Hypothesis Testing Calculator

Analyze sample data to draw conclusions about a population.

Perform Z-tests and T-tests for means and proportions. Determine statistical significance by calculating p-values, test statistics, and critical values based on your data.

Practical Examples

See how the calculator works with real-world scenarios.

Z-Test for Mean (Quality Control)

zTestMean

A factory produces bolts with a mean diameter of 10mm and population st. dev. of 0.03mm. A sample of 50 bolts has a mean of 10.01mm. Test if the mean has changed at α = 0.05.

Test: zTestMean

H₀: 10, H₁: twoTailed

α: 0.05

T-Test for Mean (New Drug Efficacy)

tTestMean

A new drug is tested to see if it lowers blood pressure. The mean reduction is claimed to be >10 mmHg. In a sample of 30 patients, the mean reduction was 12 mmHg with a st. dev. of 3. Test this claim at α = 0.05.

Test: tTestMean

H₀: 10, H₁: rightTailed

α: 0.05

Z-Test for Proportion (A/B Testing)

zTestProportion

A website wants to know if a new button design increases the click-through rate from the current 8%. Out of 1000 visitors, 95 clicked the new button. Test if the new rate is higher at α = 0.05.

Test: zTestProportion

H₀: 0.08, H₁: rightTailed

α: 0.05

Z-Test for Mean (Fuel Efficiency)

zTestMeanLeft

A car model is advertised to have a fuel efficiency of at least 30 mpg (σ = 2). A consumer group tests 40 cars and finds a mean of 29 mpg. Test if the efficiency is less than advertised at α = 0.01.

Test: zTestMean

H₀: 30, H₁: leftTailed

α: 0.01

Other Titles
Understanding Hypothesis Testing: A Comprehensive Guide
Learn the core concepts of statistical hypothesis testing, from setting up hypotheses to interpreting p-values and making sound, data-backed decisions.

What is Hypothesis Testing?

  • The Core Idea
  • Null vs. Alternative Hypothesis
  • The Role of a 'p-value'
Hypothesis testing is a fundamental procedure in statistical inference. It allows practitioners to use sample data to evaluate a claim or hypothesis about a population parameter. The process involves making a conjecture about the population, collecting sample data, and then using the data to determine whether there is enough evidence to support the conjecture.
The Core Idea
The main purpose of hypothesis testing is to decide between two competing hypotheses: the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis typically represents the status quo or a statement of no effect, while the alternative hypothesis represents the research question or the effect we want to demonstrate.
Null vs. Alternative Hypothesis
The null hypothesis (H₀) is a statement of no difference or no relationship. For example, H₀ might state that the mean weight of a product is 500 grams. The alternative hypothesis (H₁) is what we test for. It can be one-sided (e.g., the mean weight is greater than 500 grams) or two-sided (e.g., the mean weight is not equal to 500 grams).
The Role of a 'p-value'
The p-value is a crucial output of a hypothesis test. It is the probability of observing sample results as extreme as, or more extreme than, the results actually observed, assuming the null hypothesis is true. A small p-value (typically ≤ 0.05) indicates that your data are unlikely under the null hypothesis, providing evidence to reject it.

Step-by-Step Guide to Using the Hypothesis Testing Calculator

  • Selecting the Right Test
  • Defining Your Hypotheses
  • Interpreting the Results
This calculator simplifies the process of hypothesis testing. Follow these steps to get your results.
1. Selecting the Right Test

Choose your test based on your data:

  • Z-Test (Mean): Use when you know the population standard deviation (σ).
  • T-Test (Mean): Use when the population standard deviation is unknown and you must use the sample standard deviation (s).
  • Z-Test (Proportion): Use for categorical data to test a claim about a population proportion.
2. Defining Your Hypotheses and Inputs
Enter the null hypothesis value (the value you are testing against). Select the alternative hypothesis (is the parameter not equal to, greater than, or less than the null value?). Input your sample data (mean, standard deviation, sample size, etc.) and the significance level (α).
3. Interpreting the Results

The calculator provides the test statistic, p-value, and critical value. The key is the decision:

  • Reject H₀: If the p-value is less than or equal to your significance level (α), the result is statistically significant.
  • Fail to Reject H₀: If the p-value is greater than α, the result is not statistically significant.

Real-World Applications of Hypothesis Testing

  • Marketing and Business
  • Medicine and Healthcare
  • Scientific Research
Hypothesis testing is used across countless fields to make informed decisions.
Marketing and Business (A/B Testing)
Companies use hypothesis testing to determine if a change, such as a new website layout or ad campaign, leads to a statistically significant improvement in a key metric like conversion rate. This is a classic example of a Z-test for proportions.
Medicine and Healthcare
In clinical trials, researchers use t-tests or z-tests to determine if a new drug is more effective than a placebo or an existing treatment. For example, they might test if a new drug significantly lowers cholesterol levels more than the current standard drug.
Scientific Research
Scientists use hypothesis testing to validate their theories. For example, a psychologist might test if a new therapy technique reduces anxiety scores in patients, or an agricultural scientist might test if a new fertilizer increases crop yield.

Common Misconceptions and Correct Methods

  • P-Value is Not the Probability of H₀ Being True
  • Statistical vs. Practical Significance
  • The Importance of Assumptions
P-Value is Not the Probability of H₀ Being True
A common mistake is to interpret the p-value as the probability that the null hypothesis is true. The p-value is calculated assuming H₀ is true. It tells you the probability of your data, not your hypothesis.
Statistical vs. Practical Significance
A statistically significant result (a small p-value) doesn't always mean the result is practically important or meaningful. With a very large sample size, even a tiny, trivial effect can become statistically significant. Always consider the effect size and the context of your research.
This calculator simplifies the process, but remember that each test has underlying assumptions (e.g., normality of data, random sampling). Violating these assumptions can lead to incorrect conclusions. Always ensure your data meets the criteria for the test you are performing.

Mathematical Derivations and Formulas

  • Formula for Z-Test (Mean)
  • Formula for T-Test (Mean)
  • Formula for Z-Test (Proportion)
The calculations are based on standardized test statistics. Here are the formulas used by the calculator.
Formula for Z-Test (Mean)
The Z-statistic measures how many standard deviations the sample mean (x̄) is from the population mean (μ₀). The formula is: Z = (x̄ - μ₀) / (σ / √n)
Formula for T-Test (Mean)
The T-statistic is used when the population standard deviation (σ) is unknown. It's similar to the Z-statistic but uses the sample standard deviation (s). The formula is: t = (x̄ - μ₀) / (s / √n)
Formula for Z-Test (Proportion)
This test compares a sample proportion (p̂) to a population proportion (p₀). The formula for the Z-statistic is: Z = (p̂ - p₀) / √[p₀(1 - p₀) / n], where p̂ = x / n.