Binomial Distribution and Bernoulli Trials Quiz

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Questions and Answers

What is the binomial distribution?

A distribution used for normalizing data
A distribution for large sample sizes
A discrete probability distribution (correct)
A continuous probability distribution

What are the parameters of the binomial distribution?

x and y
n and p (correct)
alpha and beta
n and q

What is a Bernoulli trial?

An experiment with a fixed outcome
A trial with only one outcome (correct)
An experiment with independent outcomes
A trial with a continuous outcome

When does the binomial distribution become a good approximation even for large sample sizes?

When N is much larger than n (A) Signup and view all the answers

In general, how do we represent a random variable X following the binomial distribution with parameters n and p?

X ~ B(n, p) (A) Signup and view all the answers

What is the binomial distribution?

The binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each with a yes–no outcome and a probability of success (p). Signup and view all the answers

What is a Bernoulli trial?

A Bernoulli trial is a single success/failure experiment with a yes–no outcome, and is also called a Bernoulli experiment. Signup and view all the answers

What is the basis for the popular binomial test of statistical significance?

The binomial distribution is the basis for the popular binomial test of statistical significance. Signup and view all the answers

When is the hypergeometric distribution used instead of the binomial distribution?

The hypergeometric distribution is used when the sampling is carried out without replacement, and the draws are not independent. Signup and view all the answers

How do we represent a random variable X following the binomial distribution with parameters n and p?

We write X ~ B(n, p) to represent a random variable following the binomial distribution with parameters n and p. Signup and view all the answers

What is the binomial distribution in probability theory and statistics?

The binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each with its own Boolean-valued outcome: success or failure. Signup and view all the answers

What is a Bernoulli trial?

A single success/failure experiment, where each experiment asks a yes–no question and has its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p). Signup and view all the answers

When is the binomial distribution a good approximation even for large sample sizes?

The binomial distribution remains a good approximation for large sample sizes when the population size N is much larger than the sample size n. Signup and view all the answers

What is the basis for the popular binomial test of statistical significance?

The binomial distribution is the basis for the popular binomial test of statistical significance. Signup and view all the answers

When is the hypergeometric distribution used instead of the binomial distribution?

The hypergeometric distribution is used instead of the binomial distribution when the sampling is carried out without replacement. Signup and view all the answers

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Study Notes

Binomial Distribution Overview

Describes the number of successes in a fixed number of independent Bernoulli trials.
Each trial has two possible outcomes: success or failure.
Characterized by its discrete probability distribution.

Parameters of Binomial Distribution

n: Represents the number of trials.
p: Represents the probability of success on each trial.

Bernoulli Trial

A single experiment or process that results in one of two outcomes (success or failure).
Each trial is independent of others.
Probability of success is denoted by p, while the probability of failure is 1-p.

Approximation for Large Sample Sizes

The binomial distribution can be approximated using a normal distribution when:
- Both np and n(1-p) are greater than or equal to 5.
This approximation simplifies calculations for large n.

Representation of Random Variable

A random variable X following a binomial distribution is denoted as X ~ Binomial(n, p).
Represents the count of successes in n trials with success probability p.

Binomial Test of Statistical Significance

The basis lies in comparing the observed number of successes to the expected using the binomial distribution.
Helps assess whether the observed frequency of successes provides enough evidence against a null hypothesis.

Hypergeometric Distribution Use

Used when sampling without replacement from a finite population.
Differentiates from binomial distribution, which assumes replacement and independence of trials.
Necessary when the sample size is a significant portion of the population size.