Skip to main content
Ch. 4 - Discrete Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 4 - Discrete Probability DistributionsProblem 4.R.4
Chapter 4, Problem 4.R.4

In Exercises 3 and 4, (a) construct a probability distribution, and (b) graph the probability distribution using a histogram and describe its shape.


The number of hours students in a college class slept the previous night
hist

Verified step by step guidance
1
Step 1: Calculate the total number of students by summing the frequencies provided in the table. This will be used to compute probabilities for the probability distribution. Total students = 1 + 6 + 13 + 23 + 14 + 4 + 2.
Step 2: Construct the probability distribution by dividing the frequency of each hour by the total number of students. For example, the probability for 4 hours is calculated as P(4) = Frequency of 4 hours / Total students. Repeat this for all hours (4, 5, 6, 7, 8, 9, 10).
Step 3: Create a table for the probability distribution. The table should have two columns: 'Hours' and 'Probability'. Populate the 'Probability' column with the calculated probabilities for each hour.
Step 4: Graph the probability distribution using a histogram. On the x-axis, plot the 'Hours' (4, 5, 6, 7, 8, 9, 10), and on the y-axis, plot the corresponding probabilities. Ensure the bars are proportional to the probabilities.
Step 5: Analyze the shape of the histogram. Determine whether the distribution is symmetric, skewed left, skewed right, or uniform based on the visual representation of the histogram.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability Distribution

A probability distribution describes how the probabilities are distributed over the values of a random variable. In this context, it represents the likelihood of different hours of sleep among students. Each hour corresponds to a probability calculated by dividing the number of students who slept that many hours by the total number of students.
Recommended video:
Guided course
06:39
Calculating Probabilities in a Binomial Distribution

Histogram

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into bins or intervals. In this case, the hours of sleep will be the x-axis, and the frequency of students will be the y-axis. The height of each bar indicates how many students fall into each sleep duration category, providing a visual summary of the data.
Recommended video:
Guided course
05:54
Intro to Histograms

Shape of Distribution

The shape of a distribution refers to the visual appearance of the histogram, which can indicate patterns such as normality, skewness, or modality. Analyzing the shape helps in understanding the underlying trends in the data, such as whether most students slept fewer or more hours, and can inform conclusions about sleep habits in the class.
Recommended video:
06:06
Uniform Distribution
Related Practice
Textbook Question

In Exercises 21–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities

Thirty-six percent of Americans think there is still a need for the practice of changing their clocks for Daylight Savings Time. You randomly select seven Americans. Find the probability that the number who say there is still a need for changing their clocks for Daylight Savings Time is (c) at least six.

92
views
Textbook Question

In Exercises 17 and 18, (a) construct a binomial distribution, (b) graph the binomial distribution using a histogram and describe its shape, and (c) identify any values of the random variable x that you would consider unusual. Explain your reasoning.

Seventy-two percent of U.S adults have read a book in any format in the past year. You randomly select five U.S adults and ask them whether they have read a book in any format in the past year. The random variable represents the number of adults who have read a book in any format in the past year. (Source: Pew Research)

131
views
Textbook Question

In Exercises 9 and 10, find the expected net gain to the player for one play of the game.


It costs \(25 to bet on a horse race. The horse has a 1/8 chance of winning and a 1/4 chance of placing second or third. You win \)125 if the horse wins and receive your money back if the horse places second or third.

121
views
Textbook Question

In Exercises 19 and 20, find the mean, variance, and standard deviation of the binomial distribution for the given random variable. Interpret the results and determine any unusual values.

About 13% of U.S. drivers are uninsured. You randomly select eight U.S. drivers and ask them whether they are uninsured. The random variable represents the number who are uninsured. (Source: Insurance Research Council)

145
views
Textbook Question

In Exercises 21–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

Fourteen percent of noninstitutionalized U.S. adults smoke cigarettes. After randomly selecting ten noninstitutionalized U.S. adults, you ask them whether they smoke cigarettes. Find the probability that the first adult who smokes cigarettes is (c) not one of the first six persons selected.

77
views
Textbook Question

In Exercises 21–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities

Thirty-six percent of Americans think there is still a need for the practice of changing their clocks for Daylight Savings Time. You randomly select seven Americans. Find the probability that the number who say there is still a need for changing their clocks for Daylight Savings Time is (b) less than two

98
views