Hypothesis Testing Formulas

Hypothesis Testing Formula Reference

Complete collection of formulas for statistical hypothesis testing: test statistics, critical values, decision rules, and practical applications in statistical inference.

Comprehensive ReferencePractical ApplicationsStep-by-step Solutions

Quick Formula Reference

Essential hypothesis testing formulas for quick lookup

Fundamental Concepts

Core definitions and error probabilities in hypothesis testing

Type I Error (α)

\alpha(\theta) = P_{\theta}(\tilde{X} \in D | \theta \in \Theta_0)

Probability of rejecting H₀ when it is true (false positive)

Type II Error (β)

\beta(\theta) = P_{\theta}(\tilde{X} \in \overline{D} | \theta \in \Theta_1)

Probability of accepting H₀ when H₁ is true (false negative)

Power Function

g(\theta) = P_{\theta}(\tilde{X} \in D) = 1 - \beta(\theta)

Probability of correctly rejecting H₀ when it is false

Significance Level

\sup_{\theta \in \Theta_0} \alpha(\theta) \leq \alpha

Maximum Type I error probability under Neyman-Pearson principle

Test Statistics

Key test statistics for different scenarios

U-Test Statistic (σ² known)

U = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}} \sim N(0,1)

Standard normal test for population mean with known variance

T-Test Statistic (σ² unknown)

T = \frac{\bar{X} - \mu_0}{S/\sqrt{n}} \sim t(n-1)

t-distribution test for population mean with unknown variance

Chi-Square Test Statistic

\chi^2 = \frac{(n-1)S^2}{\sigma_0^2} \sim \chi^2(n-1)

Chi-square test for population variance

Two-Sample T-Test

T = \frac{\bar{X} - \bar{Y}}{S_w\sqrt{1/m + 1/n}} \sim t(m+n-2)

Compare means of two independent normal populations

Decision Rules

Critical values and rejection regions for common tests

Two-Sided U-Test

\text{Reject } H_0 \text{ if } |U| > u_{\alpha/2}

H₀: μ = μ₀ vs H₁: μ ≠ μ₀

Right-Sided U-Test

\text{Reject } H_0 \text{ if } U > u_\alpha

H₀: μ ≤ μ₀ vs H₁: μ > μ₀

Two-Sided T-Test

|T| > t_{\alpha/2}(n-1)

Critical region for t-test with unknown variance

P-value Decision Rule

\text{Reject } H_0 \text{ if P-value} < \alpha

General decision rule using probability values

Single Normal Population Tests

Complete formulas for testing normal population parameters

U-Test for Population Mean (σ² Known)

Hypotheses

H_0: \mu = \mu_0 \text{ vs } H_1: \mu \neq \mu_0 \text{ (two-sided)}

H_0: \mu \leq \mu_0 \text{ vs } H_1: \mu > \mu_0 \text{ (right-sided)}

H_0: \mu \geq \mu_0 \text{ vs } H_1: \mu < \mu_0 \text{ (left-sided)}

Assumptions

X_1, X_2, \ldots, X_n \sim N(\mu, \sigma^2)

\sigma^2 \text{ is known}

\text{Random sample}

Test Statistic

U = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}} \sim N(0,1) \text{ under } H_0

Rejection Regions

\text{Two-sided: } |U| > u_{\alpha/2}

\text{Right-sided: } U > u_\alpha

\text{Left-sided: } U < -u_\alpha

P-value Formulas

\text{Two-sided: } P = 2[1 - \Phi(|u|)]

\text{Right-sided: } P = 1 - \Phi(u)

\text{Left-sided: } P = \Phi(u)

where Φ(·) is the standard normal CDF

T-Test for Population Mean (σ² Unknown)

Hypotheses

H_0: \mu = \mu_0 \text{ vs } H_1: \mu \neq \mu_0

H_0: \mu \leq \mu_0 \text{ vs } H_1: \mu > \mu_0

H_0: \mu \geq \mu_0 \text{ vs } H_1: \mu < \mu_0

Assumptions

X_1, X_2, \ldots, X_n \sim N(\mu, \sigma^2)

\sigma^2 \text{ is unknown}

\text{Random sample}

Test Statistic

T = \frac{\bar{X} - \mu_0}{S/\sqrt{n}} \sim t(n-1) \text{ under } H_0

Where:

S^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2

Rejection Regions

\text{Two-sided: } |T| > t_{\alpha/2}(n-1)

\text{Right-sided: } T > t_\alpha(n-1)

