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 4.2.14
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
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 into the formula for the mean. Here, n = 316 and p = 0.82. So, μ = 316 × 0.82.
Step 3: Substitute the given values into the formula for the variance. Using n = 316 and p = 0.82, calculate σ² = 316 × 0.82 × (1 - 0.82).
Step 4: Compute the standard deviation by taking the square root of the variance. Use the formula σ = √(σ²) and substitute the value of the variance obtained in Step 3.
Step 5: Interpret the results. The mean represents the expected number of successes, the variance measures the spread of the distribution, and the standard deviation provides a measure of the average deviation from the mean.
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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 applicable 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, providing a central point around which the distribution is centered. For the given values of n and p, the mean indicates how many successes we anticipate.
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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, provides a measure of the average distance of each data point from the mean. These metrics help in understanding the variability and reliability of the outcomes in the context of the binomial distribution.
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Related Practice
Textbook Question
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
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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.
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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.
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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
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
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