Textbook Question
The five-year survival rate of people who undergo a liver transplant is 75%. The surgery is performed on six patients. (Source: Mayo Clinic)
b. Graph the binomial distribution using a histogram and describe its shape.
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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.Q.2c
The table lists the number of wireless devices per household in a small town in the United States.
c. Find the mean, variance, and standard deviation of the probability distribution and interpret the results.
Verified step by step guidance1
Step 1: Calculate the total number of households by summing the frequencies in the table. This will be used to compute the probabilities for the probability distribution. Total households = 277 + 471 + 243 + 105 + 46 + 22.
Step 2: Compute the probability for each number of wireless devices by dividing the frequency of each category by the total number of households. For example, P(0 devices) = 277 / Total households, P(1 device) = 471 / Total households, and so on.
Step 3: Calculate the mean (expected value) of the probability distribution using the formula: \( \mu = \sum (x \cdot P(x)) \), where \( x \) is the number of wireless devices and \( P(x) \) is the probability of \( x \). Multiply each \( x \) value by its corresponding probability and sum the results.
Step 4: Compute the variance using the formula: \( \sigma^2 = \sum ((x - \mu)^2 \cdot P(x)) \). Subtract the mean from each \( x \), square the result, multiply by the corresponding probability, and sum these values.
Step 5: Find the standard deviation by taking the square root of the variance: \( \sigma = \sqrt{\sigma^2} \). Interpret the mean, variance, and standard deviation in the context of the number of wireless devices per household.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Mean
The mean, or average, is a measure of central tendency that summarizes a set of values by dividing the sum of all values by the number of values. In the context of a probability distribution, the mean represents the expected value, indicating the average number of wireless devices per household. It is calculated by multiplying each value by its probability and summing the results.
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Calculating the Mean
Variance
Variance is a statistical measure that quantifies the degree of spread in a set of values. It is calculated by taking the average of the squared differences from the mean. In a probability distribution, variance helps to understand how much the number of wireless devices per household varies from the mean, providing insight into the distribution's dispersion.
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Standard Deviation
Standard deviation is the square root of the variance and provides a measure of the average distance of each data point from the mean. It is a key concept in statistics as it indicates the extent of variability in a dataset. A low standard deviation suggests that the data points are close to the mean, while a high standard deviation indicates a wider spread of values, which is crucial for interpreting the distribution of wireless devices in households.
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Related Practice
Textbook Question
The five-year survival rate of people who undergo a liver transplant is 75%. The surgery is performed on six patients. (Source: Mayo Clinic)
a. Construct a binomial distribution.
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Textbook Question
In the past year, thirty-three percent of U.S. adults have put off medical treatment because of the cost. You randomly select nine U.S. adults. Find the probability that the number who have put off medical treatment because of the cost in the past year is (a) exactly three, (b) at most four, and (c) more than five. (Source: Gallup)
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Textbook Question
An online magazine finds that the mean number of typographical errors per page is five. Find the probability that the number of typographical errors found on any given page is (a) exactly five, (b) less than five, and (c) exactly zero.
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Textbook Question
The table lists the number of wireless devices per household in a small town in the United States.
a. Construct a probability distribution.
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Textbook Question
In Exercises 1 and 2, determine whether the random variable x is discrete or continuous. Explain.
Let x represent the grade on an exam worth a total of 100 points.
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