Math Problem Statement
(a) Identify the shape of the distribution, and (b) determine the five-number summary. Assume that each number in the five-number summary is an integer. 0 10 20
A horizontal boxplot is above a number line from 0 to 20 in increments of 1 and consists of a box extending from 12 to 18 with a vertical line segment through the box at 17 and two horizontal line segments extending from the left and right sides of the box to 0 and 20, respectively. Question content area bottom Part 1 a. Choose the correct answer below for the shape of the distribution. A.The distribution is skewed right. The distribution is skewed right. B.The distribution is roughly symmetric. The distribution is roughly symmetric. C.The distribution is skewed left. The distribution is skewed left. D. The shape of the distribution cannot be determined from the boxplot. Part 2 b. The five-number summary is enter your response here, enter your response here, enter your response here, enter your response here, enter your response here.
Solution
Let's break this problem down step-by-step.
Part 1: Identify the Shape of the Distribution
We are given a boxplot with the following key information:
- The box extends from 12 to 18.
- The median (vertical line inside the box) is at 17.
- The whiskers extend from 0 (minimum) to 20 (maximum).
To determine the shape of the distribution:
- The box is not centered. It is closer to the higher end of the data range (from 12 to 18, but the median is at 17), meaning most of the data points are in the upper range.
- The right whisker is very short (from 18 to 20), while the left whisker is long (from 12 down to 0).
This suggests that there are fewer larger values and more smaller values, leading to a left-skewed distribution (skewed to the left). Thus, the correct answer is:
(a) C. The distribution is skewed left.
Part 2: Five-Number Summary
The five-number summary consists of:
- Minimum (Q₀): This is the smallest value. The whisker extends to 0, so the minimum is 0.
- First Quartile (Q₁): The lower end of the box. This is 12.
- Median (Q₂): The vertical line in the box. This is 17.
- Third Quartile (Q₃): The upper end of the box. This is 18.
- Maximum (Q₄): This is the largest value. The whisker extends to 20, so the maximum is 20.
So, the five-number summary is: (b) 0, 12, 17, 18, 20.
Summary of Results:
- (a) The distribution is skewed left.
- (b) The five-number summary is: 0, 12, 17, 18, 20.
Do you need more details or have any other questions?
Here are 5 related questions for practice:
- How does the length of the whiskers in a boxplot affect the interpretation of skewness?
- What would the boxplot look like if the distribution were symmetric?
- How would the five-number summary change if the boxplot extended from 10 to 20?
- What does the median tell you about the central tendency of the data?
- How can outliers affect the whiskers in a boxplot?
Tip: Always check the length of the whiskers in a boxplot when determining skewness; a longer whisker on one side typically indicates the direction of skewness.
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Math Problem Analysis
Mathematical Concepts
Descriptive Statistics
Boxplot
Five-number Summary
Skewness
Formulas
Five-number summary: minimum, Q1, median (Q2), Q3, maximum
Theorems
None explicitly; uses statistical concepts such as skewness and five-number summary interpretation
Suitable Grade Level
Grades 8-10