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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.7
Chapter 4, Problem 4.R.7

In Exercises 7 and 8, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.


The number of cell phones per household in a small town
cell

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Step 1: To find the mean of the probability distribution, use the formula for the expected value: \( \mu = \sum (x \cdot P(x)) \), where \( x \) represents the number of cell phones and \( P(x) \) represents the corresponding probability. Multiply each \( x \) value by its probability and sum the results.
Step 2: To calculate the variance, use the formula \( \sigma^2 = \sum [(x - \mu)^2 \cdot P(x)] \). First, subtract the mean \( \mu \) from each \( x \) value, square the result, and then multiply by the corresponding probability. Sum these values to get the variance.
Step 3: To find the standard deviation, take the square root of the variance: \( \sigma = \sqrt{\sigma^2} \). This provides a measure of the spread of the distribution.
Step 4: Interpret the mean: The mean represents the average number of cell phones per household in the town. It is a weighted average based on the probabilities provided.
Step 5: Interpret the standard deviation: The standard deviation indicates how much the number of cell phones per household varies from the mean. A smaller standard deviation suggests less variability, while a larger standard deviation indicates more variability.

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Key Concepts

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

Mean of a Probability Distribution

The mean of a probability distribution, also known as the expected value, is calculated by multiplying each outcome by its probability and summing these products. It provides a measure of the central tendency of the distribution, indicating the average number of cell phones per household in this context.
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Variance and Standard Deviation

Variance measures the spread of a probability distribution by calculating the average of the squared differences from the mean. The standard deviation, the square root of the variance, indicates how much the values typically deviate from the mean, providing insight into the variability of cell phone ownership in households.
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Interpreting Results

Interpreting the results involves analyzing the calculated mean, variance, and standard deviation to understand the distribution of cell phones per household. This includes discussing what the average number of cell phones suggests about the community and how the variability reflects differences in cell phone ownership among households.
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Related Practice
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Textbook Question

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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.

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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.

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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

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