\text{Left-sided: } T < -t_\alpha(n-1)

Chi-Square Test for Population Variance

Hypotheses

H_0: \sigma^2 = \sigma_0^2 \text{ vs } H_1: \sigma^2 \neq \sigma_0^2

H_0: \sigma^2 \leq \sigma_0^2 \text{ vs } H_1: \sigma^2 > \sigma_0^2

H_0: \sigma^2 \geq \sigma_0^2 \text{ vs } H_1: \sigma^2 < \sigma_0^2

Assumptions

X_1, X_2, \ldots, X_n \sim N(\mu, \sigma^2)

\mu \text{ is unknown}

\text{Random sample}

Test Statistic

\chi^2 = \frac{(n-1)S^2}{\sigma_0^2} \sim \chi^2(n-1) \text{ under } H_0

Rejection Regions

\text{Two-sided: } \chi^2 < \chi^2_{1-\alpha/2}(n-1) \text{ or } \chi^2 > \chi^2_{\alpha/2}(n-1)

\text{Right-sided: } \chi^2 > \chi^2_\alpha(n-1)

\text{Left-sided: } \chi^2 < \chi^2_{1-\alpha}(n-1)

Two Sample Comparison Tests

Formulas for comparing parameters between two populations

Two-Sample U-Test (σ₁², σ₂² Known)

Hypotheses

H_0: \mu_X = \mu_Y \text{ vs } H_1: \mu_X \neq \mu_Y

H_0: \mu_X \leq \mu_Y \text{ vs } H_1: \mu_X > \mu_Y

H_0: \mu_X \geq \mu_Y \text{ vs } H_1: \mu_X < \mu_Y

Assumptions

X_1, \ldots, X_m \sim N(\mu_X, \sigma_X^2), Y_1, \ldots, Y_n \sim N(\mu_Y, \sigma_Y^2)

\sigma_X^2, \sigma_Y^2 \text{ are known}

\text{Independent samples}

Test Statistic

U = \frac{\bar{X} - \bar{Y}}{\sqrt{\sigma_X^2/m + \sigma_Y^2/n}} \sim N(0,1) \text{ under } H_0

Rejection Regions

\text{Two-sided: } |U| > u_{\alpha/2}

\text{Right-sided: } U > u_\alpha

\text{Left-sided: } U < -u_\alpha

Two-Sample T-Test (Equal Variances)

Hypotheses

H_0: \mu_X = \mu_Y \text{ vs } H_1: \mu_X \neq \mu_Y

Assumptions

X_1, \ldots, X_m \sim N(\mu_X, \sigma^2), Y_1, \ldots, Y_n \sim N(\mu_Y, \sigma^2)

\sigma^2 \text{ is unknown but equal}

\text{Independent samples}

Test Statistic

T = \frac{\bar{X} - \bar{Y}}{S_w\sqrt{1/m + 1/n}} \sim t(m+n-2) \text{ under } H_0

Pooled Variance:

S_w^2 = \frac{(m-1)S_X^2 + (n-1)S_Y^2}{m+n-2}

Rejection Regions

\text{Two-sided: } |T| > t_{\alpha/2}(m+n-2)

F-Test for Variance Equality

Hypotheses

H_0: \sigma_X^2 = \sigma_Y^2 \text{ vs } H_1: \sigma_X^2 \neq \sigma_Y^2

H_0: \sigma_X^2 \leq \sigma_Y^2 \text{ vs } H_1: \sigma_X^2 > \sigma_Y^2

Assumptions

X_1, \ldots, X_m \sim N(\mu_X, \sigma_X^2), Y_1, \ldots, Y_n \sim N(\mu_Y, \sigma_Y^2)

\text{Independent samples}

Test Statistic

F = \frac{S_X^2}{S_Y^2} \sim F(m-1, n-1) \text{ under } H_0

Rejection Regions

\text{Two-sided: } F < F_{1-\alpha/2}(m-1,n-1) \text{ or } F > F_{\alpha/2}(m-1,n-1)

\text{Right-sided: } F > F_\alpha(m-1,n-1)

Generalized Likelihood Ratio Test (GLRT)

General framework for constructing hypothesis tests

Generalized Likelihood Ratio Test

Likelihood Ratio Definition

\lambda(\tilde{x}) = \frac{\sup_{\theta \in \Theta} L(\theta; \tilde{x})}{\sup_{\theta \in \Theta_0} L(\theta; \tilde{x})} = \frac{L(\hat{\theta}; \tilde{x})}{L(\hat{\theta}_0; \tilde{x})}

