Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

Statistics

5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

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2:04 minutes

Problem 4.2.14

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

Verified step by step guidance

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.

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.

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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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

Textbook Question

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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

Key Concepts

Binomial Distribution

Mean of a Binomial Distribution

Variance and Standard Deviation

Watch next

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