How to Do Hypothesis Testing Step by Step (With Examples)
Hypothesis testing is a systematic procedure for deciding whether sample data provides enough evidence to reject a claim about a population. It’s one of the most important techniques in statistics — and one of the most commonly misunderstood. This guide walks through every step with a concrete example so you can apply the process to any problem.
The 5 Steps of Hypothesis Testing
Every hypothesis test follows the same five steps, regardless of the test type. Once you know these steps, you can apply them to t-tests, z-tests, chi-square tests, F-tests, and any other test you’ll encounter in a statistics course.
- State the null hypothesis (H₀) and alternative hypothesis (H₁)
- Choose the significance level (α)
- Calculate the test statistic
- Find the p-value or critical value
- Make a decision and state the conclusion
Step 1 — State the Hypotheses
The null hypothesis (H₀) is the default assumption — it states that there is no effect, no difference, or no relationship. The alternative hypothesis (H₁) is what you’re trying to find evidence for.
The way H₁ is written determines whether the test is one-tailed or two-tailed:
- Two-tailed: H₁: μ ≠ μ₀ (testing for any difference)
- Right-tailed: H₁: μ > μ₀ (testing for an increase)
- Left-tailed: H₁: μ < μ₀ (testing for a decrease)
A teacher claims that students who use a new study method score higher than the school average of 72 on the final exam. A sample of 30 students using the new method has a mean score of 76 with a standard deviation of 10. Test this claim at α = 0.05.
H₁: μ > 72 (new method improves scores) → right-tailed test
Step 2 — Choose the Significance Level
The significance level (α) is the probability threshold below which you’ll reject H₀. The most common choice is α = 0.05, which means you’re willing to accept a 5% chance of incorrectly rejecting a true null hypothesis (a Type I error).
Some fields use stricter thresholds: medical research often uses α = 0.01, and particle physics uses α = 0.0000003. Use whatever value is specified in your problem; if none is given, α = 0.05 is the standard default.
Example continued: The problem specifies α = 0.05. Since H₁ states μ > 72, this is a right-tailed test. The critical region is in the right tail of the distribution.
Step 3 — Calculate the Test Statistic
The test statistic converts your sample result into a standardized number that tells you how far the sample is from what H₀ predicts. For a one-sample t-test:
t = (x̄ − μ₀) / (s / √n)
Where x̄ is the sample mean, μ₀ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
t = (76 − 72) / (10 / √30)
t = 4 / (10 / 5.477)
t = 4 / 1.826 = 2.19
df = n − 1 = 29
Step 4 — Find the P-Value or Critical Value
There are two equivalent ways to make the decision:
P-value approach
Find the probability of getting a test statistic as extreme as yours, assuming H₀ is true. For t = 2.19, df = 29, right-tailed: p ≈ 0.018. Since p = 0.018 < α = 0.05, reject H₀.
Critical value approach
Find the critical value t* for α = 0.05, df = 29, right-tailed: t* = 1.699. Since t = 2.19 > t* = 1.699, the test statistic falls in the critical region, so reject H₀.
t = 2.19 > t* = 1.699 → REJECT H₀
Both approaches agree: reject the null hypothesis.
Step 5 — State the Conclusion
Never just say “reject” or “fail to reject” — always connect the decision back to the original research question in plain English.
There is sufficient statistical evidence at the 5% significance level to conclude that students using the new study method score higher than the school average of 72. (t(29) = 2.19, p = .018)
Common Mistakes to Avoid
- Saying “accept H₀” — you never accept the null; you either reject it or fail to reject it
- Choosing the wrong tail — read H₁ carefully to determine one-tailed vs two-tailed
- Wrong degrees of freedom — for one-sample t-test, df = n−1
- Confusing p-value with probability that H₀ is true — p-value is about the data given H₀, not about H₀ itself
- Reporting only p < 0.05 — always report the exact p-value and test statistic
When to Use Which Test
- One-sample t-test: Compare a sample mean to a known value
- Two-sample t-test: Compare means from two independent groups
- Paired t-test: Compare means from the same subjects before and after
- z-test for proportion: Test whether a sample proportion equals a hypothesized value
- Chi-square test: Test relationships between categorical variables
- One-way ANOVA: Compare means across three or more groups
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Run the Hypothesis Test Solver →Summary
Hypothesis testing follows a consistent five-step structure: state the hypotheses, set α, compute the test statistic, find the p-value or critical value, and state the conclusion. The type of test (t-test, z-test, chi-square, etc.) changes the formula in step 3, but the framework stays the same. Once you internalize this structure, any new test type becomes a matter of finding the right formula — the logical process doesn’t change.