Let's analyze the data provided to determine the correct statement about the mean of the sums of the two spinners.
Given the sums and their respective frequencies:
| Sum | Frequency |
|-----|-----------|
| 5 | 1 |
| 7 | 2 |
| 9 | 3 |
| 11 | 4 |
| 13 | 3 |
| 15 | 2 |
| 17 | 1 |
### Step 1: Calculate the Mean of the Sums
To find the mean, we'll use the formula for the weighted mean:
[tex]\[
\text{Mean} = \frac{\sum (\text{Sum} \times \text{Frequency})}{\sum \text{Frequency}}
\][/tex]
Compute the total sum of the sums multiplied by their frequencies:
[tex]\[
5 \times 1 + 7 \times 2 + 9 \times 3 + 11 \times 4 + 13 \times 3 + 15 \times 2 + 17 \times 1 = 5 + 14 + 27 + 44 + 39 + 30 + 17 = 176
\][/tex]
Sum of the frequencies:
[tex]\[
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
\][/tex]
Now, calculate the mean:
[tex]\[
\text{Mean} = \frac{176}{16} = 11.0
\][/tex]
### Step 2: Determine the Median of the Sums
To find the median, list all the sums in order and account for the frequencies:
[tex]\[
5, 7, 7, 9, 9, 9, 11, 11, 11, 11, 13, 13, 13, 15, 15, 17
\][/tex]
Since there are 16 numbers (even), the median is the average of the 8th and 9th numbers:
[tex]\[
\text{8th number} = 11 \quad \text{and} \quad \text{9th number} = 11
\][/tex]
So the median is:
[tex]\[
\text{Median} = \frac{11 + 11}{2} = 11.0
\][/tex]
### Step 3: Calculate the Range of the Sums
The range is the difference between the maximum and minimum sums:
[tex]\[
\text{Range} = 17 - 5 = 12
\][/tex]
### Step 4: Compare the Mean to the Median and the Range
- The mean is [tex]\( 11.0 \)[/tex]
- The median is [tex]\( 11.0 \)[/tex]
- The range is [tex]\( 12 \)[/tex]
### Conclusion
The mean is the same as the median but different from the range.
Therefore, the correct statement is:
The mean is the same as the median.
