Hypothesis Test Calculator

Hypothesis testing is a formal statistical procedure for deciding whether sample data provide sufficient evidence to reject a claim about a population parameter.

The Two Hypotheses

• Null hypothesis $H_0$: the default claim — assumes no effect, no difference, or a specific parameter value (e.g., $\mu = 50$).
• Alternative hypothesis $H_a$ (or $H_1$): the claim you want to support — can be two-sided ($\neq$), left-tailed ($<$), or right-tailed ($>$).

The Logic

Assume $H_0$ is true. Compute how extreme the sample result is if $H_0$ were true — this probability is the p-value. A very small p-value means the data would be highly unlikely under $H_0$, so we reject $H_0$ in favor of $H_a$.

Significance Level $\alpha$

$\alpha$ is the threshold for rejection. The most common choices are $\alpha = 0.05$ (5%) and $\alpha = 0.01$ (1%). If $p\text{-value} < \alpha$, you reject $H_0$.

Type I and Type II Errors

Decision	$H_0$ is true	$H_0$ is false
Reject $H_0$	Type I error (false positive), prob. $= \alpha$	Correct (power $= 1 - \beta$)
Fail to reject $H_0$	Correct (prob. $= 1 - \alpha$)	Type II error (false negative), prob. $= \beta$

One-Sample Z-Test (known $\sigma$)

Tests whether the population mean equals a specified value when the population standard deviation $\sigma$ is known:

$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

Compare $z$ to a standard normal critical value $z^*$ (e.g., $\pm 1.96$ for two-sided $\alpha = 0.05$).

One-Sample T-Test (unknown $\sigma$)

The most common test in practice — uses the sample standard deviation $s$:

$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}, \quad df = n - 1$

Compare $t$ to a t-distribution critical value. For large $n$, the t-distribution approaches the standard normal.

Two-Sample T-Test

Tests whether two independent population means are equal:

$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}}$

Degrees of freedom are estimated with the Welch–Satterthwaite approximation when $\sigma_1 \neq \sigma_2$ is not assumed.

Z-Test for a Proportion

Tests whether a population proportion equals a specified value $p_0$:

$z = \frac{\hat{p} - p_0}{\sqrt{p_0(1 - p_0)/n}}$

Valid when $np_0 \geq 10$ and $n(1 - p_0) \geq 10$.

Critical Values Quick Reference

Test type	$\alpha = 0.05$	$\alpha = 0.01$
Two-sided z	$\pm 1.96$	$\pm 2.576$
Right-tailed z	$1.645$	$2.326$
Left-tailed z	$-1.645$	$-2.326$

Follow these five steps for any hypothesis test:

1. State the hypotheses: Write $H_0$ and $H_a$ in terms of the population parameter. Identify the tail direction (two-sided, left, or right).

2. Choose the test and check conditions: Pick the appropriate test statistic (z or t). Verify sample size conditions (normality, independence).

3. Compute the test statistic: Plug in the sample values.

4. Find the p-value: Using the test statistic and the sampling distribution, compute the probability of observing a result at least as extreme as yours under $H_0$. For two-sided tests, double the one-tail area.

5. State the conclusion: If $p\text{-value} < \alpha$, reject $H_0$ — the data provide statistically significant evidence for $H_a$. Otherwise, fail to reject $H_0$ (this is NOT the same as accepting $H_0$).