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 4.2.11
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 = 50, p = 0.4
Verified step by step guidance1
Step 1: Recall the formulas for the mean, variance, and standard deviation of a binomial distribution. The mean (μ) is given by μ = n * p, the variance (σ²) is given by σ² = n * p * (1 - p), and the standard deviation (σ) is the square root of the variance, σ = √(σ²).
Step 2: Substitute the given values of n = 50 and p = 0.4 into the formula for the mean. Compute μ = 50 * 0.4.
Step 3: Substitute the same values of n = 50 and p = 0.4 into the formula for the variance. Compute σ² = 50 * 0.4 * (1 - 0.4).
Step 4: Use the result from Step 3 to calculate the standard deviation. Compute σ = √(σ²), where σ² is the variance obtained in the previous step.
Step 5: Summarize the results by stating the mean, variance, and standard deviation of the binomial distribution based on the calculations from the previous steps.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: n (the number of trials) and p (the probability of success on each trial). This distribution is useful in scenarios where there are two possible outcomes, such as success or failure.
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Mean & Standard Deviation of Binomial Distribution
Mean of a Binomial Distribution
The mean of a binomial distribution, also known as the expected value, is calculated using the formula μ = n * p. This value represents the average number of successes expected in n trials. For the given parameters n = 50 and p = 0.4, the mean provides a central value around which the distribution of outcomes is centered.
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Variance and Standard Deviation
Variance measures the spread of a distribution and is calculated for a binomial distribution using the formula σ² = n * p * (1 - p). The standard deviation, which is the square root of the variance, indicates how much the outcomes deviate from the mean. These measures are essential for understanding the variability in the number of successes in the given binomial trials.
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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
Finding Binomial Probabilities In Exercises 19–26, find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B.
Responsible Consumption Forty-five percent of consumers say it is important that the clothing they buy is made without child labor. You randomly select 16 consumers. Find the probability that the number of of consumers who say it is important that the clothing they buy is made without child labor is (a) e
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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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Textbook Question
Is the expected value of the probability distribution of a random variable always one of the possible values of x? Explain.v
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Textbook Question
"True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.
The mean of the random variable of a probability distribution describes how the outcomes vary."
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