To solve this problem, we need to determine the mean and median of the given data set of weekly coffee consumption and then decide which measure is the better indicator of central tendency.
### Step-by-Step Solution:
1. List the Data Set:
Given data set is:
[tex]\[1, 33, 6, 3, 5, 9, 8, 4, 9, 9, 3, 9, 3, 3, 5, 2\][/tex]
2. Calculate the Mean:
The mean is the sum of all data values divided by the number of data values.
- Sum of all values: [tex]\(1 + 33 + 6 + 3 + 5 + 9 + 8 + 4 + 9 + 9 + 3 + 9 + 3 + 3 + 5 + 2\)[/tex]
- Number of values: [tex]\(16\)[/tex]
The mean [tex]\( \mu \)[/tex] is calculated as follows:
[tex]\[
\mu = \frac{\sum \text{data values}}{\text{number of values}} = \frac{111}{16} = 6.9375 \approx 7.0
\][/tex]
3. Calculate the Median:
The median is the middle value when the data set is ordered from least to greatest.
- First, sort the data set:
[tex]\[1, 2, 3, 3, 3, 3, 4, 5, 5, 6, 8, 9, 9, 9, 9, 33\][/tex]
- With 16 values, the median is the average of the 8th and 9th values:
[tex]\[
\text{Median} = \frac{5 + 5}{2} = 5.0
\][/tex]
4. Determine the Better Measure of Central Tendency:
- Mean = 7.0
- Median = 5.0
When choosing between the mean and median, we considered the potential impact of outliers. In this data set, the value 33 is significantly higher than most of the other values, which skews the mean upwards. The median is less affected by extreme values and provides a better measure of the central tendency for this data set.
Hence, the better measure of central tendency is the one that is less influenced by the outlier, which in this case is the median.
5. Final Result:
The mean is 7.0, and the median is 5.0. Since the mean is influenced by the outlier, the median is the better measure of central tendency for this data set.
So, the correct answer is:
[tex]\[ \text{Mean } = 7, \text{ Median } = 5, \text{ The mean is the better measure of central tendency.} \][/tex]
