Answer :
To determine the correct statement about the mean of the sums of the two spinners, let's analyze and calculate the mean, median, and range step by step.
### Step 1: Calculate the Mean
The mean (or average) is calculated by taking the sum of each possible sum multiplied by its frequency, and then dividing by the total frequency.
Sums and Frequencies:
- Sum = 5, Frequency = 1
- Sum = 7, Frequency = 2
- Sum = 9, Frequency = 3
- Sum = 11, Frequency = 4
- Sum = 13, Frequency = 3
- Sum = 15, Frequency = 2
- Sum = 17, Frequency = 1
Total sum:
[tex]\[ \sum_{i=1}^{7} (\text{sum}_i \times \text{frequency}_i) = (5 \times 1) + (7 \times 2) + (9 \times 3) + (11 \times 4) + (13 \times 3) + (15 \times 2) + (17 \times 1) = 5 + 14 + 27 + 44 + 39 + 30 + 17 = 176 \][/tex]
Total frequency:
[tex]\[ \sum_{i=1}^{7} \text{frequency}_i = 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 \][/tex]
Mean:
[tex]\[ \text{Mean} = \frac{\text{Total sum}}{\text{Total frequency}} = \frac{176}{16} = 11.0 \][/tex]
### Step 2: Calculate the Median
The median is the middle value of a data set ordered from least to greatest.
To find the median, extend each sum by its frequency.
Ordered data set:
[tex]\[ \{5, 7, 7, 9, 9, 9, 11, 11, 11, 11, 13, 13, 13, 15, 15, 17\} \][/tex]
Since there are 16 values (an even number), the median is the average of the 8th and 9th values in the ordered data.
8th value is 11 and 9th value is also 11.
Median:
[tex]\[ \text{Median} = \frac{11 + 11}{2} = 11.0 \][/tex]
### Step 3: Calculate the Range
The range is the difference between the maximum and minimum values of the sums.
Minimum sum = 5
Maximum sum = 17
Range:
[tex]\[ \text{Range} = 17 - 5 = 12 \][/tex]
### Conclusion
Based on the calculations:
- The mean is 11.0
- The median is 11.0
- The range is 12
The correct statement from the options is:
[tex]\[ \textbf{The mean is the same as the median.} \][/tex]
### Step 1: Calculate the Mean
The mean (or average) is calculated by taking the sum of each possible sum multiplied by its frequency, and then dividing by the total frequency.
Sums and Frequencies:
- Sum = 5, Frequency = 1
- Sum = 7, Frequency = 2
- Sum = 9, Frequency = 3
- Sum = 11, Frequency = 4
- Sum = 13, Frequency = 3
- Sum = 15, Frequency = 2
- Sum = 17, Frequency = 1
Total sum:
[tex]\[ \sum_{i=1}^{7} (\text{sum}_i \times \text{frequency}_i) = (5 \times 1) + (7 \times 2) + (9 \times 3) + (11 \times 4) + (13 \times 3) + (15 \times 2) + (17 \times 1) = 5 + 14 + 27 + 44 + 39 + 30 + 17 = 176 \][/tex]
Total frequency:
[tex]\[ \sum_{i=1}^{7} \text{frequency}_i = 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 \][/tex]
Mean:
[tex]\[ \text{Mean} = \frac{\text{Total sum}}{\text{Total frequency}} = \frac{176}{16} = 11.0 \][/tex]
### Step 2: Calculate the Median
The median is the middle value of a data set ordered from least to greatest.
To find the median, extend each sum by its frequency.
Ordered data set:
[tex]\[ \{5, 7, 7, 9, 9, 9, 11, 11, 11, 11, 13, 13, 13, 15, 15, 17\} \][/tex]
Since there are 16 values (an even number), the median is the average of the 8th and 9th values in the ordered data.
8th value is 11 and 9th value is also 11.
Median:
[tex]\[ \text{Median} = \frac{11 + 11}{2} = 11.0 \][/tex]
### Step 3: Calculate the Range
The range is the difference between the maximum and minimum values of the sums.
Minimum sum = 5
Maximum sum = 17
Range:
[tex]\[ \text{Range} = 17 - 5 = 12 \][/tex]
### Conclusion
Based on the calculations:
- The mean is 11.0
- The median is 11.0
- The range is 12
The correct statement from the options is:
[tex]\[ \textbf{The mean is the same as the median.} \][/tex]
