To calculate the 5-number summary of the dataset \(15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29\), we need to identify the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum of the dataset.
- Minimum (Min): The smallest value in the dataset, which is \(15\).
- Maximum (Max): The largest value in the dataset, which is \(29\).
- Median (Q2): The median is the middle number when the dataset is ordered. With 11 numbers, the median is the 6th number in the ordered list, which is \(18\).
- First Quartile (Q1): To find Q1, we take the median of the first half of the data: \(15, 16, 16, 17, 17\). The median of this subset is the 3rd number, which is \(16\).
- Third Quartile (Q3): To find Q3, we take the median of the second half of the data: \(18, 18, 19, 19, 29\). The median of this subset is the 3rd number, which is \(19\).
Now, we can summarize the results:
- Minimum: \(15\)
- Q1: \(16\)
- Median (Q2): \(18\)
- Q3: \(19\)
- Maximum: \(29\)
The 5-number summary is:
- Minimum: \(15\)
- Q1: \(16\)
- Median: \(18\)
- Q3: \(19\)
- Maximum: \(29\)
Now, for the box plot interpretation and to explain how the dataset is skewed, let's look at the whiskers. The right whisker (from Q3 to the maximum) is much longer than the left whisker (from the minimum to Q1), which indicates that there are more data values further away from the median on the right side, suggesting that the data is positively skewed.
Therefore, the appropriate explanation for the skewness of the dataset is:
The right whisker is much longer than the left whisker.