/*! 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 101 What shape do you expect? For th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 101

What shape do you expect? For the following variables, indicate whether you would expect its histogram to be bell shaped, skewed to the right, or skewed to the left. Explain why. a. Number of times arrested in past year b. Time needed to complete difficult exam (maximum time is 1 hour \()\) c. Assessed value of home d. Age at death

Short Answer

Expert verified
Expect skewed right for arrests and home value; skewed left for exam time; bell-shaped for age at death.

Step by step solution

01

Number of Times Arrested

Think about the distribution of how often people are arrested in a year. The majority of people are not arrested at all in a year, leading to a peak at zero. As the number of arrests increases, fewer people will have those higher counts, which creates a long tail to the right. Thus, the histogram for the number of times arrested is expected to be **skewed to the right**.
02

Time to Complete Exam

Consider how people complete a difficult exam with a time limit of 1 hour. Most students will use a significant portion of the available time, but very few will use almost none of it. This means most data points will be near the maximum time limit (1 hour), with fewer completing significantly quicker, creating a longer tail on the left. Thus, we expect the histogram to be **skewed to the left**.
03

Assessed Home Value

Assess the distribution of home values, where there are many homes with an average or slightly below average value, and fewer extremely high-value homes. This distribution tends to have a long tail on the right due to the few extraordinarily high-value homes. As such, the histogram for home values is likely **skewed to the right**.
04

Age at Death

For age at death, we expect most people to die at older ages, with fewer dying significantly younger or older. This often results in a normal distribution centered around the average life expectancy, forming a bell-shaped curve due to the symmetry around the mean. Therefore, the histogram for age at death is expected to be **bell-shaped**.

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.

Histogram Shapes
A histogram is a type of graph that visually represents the distribution of numerical data. It is a helpful tool for understanding how often different values occur in a dataset. The x-axis of a histogram represents the various values of the data, while the y-axis shows the frequency of these values.

Based on the data distribution, histograms can take on different shapes, each providing insight into the nature and tendencies of the dataset. The shape of a histogram can be:
  • Bell-shaped: Indicates a symmetrical distribution around a central value.
  • Right-skewed: Shows a long tail on the right side, with most data points clustered on the left.
  • Left-skewed: Has a long tail on the left, with the majority of data points on the right.
Interpreting the shape of a histogram helps in understanding the general pattern of data, identifying any anomalies, and guiding decisions about data analysis and further statistical methods.
Right-Skewed Distribution
A right-skewed distribution, also known as positively skewed, occurs when the majority of data points are concentrated on the lower end of the range, but there are a few higher values that stretch the tail to the right.

Examples of right-skewed distributions include:
  • Number of times arrested in a year: Most individuals are not arrested, creating a peak at zero, while fewer people have higher arrest counts, producing a long right tail.
  • Assessed home value: Most homes are valued at or slightly below average, while fewer homes are remarkably valuable, thus extending the right tail.
The presence of a right-skewed distribution often indicates data with some high outliers, suggesting the use of median and interquartile range as better measures than mean and standard deviation for summarizing the data.
Left-Skewed Distribution
In a left-skewed distribution, or negatively skewed distribution, the majority of data points cluster at the higher end with a few lower values extending the tail to the left.

One example of a left-skewed distribution is the time needed to complete a difficult exam with a set time limit:
  • Time to complete an exam: Many students will take most of the allotted time, creating a concentration near the maximum, with fewer finishing quickly and contributing to the left tail.
This type of distribution is less common but occurs in scenarios where most data are higher, with fewer low outliers. Using measures of central tendency that are resilient to skews, such as median, can provide more accurate insights.
Bell-Shaped Curve
A bell-shaped curve, or normal distribution, is characterized by its symmetry around the mean. This ideal distribution portrays a dataset where most values cluster around the center and gradually taper off into both tails.

The age at death is a classic example of a bell-shaped curve:
  • Age at death: Most individuals pass away around the average life expectancy, with fewer reaching significantly younger or older ages, resulting in a bell shape.
Normal distributions are significant in statistics as they allow the application of various statistical processes and hypothesis testing. They also provide a standard for comparison when evaluating the nature of other distributions.

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

Computer use During a recent semester at the University of Florida, students having accounts on a mainframe computer had storage space use (in kilobytes) described by the five-number summary, minimum \(=4, \mathrm{Q} 1=256\), median \(=530, \mathrm{Q} 3=1105,\) and maximum \(=320,000\) a. Would you expect this distribution to be symmetric, skewed to the right, or skewed to the left? Explain. b. Use the \(1.5 \times\) IQR criterion to determine if any potential outliers are present.

Great pay (on the average) The six full-time employees of Linda's Tanning Salon near campus had annual incomes last year of \(\$ 8900, \$ 9200, \$ 9200, \$ 9300, \$ 9500,\) \(\$ 9800 .\) Linda herself made \(\$ 250,000\). a. For the seven annual incomes at Linda's Salon, report the mean and median. b. Why is it misleading for Linda to boast to her friends that the average salary at her business is more than \(\$ 40,000 ?\)

Air pollution Example 18 discussed EU carbon dioxide emissions, which had a mean of 8.3 and standard deviation of 3.6 a. Canada's observation was \(16.5 .\) Find its \(z\) -score relative to the distribution of values for the EU nations, and interpret. b. Sweden's observation was \(5.0 .\) Find its \(z\) -score, and interpret.

Variability of cigarette taxes Here's the five-number summary for the distribution of cigarette taxes (in cents) among the 50 states and Washington, D.C. in the United States. $$ \begin{array}{c} \text { Minimum }=2.5, \mathrm{Q} 1=36, \text { Median }=60 \\ \mathrm{Q} 3=100, \text { Maximum }=205 \end{array} $$ a. About what proportion of the states have cigarette taxes (i) greater than 36 cents and (ii) greater than \(\$ 1 ?\) b. Between which two values are the middle \(50 \%\) of the observations found? c. Find the interquartile range. Interpret it. d. Based on the summary, do you think that this distribution was bell shaped? If so, why? If not, why not, and what shape would you expect?

Shape of home prices? According to the National Association of Home Builders, the median selling price of new homes in the United States in January 2007 was \(\$ 239,800\). Which of the following is the most plausible value for the standard deviation: $$ -\$ 15,000, \$ 1000, \$ 60,000, \text { or } \$ 1,000,000 ? \text { Why? } $$ Explain what's unrealistic about each of the other values.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.