To calculate the 5-number summary (minimum, first quartile, median, third quartile, and maximum) for the dataset \(15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29\), we can follow these steps:
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Minimum: The smallest value in the dataset is \(15\).
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Maximum: The largest value in the dataset is \(29\).
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Median: The median is the middle value of the dataset. Since there are \(11\) numbers (an odd number), the median is the \(6^{th}\) value: \[ \text{Median} = 18 \]
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First Quartile (Q1): The first quartile is the median of the first half of the data (the first \(5\) values: \(15, 16, 16, 17, 17\)). The median of these five values: \[ Q1 = 16 \]
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Third Quartile (Q3): The third quartile is the median of the second half of the data (the last \(5\) values: \(18, 18, 19, 19, 29\)). The median of these five values: \[ Q3 = 19 \]
Now we can summarize the 5-number summary:
- Minimum: \(15\)
- Q1: \(16\)
- Median: \(18\)
- Q3: \(19\)
- Maximum: \(29\)
The 5-number summary is therefore: \[ 15, 16, 18, 19, 29 \]
Reason for Skewness:
To explain the reason the dataset is skewed when constructing the box plot, we observe the following options:
- The whiskers are the same length.
- The median is exactly in the middle of the box.
- The 5 numbers in the summary are equally spread apart.
- The right whisker is much longer than the left whisker.
The correct response is: The right whisker is much longer than the left whisker.
This indicates that there are extreme values (in this case, \(29\)) that pull the distribution towards the higher values, leading to a right (or positive) skew in the dataset.
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