In statistics, the null and alternative hypotheses are two mutually exclusive and exhaustive hypotheses used in hypothesis testing to evaluate the evidence in a sample.

The null hypothesis represents the default assumption that no significant difference or relationship exists between the studied variables. In contrast, the alternative hypothesis represents the claim or hypothesis the researcher is testing.

## Null and Alternate Hypotheses: Example of the Court of Law

In a criminal trial, the null hypothesis would represent the default assumption that the defendant is not guilty. In contrast, the alternative hypothesis would represent the claim that the defendant is guilty.

For example, if a defendant is charged with robbery, the null hypothesis would be that the defendant is not guilty of robbery. The alternative hypothesis would be that the defendant is guilty of robbery. The trial's goal would be to determine whether sufficient evidence exists to reject the null hypothesis in favour of the alternative hypothesis.

To do this, the prosecution and the defence would present evidence and make arguments to support their respective hypotheses. The judge or jury would then evaluate the evidence and decide whether the prosecution has provided sufficient evidence to reject the null hypothesis and conclude that the defendant is guilty.

If the prosecution can provide convincing evidence that supports the alternative hypothesis, the null hypothesis will be rejected, and the defendant will be found guilty. If the evidence is insufficient to reject the null hypothesis, the defendant would be found not guilty. (Please note it is "not guilty" and NOT "innocent")

In this example, the null and alternative hypotheses represent the trial's two possible conclusions, and the trial's goal is to determine which hypothesis is supported by the evidence.

## Null and Alternate Hypotheses: Example of a New Drug

For example, if a researcher is studying the effect of a new medication on blood pressure, the null hypothesis might be that the medication has no effect on blood pressure, while the alternative hypothesis might be that the medication lowers blood pressure. The study's goal would be to determine whether sufficient evidence exists in the sample data to reject the null hypothesis in favour of the alternative hypothesis.

To do this, the researcher would collect a sample of data from patients taking the medication and compare their blood pressure to a control group of patients who are not. The researcher would then use statistical tests to evaluate the evidence and determine whether the observed difference in blood pressure is statistically significant.

If the observed difference is statistically significant, it will provide evidence to reject the null hypothesis and support the alternative hypothesis. If the observed difference is not statistically significant, it will not provide sufficient evidence to reject the null hypothesis, and the null hypothesis will be retained.

In this example, the null and alternative hypotheses represent the two possible conclusions of the study, and the study's goal is to determine which hypothesis is supported.

## Null and Alternate Hypotheses: In Statistical Terms

In statistical terms, you **either reject the null hypothesis or fail to reject the null hypothesis**.

In hypothesis testing, the null hypothesis is typically written with an equal sign (=) to indicate no difference or relationship between the studied variables.

For example, if a researcher is studying the effect of a new medication on blood pressure, the null hypothesis might be stated as follows:

"The new medication has no effect on blood pressure" (H0: μ1 = μ2).

In this case, the equal sign indicates that the mean blood pressure of the group taking the medication is the same as the mean blood pressure of the control group.

The alternate hypothesis in the above examples will contain ≠ signs. For example:

"The new medication has an effect on blood pressure" (Ha: μ1 ≠ μ2)

These two hypotheses show that the null and alternate hypotheses are mutually exclusive and exhaustive. They do not overlap each other (mutually exclusive), and together they cover all possibilities (exhaustive). The above two examples were for "two-tail tests." The null hypothesis will get rejected if μ1 > μ2 or μ1 < μ2. The null hypothesis can be rejected on the left tail or right tail.

## Null and Alternate Hypotheses: Examples of Testing Equality of Means

### 1. Left-tail test:

**Null hypothesis: **The population mean is greater than or equal to 10. (H0: μ ≥ 10)

**Alternative hypothesis:** The population mean is less than 10. (H1: μ < 10)

### 2. Right-tail test:

**Null hypothesis:** The population mean is less than or equal to 10. (H0: μ ≤ 10)

**Alternative hypothesis: **The population mean is greater than 10. (H1: μ > 10)

### 3. Two-tail test:

**Null hypothesis:** The population mean is equal to 10. (H0: μ = 10)

**Alternative hypothesis: **The population mean is not equal to 10. (H1: μ ≠ 10)

## Null and Alternate Hypotheses: Examples of Regression Analysis

**Null hypothesis:** The slope of the regression line is equal to 0. (H0: β1 = 0)

**Alternative hypothesis: **The slope of the regression line is not equal to 0. (H1: β1 ≠ 0)

## Null and Alternate Hypotheses: The Test for Normality

**Null hypothesis: **The population is normally distributed

**Alternative hypothesis:** The population is not normally distributed

In these examples, the null hypothesis represents the default assumption that there is no significant difference or relationship between the studied variables. In contrast, the alternative hypothesis represents the claim or hypothesis the researcher is testing. The form of the null and alternative hypotheses depends on the test type and the research question being addressed.