# 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(H**in binomial tests often states that_{0})*p*equals a specified value, say*p*._{0}- The
**Alternative Hypothesis(H**posits that_{1})*p*does not equal the specified value, or that it’s less than or greater than*p*._{0}

**Steps in Conducting a Binomial Test**

- Step 1: Determine and state your
**Null Hypothesis(H**and_{0})**Alternative Hypothesis(H**._{1}) - 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 H_{0}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
**H**if_{0}**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.