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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 4.1.31
Chapter 4, Problem 4.1.31

Finding the Mean, Variance, and Standard Deviation In Exercises 29–34, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.
Machine Parts The number of defects per 1000 machine parts inspected
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Step 1: To find the mean (expected value) of the probability distribution, use the formula: μ=∑xxP(x). Multiply each defect value (x) by its corresponding probability P(x), then sum all the products.
Step 2: To find the variance, use the formula: σ²=∑x(x−μ)2P(x). First, calculate the squared difference between each defect value and the mean, then multiply each squared difference by its corresponding probability, and finally sum all the products.
Step 3: To find the standard deviation, take the square root of the variance using the formula: σ=σ². This provides a measure of the spread of the distribution.
Step 4: Interpret the mean. The mean represents the average number of defects per 1000 machine parts inspected. It provides a central value for the distribution.
Step 5: Interpret the standard deviation. The standard deviation indicates the variability or spread of the number of defects around the mean. A smaller standard deviation suggests that the defects are more consistently close to the mean, while a larger standard deviation indicates greater variability.

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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 expected value, of a probability distribution is calculated by multiplying each possible outcome by its probability and summing these products. It represents the average number of defects expected per 1000 machine parts inspected. In this case, it provides a central value around which the number of defects is likely to cluster.
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Variance

Variance measures the spread of a probability distribution by calculating the average of the squared differences from the mean. It quantifies how much the number of defects varies from the mean value. A higher variance indicates a wider spread of defect counts, while a lower variance suggests that the defect counts are more closely clustered around the mean.
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Standard Deviation

Standard deviation is the square root of the variance and provides a measure of dispersion in the same units as the data. It indicates how much the number of defects typically deviates from the mean. A smaller standard deviation implies that the defect counts are consistently close to the mean, while a larger standard deviation indicates greater variability in the number of defects.
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