Ask a New Question
Menu

Calculate the 5-number summary of the skewed dataset 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29. When constructing the box plot on a number line, which explains a reason why the dataset is skewed?(1 point)

Responses

The right whisker is much longer than the left whisker.
The right whisker is much longer than the left whisker.

The whiskers are the same length.
The whiskers are the same length.

The median is exactly in the middle of the box.
The median is exactly in the middle of the box.

The 5 numbers in the summary are equally spread apart.

1 answer

To calculate the 5-number summary of the dataset \(15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29\), we need to determine the following:

  1. Minimum: The smallest number in the dataset.
  2. Maximum: The largest number in the dataset.
  3. Median: The middle value when the numbers are sorted.
  4. First Quartile (Q1): The median of the lower half of the dataset.
  5. Third Quartile (Q3): The median of the upper half of the dataset.

Let's compute each of these values step-by-step:

Step 1: Minimum

  • The minimum value is \(15\).

Step 2: Maximum

  • The maximum value is \(29\).

Step 3: Median

  • The dataset has \(11\) values. The median is the \(6^{th}\) value:
    • Sorted dataset: \(15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 29\)
    • Median is \(18\).

Step 4: First Quartile (Q1)

  • The lower half (not including the median) is \(15, 16, 16, 17, 17\). The median of this set is \(16\) (the \(3^{rd}\) value).

Step 5: Third Quartile (Q3)

  • The upper half (not including the median) is \(18, 18, 19, 19, 29\). The median of this set is \(19\) (the \(3^{rd}\) value).

Summary of 5-Number Summary

  • Minimum: \(15\)
  • Q1: \(16\)
  • Median (Q2): \(18\)
  • Q3: \(19\)
  • Maximum: \(29\)

The 5-number summary is:

\[ 15, 16, 18, 19, 29 \]

Identifying Skewness

A box plot of this dataset would have the following characteristics:

  • The left whisker extends from \(15\) to \(16\) (length \(1\)).
  • The right whisker extends from \(19\) to \(29\) (length \(10\)).
  • The box spans from \(16\) to \(19\) with the median at \(18\).

Conclusion on Skewness

A reason why the dataset is skewed is:

The right whisker is much longer than the left whisker.

This indicates that there are outliers on the right side, causing the right tail to be longer, which is characteristic of a positively skewed distribution (right skew).

Final Selection: The first response is correct.