To analyze the given data sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we need to determine their mean and median and analyze their symmetry and skewness properties.
### Data Set [tex]\( A \)[/tex]
Given [tex]\( A = \{10, 12, 12, 6, 8, 5, 4, 10, 8, 10, 12, 12, 6\} \)[/tex]:
1. Mean of [tex]\( A \)[/tex]: The mean is calculated by adding all the data points and dividing by the number of data points.
[tex]\[
\text{Mean of } A = 8.846153846153847
\][/tex]
2. Median of [tex]\( A \)[/tex]: The median is the middle value when the data points are sorted in ascending order. For an odd number of data points, it's the central number.
[tex]\[
\text{Median of } A = 10.0
\][/tex]
3. Symmetry of [tex]\( A \)[/tex]: A data set is symmetrical if the mean is equal to the median. For [tex]\( A \)[/tex],
[tex]\[
\text{Mean of } A \neq \text{Median of } A
\][/tex]
Hence, [tex]\( A \)[/tex] is not symmetrical.
4. Skewness of [tex]\( A \)[/tex]: If the median is greater than the mean, the data set is left skewed; if the mean is greater than the median, it is right skewed.
[tex]\[
\text{Median of } A (10.0) > \text{Mean of } A (8.846153846153847)
\][/tex]
Therefore, [tex]\( A \)[/tex] is left skewed.
### Data Set [tex]\( B \)[/tex]
Given [tex]\( B = \{8, 8, 8, 7, 9, 7, 9, 10, 6, 8\} \)[/tex]:
1. Mean of [tex]\( B \)[/tex]: The mean is calculated similarly,
[tex]\[
\text{Mean of } B = 8.0
\][/tex]
2. Median of [tex]\( B \)[/tex]: The median is the middle value,
[tex]\[
\text{Median of } B = 8.0
\][/tex]
3. Symmetry of [tex]\( B \)[/tex]: For [tex]\( B \)[/tex],
[tex]\[
\text{Mean of } B = \text{Median of } B
\][/tex]
Hence, [tex]\( B \)[/tex] is symmetrical.
4. Skewness of [tex]\( B \)[/tex]: Given that the mean and median are equal, the data set is not skewed.
### Conclusion
Based on the analysis:
- Set [tex]\( A \)[/tex] is not symmetrical.
- Set [tex]\( B \)[/tex] has the same mean and median.
- Set [tex]\( B \)[/tex] is not right skewed; it is symmetrical.
- Set [tex]\( A \)[/tex] is left skewed.
- The median of set [tex]\( A \)[/tex] is larger than the mean of set [tex]\( A \)[/tex].
Therefore, the true statements are:
- [tex]\(\operatorname{Set} B\)[/tex] has the same mean and median.
- Set [tex]\( A \)[/tex] is left skewed.
- The median of set [tex]\( A \)[/tex] is larger than the mean of set [tex]\( A \)[/tex].
