# Hypothesis Testing for a Binomial Probability

Hypothesis Testing for a Binomial Probability

Binomial Distribution Basics

• A binomial distribution is utilised when an experiment is repeated a fixed number of times, each repetition being independent and has two possible outcomes or categories (success and failure).
• The binomial probability refers to the probability of getting a certain number of successes in a fixed number of trials of a binomial experiment.
• The binomial distribution is represented by two parameters: n (the number of trials) and p (the probability of success on any given trial).

Hypothesis Testing with Binomial Distribution

• The main objective of a binomial test is to test whether the probability of success, p, equals a specified value.
• Null Hypothesis(H0) in binomial tests often states that p equals a specified value, say p0.
• The Alternative Hypothesis(H1) posits that p does not equal the specified value, or that it’s less than or greater than p0.

Steps in Conducting a Binomial Test

• Step 1: Determine and state your Null Hypothesis(H0) and Alternative Hypothesis(H1).
• Step 2: Set your significance level. This is often set at 0.05.
• Step 3: Compute the test statistic. In binomial tests, this is the observed number of successes in n trials.
• Step 4: Compute the p-value, the probability of observing as many or more successes given that H0 is true.
• Step 5: Compare the p-value with the significance level. A p-value smaller than the significance level indicates strong evidence against the null hypothesis, and thus it can be rejected.

Interpreting the Result of the Binomial Test

• Reject H0 if p-value is less than the significance level. This shows significant evidence against null hypothesis.
• If the p-value is more than the significance level, then the null hypothesis cannot be rejected. It means that there isn’t enough evidence against it.

Assumptions for Binomial Tests

• The binomial hypothesis test assumes that each trial of the experiment is independent of each other.
• The probability of success is the same for each trial.
• There are a fixed number of trials.

Error Types in Binomial Tests

• Type I error is rejecting the null hypothesis when it is true, indicating a false positive.
• Type II error is failing to reject the null hypothesis when it is false, indicating a false negative.