Textbook Question
In Exercises 1–4, find the indicated probability using the geometric distribution.
Find P(3) when p = 0.65
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Ch. 4 - Discrete Probability Distributions
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 4, Problem 4.1.19
Constructing and Graphing Discrete Probability Distributions In Exercises 19 and 20, (a) construct a probability distribution, and (b) graph the probability distribution using a histogram and describe its shape.
Televisions The number of high-definition (HD) televisions per household in a small town
Verified step by step guidance1
Step 1: Calculate the total number of households by summing the values in the 'Households' row. This will be used to determine the probabilities for the probability distribution.
Step 2: For each category of televisions (0, 1, 2, 3 or more), calculate the probability by dividing the number of households in that category by the total number of households. Use the formula: P(X = x) = (Number of households for x televisions) / (Total number of households).
Step 3: Construct the probability distribution table. The table should include two columns: one for the number of televisions (0, 1, 2, 3 or more) and one for the corresponding probabilities calculated in Step 2.
Step 4: Create a histogram to graph the probability distribution. On the x-axis, plot the number of televisions (0, 1, 2, 3 or more), and on the y-axis, plot the 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 heights of the bars.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Discrete Probability Distribution
A discrete probability distribution describes the likelihood of each possible outcome in a discrete sample space. In this context, it represents the number of high-definition televisions per household, where each outcome (0, 1, 2, or 3 or more televisions) has a corresponding probability calculated by dividing the number of households with that outcome by the total number of households.
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Variance & Standard Deviation of Discrete Random Variables
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 histogram will display the frequency of households for each category of televisions, allowing for a visual interpretation of the probability distribution and helping to identify patterns or trends in the data.
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Shape of the Distribution
The shape of a probability distribution provides insights into the data's characteristics, such as symmetry, skewness, and modality. For the given data on televisions per household, analyzing the histogram's shape will help determine whether the distribution is uniform, normal, skewed, or has multiple peaks, which can inform conclusions about household television ownership in the small town.
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Related Practice
Textbook Question
Mean, Variance, and Standard Deviation In Exercises 11–14, find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p.
n = 316, p = 0.82
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Textbook Question
In your own words, describe the difference between the value of x in a binomial distribution and in the Poisson distribution.
Textbook Question
Identifying Probability Distributions In Exercises 27 and 28, determine whether the distribution is a probability distribution. If it is not a probability distribution, explain why.
Textbook Question
Finding an Expected Value In Exercises 37 and 38, find the expected value E(x) to the player for one play of the game. If x is the gain to a player in a game of chance, then E(x) is usually negative. This value gives the average amount per game the player can expect to lose.
In American roulette, the wheel has the 38 numbers, 00, 0, 1, 2, . . ., 34, 35, and 36, marked on equally spaced slots. If a player bets \$1 on a number and wins, then the player keeps the dollar and receives an additional \$35. Otherwise, the dollar is lost.
Textbook Question
Graphical Analysis In Exercises 3–5, the histogram represents a binomial distribution with five trials. Match the histogram with the appropriate probability of success p. Explain your reasoning.
a. p = 0.25
b. p = 0.50
c. p = 0.75
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