Hypothesis Testing Calculator

Perform Z-tests and T-tests to evaluate null and alternative hypotheses with statistical significance

Hypothesis Test Setup

Choose based on sample size and known parameters

Direction of the alternative hypothesis

Value specified in the null hypothesis (H₀: μ = μ₀)

Probability of Type I error (rejecting true H₀)

Average value of your sample data

Number of observations in your sample

Sample standard deviation (uses T-test)

Example: Factory Weight Claim

Problem Statement

A factory claims that the average weight of its products is 500g. A quality control officer takes a sample of 30 products.

Sample results: Sample mean = 495g, Sample SD = 10g

Question: Test at α = 0.05 whether the factory's claim is true.

Solution Steps

1. Hypotheses: H₀: μ = 500, H₁: μ ≠ 500 (two-tailed)

2. Test statistic: Z = (495 - 500) / (10/√30) = -2.74

3. Critical value: ±1.96 (for α = 0.05, two-tailed)

4. Decision: |−2.74| > 1.96 → Reject H₀

5. Conclusion: Sufficient evidence that mean weight ≠ 500g

When to Use Which Test

Z-Test

  • • Population SD (σ) is known
  • • Sample size n ≥ 30
  • • Population is normally distributed
  • • More powerful than t-test

T-Test

  • • Population SD (σ) is unknown
  • • Any sample size (especially n < 30)
  • • Uses sample SD (s) as estimate
  • • More conservative than z-test

Types of Alternative Hypotheses

Two-tailed (μ ≠ μ₀)

Tests if mean is different from hypothesized value

Right-tailed (μ > μ₀)

Tests if mean is greater than hypothesized value

Left-tailed (μ < μ₀)

Tests if mean is less than hypothesized value

Understanding Hypothesis Testing

What is Hypothesis Testing?

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It provides a structured approach to determine whether observed differences are statistically significant or due to random chance.

Key Components

  • Null Hypothesis (H₀): Statement of no effect or difference
  • Alternative Hypothesis (H₁): Statement of effect or difference
  • Test Statistic: Standardized measure of evidence
  • P-value: Probability of observing data if H₀ is true

Test Formulas

Z-test Statistic:

Z = (x̄ - μ₀) / (σ / √n)

When population standard deviation σ is known

T-test Statistic:

t = (x̄ - μ₀) / (s / √n)

When population standard deviation σ is unknown

Decision Rules

  • Critical Value: Reject H₀ if |test statistic| > critical value
  • P-value: Reject H₀ if p-value < α
  • Confidence: Higher confidence → larger critical values

Types of Errors

Type I Error (α)

  • • Rejecting true null hypothesis
  • • False positive result
  • • Controlled by significance level
  • • More serious in some contexts

Type II Error (β)

  • • Failing to reject false null hypothesis
  • • False negative result
  • • Related to statistical power (1-β)
  • • Reduced by larger sample sizes