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, maximum, median (Q2), first quartile (Q1), and third quartile (Q3).
- Minimum: The smallest number in the dataset is 15.
- Maximum: The largest number in the dataset is 29.
- Median (Q2): The median is the middle number. Since there are 11 numbers (odd), the median is the 6th number, which is 18.
- Q1 (First Quartile): This is the median of the first half of the data (the first 5 numbers): \(15, 16, 16, 17, 17\). The median of these numbers is \(16\) (the middle value).
- Q3 (Third Quartile): This is the median of the second half of the data (the last 5 numbers): \(18, 18, 19, 19, 29\). The median of these numbers is \(19\) (the middle value).
Thus, the 5-number summary is:
- Minimum: \(15\)
- Q1: \(16\)
- Median (Q2): \(18\)
- Q3: \(19\)
- Maximum: \(29\)
In summary, the 5-number summary is:
- 15, 16, 18, 19, 29
To answer the second part of your question regarding the box plot and explaining the skewness of the dataset, let's analyze the options provided:
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The whiskers are the same length.
- This would indicate a symmetric distribution, not skewed.
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The right whisker is much longer than the left whisker.
- This indicates that the dataset has outliers or a longer tail on the right side, which is a sign of right skewness (positive skewness).
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The median is exactly in the middle of the box.
- This suggests symmetry, implying no skewness.
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The 5 numbers in the summary are equally spread apart.
- This would also suggest symmetry, not skewness.
The correct response that provides a reason why the dataset is skewed is:
The right whisker is much longer than the left whisker.