
5, 7, 7, 9, 9, 9, 11, 11, 11, 11, 13, 13, 13, 15, 15, 17
\][/tex]
Since there are 16 values, the median will be the average of the 8th and 9th values. Both are 11, so the median is 11. Since the mean (11.0) is equal to the median (11), this statement is true.
- The mean is the same as the range:
The range is calculated as the difference between the highest and lowest sums:
[tex]\[
\text{Range} = 17 - 5 = 12
\][/tex]
Since the range is 12 and the mean (11.0) is not equal to the range, this statement is false.
Based on these calculations, the correct statement is:
The mean is the same as the median.
