/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 71 The dotplot shows the distributi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 71

The dotplot shows the distribution of the heights (in feet) of a sample of roller coasters. The five-number summary of the data is given in the following table. Sketch a boxplot of the data. Explain how you determined the length of the whiskers. $$ \begin{array}{lclcl} \text { Minimum } & \text { Q1 } & \text { Median } & \text { Q3 } & \text { Maximum } \\ \hline 2.438 & 8.526 & 18.288 & 33.223 & 128.016 \end{array} $$

Short Answer

Expert verified
A boxplot is drafted with the box extending from Q1 to Q3, a line at the median, and whiskers stretching from Minimum to Q1 and from Q3 to Maximum. The lengths of the whiskers are Q1 minus Minimum and Maximum minus Q3.

Step by step solution

01

Understand boxplot elements

A boxplot is a statistical graph that gives a visual representation of the dataset's five-number summary. It consists of a box and two whiskers. The box represents the interquartile range (IQR), stretching from Q1 to Q3, with a line inside it for the median. The whiskers extend from the edges of the box to the minimum and maximum values in the data, if no outliers present.
02

Sketch the Boxplot

First, draw a horizontal line for each of the five numbers: Minimum (2.438), Q1 (8.526), Median (18.288), Q3 (33.223), and Maximum (128.016). Connect these lines parallel to the value axis to form the box and the whiskers. The box should stretch from Q1 to Q3, and the whiskers should reach out to the minimum and maximum values.
03

Determine the Length of the Whiskers

The lengths of the whiskers can be found from the boxplot. The length of the lower whisker is the difference between Q1 and the minimum value (i.e., 8.526 - 2.438). The length of the upper whisker is the difference between the maximum value and Q3 (i.e., 128.016 - 33.223).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Five-Number Summary
The five-number summary is a simple yet powerful tool in statistics.
It provides a quick overview of a data set by highlighting key statistics.
The five numbers include:
  • Minimum: The smallest value in the dataset. It sets the lower bound for the data distribution.
  • Q1 (First Quartile): This is the median of the first half of the data. It indicates that 25% of the data points are below this value.
  • Median (Q2): This is the central value of the dataset, dividing it into two equal parts.
  • Q3 (Third Quartile): The median of the second half of the data, with 75% of the data points below this value.
  • Maximum: The largest value in the dataset, setting the upper bound for the data distribution.
These five key values help us understand the distribution and range of the data.
They give insight into where the data points lie in relation to each other and the spread across the range.
Interquartile Range
The Interquartile Range (IQR) is a critical measure in descriptive statistics.
It represents the spread of the middle 50% of your data.
Calculating the IQR is simple: just subtract the first quartile (Q1) from the third quartile (Q3).The formula is:\[ IQR = Q3 - Q1 \]In our exercise, with Q3 at 33.223 and Q1 at 8.526, the IQR is calculated as follows:\[ IQR = 33.223 - 8.526 = 24.697 \]This value is important because it tells us how spread out the central range of data is.
A larger IQR indicates more variability among the middle half of data points,
while a smaller IQR suggests that these data points are more clustered.
The IQR is also valuable in identifying potential outliers, as it forms the basis to determine the whiskers in a boxplot.
Outliers
Outliers are data points that differ significantly from other observations.
Identifying outliers is crucial as they can skew your data analysis.
They may indicate variability in your measurement or an experimental error.In boxplots, we use the interquartile range to identify outliers.
  • Lower Bound: Any data point below \( Q1 - 1.5 \times IQR \).
  • Upper Bound: Any data point above \( Q3 + 1.5 \times IQR \).
This calculation helps determine if there are extreme values that lie outside the expected range of variability.
Outliers are typically plotted as individual points beyond the whiskers of the boxplot.
Correctly identifying these can help you better understand your data and make more informed decisions
about your dataset.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

From Amazon.com, the prices of 10 varieties of orange juice \((59-\) to 64 -ounce containers) sold were recorded: \(\$ 3.88, \$ 2.99, \$ 3.99, \$ 2.99, \$ 3.69, \$ 2.99, \$ 4.49\), \(\$ 3.69, \$ 3.89, \$ 3.99 .\) a. Find and interpret the mean price of orange juice sold on this site. Round to the nearest cent. b. Find the standard deviation for the prices. Round to the nearest cent. Explain what this value means in the context of the data.

The dotplot shows heights of college women; the mean is 64 inches \((5\) feet 4 inches), and the standard deviation is 3 inches. a. What is the \(z\) -score for a height of 58 inches ( 4 feet 10 inches)? b. What is the height of a woman with a z-score of 1 ?

Three-year-old boys in the United States have a mean height of 38 inches and a standard deviation of 2 inches. How tall is a three-year-old boy with a \(z\) -score of \(-1.0\) ? (Source: www.kidsgrowth.com)

In the most recent summer Olympics, do you think the standard deviation of the running times for all men who ran the 100 -meter race would be larger or smaller than the standard deviation of the running times for the men's marathon? Explain.

The top seven movies based on DC comic book characters for the U.S. box office as of fall 2017 are shown in the following table, rounded to the nearest hundred million. (Source: ultimatemovieranking.com) a. Find and interpret the median in context. b. Find and interpret the IQR in context. c. Find the range of the data. Explain why the IQR is preferred over the range as a measure of variability. $$ \begin{array}{|lc|} \hline \text { Movie } & \begin{array}{c} \text { Adjusted Domestic } \\ \text { Gross (\$ millions) } \end{array} \\ \hline \text { The Dark Knight (2008) } & \$ 643 \\ \text { Batman (1989) } & \$ 547 \\ \hline \text { Superman (1978) } & \$ 543 \\ \hline \text { The Dark Knight Rises (2012) } & \$ 487 \\ \text { Wonder Woman (2017) } & \$ 407 \\ \text { Batman Forever (1995) } & \$ 366 \\ \hline \text { Superman II (1981) } & \$ 346 \\ \hline \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.