To solve this question, we will calculate the mean and determine its relationship with other statistics such as the median and the range. Let’s go through the steps one by one.
### Step 1: Calculate the Mean
The mean (average) of the sums can be calculated by taking the weighted sum of all possible sums divided by the total frequency of occurrences.
1. Sum of each product of sum and frequency:
- \( 5 \times 1 = 5 \)
- \( 7 \times 2 = 14 \)
- \( 9 \times 3 = 27 \)
- \( 11 \times 4 = 44 \)
- \( 13 \times 3 = 39 \)
- \( 15 \times 2 = 30 \)
- \( 17 \times 1 = 17 \)
Add these products together:
[tex]\[
5 + 14 + 27 + 44 + 39 + 30 + 17 = 176
\][/tex]
2. Sum of the frequencies:
[tex]\[
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
\][/tex]
3. Mean:
[tex]\[
\text{Mean} = \frac{\text{Total Sum of products}}{\text{Total Frequency}} = \frac{176}{16} = 11.0
\][/tex]
### Step 2: Calculate the Median
The median is the middle value when the data is arranged in ascending order. Given the frequencies, we can determine the median by cumulative counting:
- The middle position in a dataset with 16 entries is between the 8th and 9th values.
- Accumulating frequencies:
- First-cumulative count for \(5\) = 1
- For \(7\) = 3 (1 + 2)
- For \(9\) = 6 (3 + 3)
- For \(11\) = 10 (6 + 4)
- Since the 8th and 9th values lie within the four occurrences of 11, the median is 11.
### Step 3: Calculate the Range
The range is the difference between the maximum and minimum sums:
[tex]\[
\text{Range} = 17 - 5 = 12
\][/tex]
### Step 4: Compare Mean, Median, and Range
Given the results:
- Mean: 11.0
- Median: 11
- Range: 12
Clearly, the mean of the sums (11.0) is the same as the median (11).
### Conclusion
Thus, the true statement about the mean of the sums of the two spinners is:
- The mean is the same as the median.
