To find the mean (average) of the sums from Violet's spinners, we use the formula for the weighted mean. Here's a detailed step-by-step explanation on how this is calculated, broken down clearly:
1. List the possible sums and their respective frequencies:
We have the sums: [tex]\(5, 7, 9, 11, 13, 15, 17\)[/tex]
And their corresponding frequencies: [tex]\(1, 2, 3, 4, 3, 2, 1\)[/tex]
2. Calculate the total number of observations:
The total frequency is the sum of all the frequencies.
[tex]\[
\text{Total Frequency} = 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
\][/tex]
3. Calculate the weighted sum of the sums:
Multiply each sum by its frequency and then add all these products together:
[tex]\[
\text{Weighted Sum} = (5 \times 1) + (7 \times 2) + (9 \times 3) + (11 \times 4) + (13 \times 3) + (15 \times 2) + (17 \times 1)
\][/tex]
Simplifying this gives:
[tex]\[
= 5 + 14 + 27 + 44 + 39 + 30 + 17 = 176
\][/tex]
4. Calculate the mean:
Now, divide the weighted sum by the total frequency to find the mean:
[tex]\[
\text{Mean} = \frac{\text{Weighted Sum}}{\text{Total Frequency}} = \frac{176}{16} = 11.0
\][/tex]
Therefore, the mean of the sums of the two spinners is [tex]\(11.0\)[/tex].
Comparing the options with our calculated mean:
- The mean is not 12.
- The mean is not 16.
- To determine if the mean is the same as the median:
- The median for this distribution, given the frequencies, would involve finding the middle value in the ordered set. The cumulative frequencies help us find that:
- The 8th and 9th values are both 11 (since four 11s fall in the middle of the data set).
Thus, the median is also 11.
So, the mean is indeed the same as the median.
- The range is the difference between the highest and lowest sums:
[tex]\[
\text{Range} = 17 - 5 = 12
\][/tex]
Therefore, the mean is not the same as the range.
Correct Statement: The mean is the same as the median.
