Hypothesis Testing Calculator | t-Test & z-Test Engine

HYPOTHESIS TESTING ENGINE ($t$-TEST & $z$-TEST)

1-Sample $\cdot$ 2-Sample Independent $\cdot$ Paired $t$-Test $\cdot$ $p$-Values

Inferential Statistics

Sample Mean ($\bar{x}$):

Hypothesized Mean ($\mu_0$):

Std Dev ($s$ or $\sigma$):

Sample Size ($n$):

Sample 1 Mean ($\bar{x}_1$):

Sample 1 Std Dev ($s_1$):

Sample 1 Size ($n_1$):

Sample 2 Mean ($\bar{x}_2$):

Sample 2 Std Dev ($s_2$):

Sample 2 Size ($n_2$):

Mean Difference ($\bar{d}$):

Std Dev of Diff ($s_d$):

Pairs Count ($n$):

Hypothesized Diff ($\mu_d$):

Significance Level ($\alpha$):

Alternative Hypothesis ($H_a$):

Distribution Assumption:

Statistical Outputs

Test Statistic ($t$ / $z$) 1.8257

Degrees of Freedom ($df$) 29

Standard Error ($SE$) 2.7386

Exact $p$-Value 0.0782

Critical Value ($t^*$ / $z^*$) $\pm 2.0452$

Decision ($\alpha = 0.05$) Fail to Reject $H_0$

Distribution & Rejection Region Plot

Hypothesis Testing Engine ($t$-Test & $z$-Test)

Hypothesis testing evaluates sample data to infer whether an observed pattern represents a statistically significant effect or mere chance variations. It provides a formal framework to reject or fail to reject a Null Hypothesis ($H_0$).

1. Mathematical Formulations by Test Type

Test Configuration	Null Hypothesis ($H_0$)	Test Statistic Formula	Degrees of Freedom ($df$)
1-Sample $t$-Test	$\mu = \mu_0$	$$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$	$n - 1$
2-Sample Welch's $t$-Test	$\mu_1 = \mu_2$	$$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$	$$\frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1 - 1} + \frac{(s_2^2/n_2)^2}{n_2 - 1}}$$
Paired $t$-Test	$\mu_d = 0$	$$t = \frac{\bar{d} - \mu_d}{s_d / \sqrt{n}}$$	$n - 1$
1-Sample $z$-Test	$\mu = \mu_0$	$$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$$	$\infty$ (Known $\sigma$)

2. Understanding $p$-Values and Decision Rules

The $p$-Value: The probability of obtaining a test statistic at least as extreme as the observed value, assuming the null hypothesis ($H_0$) is true.
Significance Level ($\alpha$): The predefined threshold for Type I error (commonly set to $0.05$ or $5\%$).
Rejection Decision: If $p\text{-value} \le \alpha$, reject $H_0$ (statistically significant result). If $p\text{-value} > \alpha$, fail to reject $H_0$.

Frequently Asked Questions

When should I select a $z$-test instead of a $t$-test?

Use a $z$-test when the true population standard deviation ($\sigma$) is known or when the sample size is exceptionally large ($n > 300$). Use a $t$-test when estimating the population standard deviation via sample standard deviation ($s$).

What makes Welch's $t$-test preferable for two independent samples?

Welch's $t$-test does not assume equal sample variances ($\sigma_1^2 = \sigma_2^2$), making it far more robust against unequal sample sizes and heteroscedasticity compared to Student's pooled two-sample $t$-test.

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Hypothesis Testing Calculator | t-Test & z-Test Engine