How to Write and Test a Null Hypothesis (With Examples)
The concept of null hypothesis plays a key role in mathematics and statistics and is widely applicable to both real-life situations and exam scenarios. Understanding null hypothesis is essential for data analysis, research studies, and competitive exams like JEE, NEET, or CBSE board assessments.
What Is Null Hypothesis?
A null hypothesis (symbol: H0) is defined as a precise mathematical statement assuming there is no effect, difference, or relationship between variables or groups in a study. It is the foundation of hypothesis testing in statistics, where we check if any observed result is just due to chance. You’ll find this concept applied in areas such as sample data analysis, chi-square test, and real-life research experiments. The null hypothesis is always tested indirectly, and we never “accept” it — it can only be “rejected” or “not rejected” based on evidence.
Null Hypothesis Symbol & Key Formula
The standard way to write a null hypothesis is to use the symbol H0 followed by an equality or “no effect” statement:
| Type | Mathematical Statement | Example |
|---|---|---|
| Mean (μ) | H0: μ = μ0 | H0: μ = 60 |
| Proportion (p) | H0: p = p0 | H0: p = 0.5 |
| Two Means | H0: μ1 = μ2 | H0: Class A mean = Class B mean |
Here’s the standard formula for the null hypothesis (mean):
H0: μ = μ0
Cross-Disciplinary Usage
Null hypothesis is not only useful in Maths but also plays an important role in Physics (error analysis), Computer Science (algorithm testing), Economics (market studies), and even daily logical reasoning. Students preparing for JEE, NEET, or board exams will see its relevance in statistical questions, especially those on significance testing and p-value calculations.
How to Write a Null Hypothesis: Step-by-Step Illustration
- Identify the claim/problem: “Is the average score in Class 10A 70 marks?”
We need to check if the average is 70 or not. - State the statistical parameter: Use μ for “population mean.”
Parameter = μ (average marks) - Set up the null hypothesis: Use equality sign (“=”).
H0: μ = 70 - Write the alternative hypothesis (for comparison): Use ≠, <, or >.
H1: μ ≠ 70
Null Hypothesis vs Alternative Hypothesis
| Feature | Null Hypothesis (H0) | Alternative Hypothesis (H1) |
|---|---|---|
| Meaning | No effect, no difference, “status quo” | Presence of an effect or difference |
| Mathematical Notation | “=” (equal) | “≠”, “<”, “>” |
| Goal in Testing | Try to reject H0 with evidence | Accepted if H0 is rejected |
| Example | H0: μ = 70 | H1: μ ≠ 70 |
Example Problems and Solutions
Example 1: Testing the Mean
Problem: A sample of student marks is taken to check if the average mark equals 60. State the null hypothesis.
1. The question asks about the average (mean), so use μ.2. Null hypothesis assumes “no effect” (average is 60): H0: μ = 60
3. If evidence shows a significant difference, we reject H0.
Example 2: Chi Square Test
Problem: In a genetics experiment, expected ratio is 3:1. Observed results differ slightly. Write the null hypothesis for a chi-square test.
1. Chi-square checks association/categorical fit.2. Null hypothesis: “There is no significant difference between observed and expected results.”
3. Mathematically: H0: Observed = Expected ratios.
Try These Yourself
- Write the null and alternative hypotheses for: “Is the number of left-handed students equal in two different classes?”
- Express the null hypothesis for: “A coin is fair.”
- Given H0: p = 0.5, what does this mean in words?
- Find the null hypothesis for the claim: “The new medicine has no effect.”
Frequent Errors and Misunderstandings
- Stating H0 as an inequality: Null hypothesis must use “=”.
- Thinking “fail to reject” means we “accept” H0. In reality, we do not.
- Confusing null with “zero” — it just means “no difference,” not “value is zero.”
- Mixing up p-value interpretation: Small p-value (p < 0.05) means we reject H0.
Relation to Other Concepts
The idea of null hypothesis is connected closely with Type I and Type II errors, p-value calculation, and statistical inference. Mastering null hypothesis logic helps you understand critical ideas in data interpretation and inferential statistics.
Classroom Tip
A quick way to remember null hypothesis: “Assume nothing happened—prove otherwise!” Vedantu’s teachers often teach students to recite “Equal, Effect-less, Evidence-based” as a memory key: “EEE = H0”. This makes hypothesis questions easy to tackle during exams.
We explored null hypothesis—from definition, notation, formula, step-wise application, common mistakes, and how it links to other stats topics. Continue practicing with Vedantu to become confident in hypothesis testing and statistical reasoning! Live sessions and resources also include practice on chi square test and normal distribution to build your expertise.
FAQs on Null Hypothesis in Statistics: Meaning, Steps & Examples
1. What is a null hypothesis in statistics?
A null hypothesis, denoted as H₀, is a statement that assumes there is no significant difference or relationship between variables being studied. It's the starting point for hypothesis testing, where we aim to determine if observed data provides enough evidence to reject this initial assumption.
2. How do you write a null hypothesis?
A null hypothesis is written using the symbol H₀ followed by an equality statement about the population parameter(s). For example: H₀: μ = 10 (the population mean is equal to 10), or H₀: μ₁ = μ₂ (the population means of two groups are equal).
3. What is the symbol of the null hypothesis?
The symbol for the null hypothesis is H₀ (H naught).
4. What is an example of a null hypothesis?
Example: "There is no difference in average height between men and women." Mathematically, this could be written as H₀: μmen = μwomen.
5. How is the null hypothesis used in chi-square tests?
In a chi-square test, the null hypothesis typically assumes there's no association between two categorical variables. The test then determines if the observed frequencies differ significantly from the frequencies expected if the null hypothesis were true.
6. Why do we assume the null hypothesis is true at first?
Assuming the null hypothesis is true provides a baseline against which to compare our observed data. This allows us to determine whether the data provides sufficient evidence to reject the initial assumption of no effect or no difference.
7. How is the null hypothesis related to the p-value?
The p-value represents the probability of observing data as extreme as, or more extreme than, the data obtained, *assuming the null hypothesis is true*. A low p-value suggests that the observed results are unlikely to have occurred by chance alone, providing evidence to reject the null hypothesis.
8. Can a null hypothesis state an inequality?
No, the null hypothesis always states equality (=) or 'no effect'. Inequalities (<, >, ≠) are used in the alternative hypothesis (H₁).
9. What happens if we “fail to reject” the null hypothesis?
Failing to reject the null hypothesis means there isn't enough statistical evidence from the data to conclude a significant effect or relationship. It does *not* mean the null hypothesis is proven true; it simply means we lack sufficient evidence to reject it.
10. How does a Type I error relate to the null hypothesis?
A Type I error occurs when we incorrectly reject a true null hypothesis (a false positive). This is often represented by the significance level (α), typically set at 0.05.
11. What is the difference between a null hypothesis and an alternative hypothesis?
The null hypothesis (H₀) proposes no effect or difference, while the alternative hypothesis (H₁) proposes an effect or difference. They are mutually exclusive; if one is true, the other is false. Hypothesis testing aims to determine which hypothesis is more likely supported by the evidence.
12. What is the importance of the null hypothesis in research?
The null hypothesis is crucial in research because it provides a starting point for testing claims. By attempting to disprove the null hypothesis, researchers can build stronger evidence for alternative explanations and contribute to a deeper understanding of the phenomena under investigation.
