Module 5 · Inference · Topic 8

Null & Alternative Hypotheses

Null Hypothesis (H₀). The null hypothesis is the default claim of no effect or no difference. It always contains an equality (=, ≤, or ≥).

Alternative Hypothesis (H₁). The alternative hypothesis is what the researcher wants to demonstrate. It is the claim that there is an effect, a difference, or a change.

One-Tailed vs 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.

Open the interactive lesson All topics

Key formulas

From the Inference formula sheet

Confidence Interval for a Mean: xbar +/- critical value * standard error — Estimate plus or minus a margin of error.
One-Sample Z Test: z = (xbar - mu_0) / (sigma / sqrt(n)) — Tests a sample mean against a null mean when population sigma is known.
Power: Power = 1 - beta — Probability of correctly rejecting a false null hypothesis.

See the full formula reference

Worked example

Formulating Hypotheses

A company claims a battery lasts 500 hours. You suspect it lasts less. H₀: μ = 500, H₁: μ < 500 (left-tailed test).

Related glossary terms

P-value: The probability of results this extreme or more extreme, assuming the null hypothesis is true. Example: A p-value of 0.03 is evidence against H0 at alpha = 0.05.
Confidence Level: The long-run success rate of a confidence interval procedure. Example: A 95% method captures the true parameter in about 95% of repeated samples.
Type I Error: Rejecting the null hypothesis when it is actually true. Example: The significance level alpha is the Type I error rate.
Type II Error: Failing to reject the null hypothesis when the alternative hypothesis is true. Example: A false negative in a hypothesis test is a Type II error.
Power: The probability of rejecting the null hypothesis when it is false. Example: Power equals 1 - beta, where beta is the Type II error probability.

Browse the full glossary

Go interactive

Work this topic in StatRise

Start with the free descriptive-statistics module and daily practice; this module's tracked lesson unlocks with Premium.

Inference lesson (Premium)Hypothesis Tests calculators Confidence Intervals calculators CI Coverage simulation Type I & II Errors simulation Practice questions

Keep reading