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Statistics

Hypothesis testing calculator

Test a claim about a population mean or proportion. Enter the sample summary, pick the alternative hypothesis and α, and the calculator states H₀ and H₁, computes the test statistic, p-value and critical value, and writes the conclusion in plain English.

Decision Reject H₀
H₀μ = 50
H₁μ ≠ 50
t statistic2.3333
p-value0.02834
Critical value (α = 0.05)±2.0639
Standard error0.9
Degrees of freedom24

Conclusion. At the 0.05 significance level there is enough evidence to conclude that the population mean differs from 50 (p = 0.02834).

−4−3−2−101234t = 2.33t

Shaded: rejection region (total area α = 0.05). Dot: your t statistic.

Show the working, step by step

  1. Standard error of the mean.

    SE = s ÷ √n = 4.5 ÷ √25 = 0.9

  2. Test statistic, with df = n − 1 = 24.

    t = (x̄ − μ₀) ÷ SE = (52.1 − 50) ÷ 0.9 = 2.3333

  3. p-value for H₁: μ ≠ 50.

    p = 2 × P(T(24) > |2.3333|) = 0.02834

  4. Critical value at α = 0.05: reject H₀ when |t| > 2.064.

    2.333 is in the rejection region, and p = 0.02834 < 0.05: reject H₀.

The p-value and the critical-value methods always agree: the statistic falls in the shaded region exactly when p is below α.

The formulas

t = (x̄ − μ₀) ÷ (s / √n), df = n − 1 z = (x̄ − μ₀) ÷ (σ / √n) z = (p̂ − p₀) ÷ √(p₀(1 − p₀) / n)

Each statistic measures how many standard errors the sample result sits from the value claimed by the null hypothesis. The proportion test uses p₀ in the standard error, not p̂, because the calculation assumes H₀ is true.

The five steps of a hypothesis test

  1. State the hypotheses. H₀ gives a single value (μ = 50); H₁ says the parameter is different, smaller or larger.
  2. Choose α, the risk of rejecting a true H₀ you will accept. 0.05 is the convention, not a law.
  3. Compute the test statistic from the sample.
  4. Find the p-value, or compare the statistic with the critical value.
  5. Decide, and state the conclusion in the language of the problem.

Worked example

A bottling line should fill cartons with 50 ml of concentrate on average. A sample of 25 cartons has mean x̄ = 52.1 ml and standard deviation s = 4.5 ml. Is the machine off target? The standard deviation comes from the sample, so this is a t-test with 24 degrees of freedom.

  • H₀: μ = 50, H₁: μ ≠ 50 (two-tailed), α = 0.05.
  • Standard error: 4.5 ÷ √25 = 0.9.
  • Test statistic: t = (52.1 − 50) ÷ 0.9 = 2.333.
  • p-value: 2 × P(T₂₄ > 2.333) = 0.0283.
  • Critical value: t₀.₀₂₅,₂₄ = 2.064, so the rejection region is |t| > 2.064.

2.333 is beyond 2.064 and 0.0283 is below 0.05, so both methods reject H₀. In words: at the 5% level there is enough evidence that the mean fill differs from 50 ml. The chart shows the t curve with both tails shaded; the statistic sits inside the right-hand shaded area.

At α = 0.01 the critical value rises to 2.797 and the same data no longer reject H₀. That is why α has to be fixed before looking at the result.

Interpreting the result

The p-value is the probability, assuming H₀ is true, of a statistic at least as far from the null value as the one observed. It is not the probability that H₀ is true, and 1 − p is not the probability that H₁ is true. A small p-value says the data would be surprising if H₀ held.

Statistical significance is also not practical importance. With thousands of observations a difference of 0.1 ml can be significant and still not worth fixing. Report the size of the difference, here 2.1 ml, next to the p-value.

Common mistakes

  • Using the z-test with a sample standard deviation. Unless σ is genuinely known, use t.
  • Picking a one-tailed test after seeing which way the data went.
  • Writing "accept H₀". The correct wording is "do not reject H₀".
  • Using p̂ instead of p₀ in the standard error of the proportion test.
  • Running a one-sample test on paired or two-group data. Paired data need differences first; two groups need a two-sample test.
Hypothesis testing calculator: the worked example on this page, with its result and chart
Hypothesis testing calculator: the worked example above, at a glance.

Common questions

How do I choose between the z-test and the t-test for a mean?

Use the t-test when the standard deviation comes from your sample, which is almost always. The z-test is for the rare case where the population standard deviation σ is known in advance, for example from a long-running manufacturing process. With large samples the two give nearly the same answer; with small samples the t-test's wider tails matter.

What is the difference between the p-value and the critical value approach?

They are two views of the same decision. The critical value approach asks whether the test statistic falls in the rejection region (beyond ±2.064 in the default example). The p-value approach asks whether the probability of a result at least this extreme is below α. One says reject exactly when the other does.

When should I use a one-tailed test?

Only when the direction was fixed before you saw the data and a result in the other direction would lead to the same action as no result. A one-tailed test puts all of α in one tail, so it halves the p-value for results in the predicted direction. Choosing the tail after looking at the data doubles the real false-positive rate.

Does "do not reject H₀" mean H₀ is true?

No. It means the data are not inconsistent with H₀ at the chosen level. A small sample can fail to detect a real difference. To say something positive about a small effect, look at a confidence interval or run a power analysis before collecting data.

What conditions does the proportion test need?

A random sample, independent observations, and at least 10 expected successes and 10 expected failures under H₀ (np₀ ≥ 10 and n(1 − p₀) ≥ 10). The calculator warns when these fail. For smaller samples use an exact binomial test.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.