Test Rule:

\text{Reject } H_0 \text{ if } \lambda(\tilde{X}) > C

Critical Value:

\text{Choose }

\text{ such that }

\sup_{\theta \in \Theta_0}

\theta

(

\lambda

(

\tilde{X}

) > C)

\leq

\alpha

Large Sample Result (Wilks' Theorem)

2\log\lambda(\tilde{X}) \xrightarrow{d} \chi^2(r) \text{ as } n \to \infty

where r = $\dim$ ( $\Theta$ ) - $\dim$ ( $\Theta_0$ )

Examples

Normal Mean (σ² unknown)

Hypotheses: H_0:

\mu

\mu_0

\text{ vs }

H_1:

\mu

\neq

\mu_0

Likelihood Ratio:

\lambda = \left(1 + \frac{t^2}{n-1}\right)^{n/2}

Equivalent Test:

\text{Equivalent to t-test: }

|t| > t_{

\alpha

/2}(n-1)

Normal Variance

Hypotheses: H_0:

\sigma

^2 =

\sigma_0

\text{ vs }

H_1:

\sigma

\neq

\sigma_0

Likelihood Ratio:

\lambda = \left(\frac{\hat{\sigma}^2}{\sigma_0^2}\right)^{n/2} \exp\left(\frac{n}{2}\left(1 - \frac{\hat{\sigma}^2}{\sigma_0^2}\right)\right)

Equivalent Test:

\text{Equivalent to chi-square test}

Single Parameter Exponential Family Tests

Tests for exponential family distributions

Binomial Distribution B(n,p)

Hypotheses:

H_0: p = p_0 \text{ vs } H_1: p \neq p_0

Test Statistic:

U = \sum_{i=1}^n X_i \sim B(n, p_0) \text{ under } H_0

Normal Approximation:

Z = \frac{U - np_0}{\sqrt{np_0(1-p_0)}} \sim N(0,1) \text{ (large n)}

Rejection Region:

U < C_1 \text{ or } U > C_2

Poisson Distribution P(λ)

Hypotheses:

H_0: \lambda = \lambda_0 \text{ vs } H_1: \lambda \neq \lambda_0

Test Statistic:

U = \sum_{i=1}^n X_i \sim \text{Poisson}(n\lambda_0) \text{ under } H_0

Normal Approximation:

Z = \frac{U - n\lambda_0}{\sqrt{n\lambda_0}} \sim N(0,1) \text{ (large n)}

Rejection Region:

U < C_1 \text{ or } U > C_2

Exponential Distribution Exp(θ⁻¹)

Hypotheses:

H_0: \theta \geq \theta_0 \text{ vs } H_1: \theta < \theta_0

Test Statistic:

\frac{2n\bar{X}}{\theta_0} \sim \chi^2(2n) \text{ under } H_0

Rejection Region:

\frac{2n\bar{X}}{\theta_0} < \chi^2_{1-\alpha}(2n)

\text{Use when testing if parameter is below threshold}

Confidence Intervals & Hypothesis Testing Duality

Mathematical relationship between interval estimation and hypothesis testing

Duality Relationships

Test → Interval

C(\tilde{x}) = \{\theta_0 : \tilde{x} \in A(\theta_0)\}

Confidence set contains all parameter values that would not be rejected

Interval → Test

A(\theta_0) = \{\tilde{x} : \theta_0 \in C(\tilde{x})\}

Accept H₀: θ = θ₀ if θ₀ lies within confidence interval

Examples

Normal Mean (σ unknown)

Confidence Interval:

[\bar{x} - t_{\alpha/2}(n-1)\frac{s}{\sqrt{n}}, \bar{x} + t_{\alpha/2}(n-1)\frac{s}{\sqrt{n}}]

Hypothesis Test:

\text{Reject }

H_0:

\mu

\mu_0

\text{ if }

\mu_0

\notin

\text{CI}

Equivalence: 95\% \text{ CI corresponds to } \alpha = 0.05 \text{ test}

Normal Variance

Confidence Interval:

\left[\frac{(n-1)s^2}{\chi^2_{\alpha/2}(n-1)}, \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}(n-1)}\right]

Hypothesis Test:

\text{Reject }

H_0:

\sigma

^2 =

\sigma_0

\text{ if }

\sigma_0

\notin

\text{CI}

Equivalence: \text{Same } \alpha \text{ level for both procedures}

Practical Applications

Essential formulas for sample size planning and multiple testing corrections

Sample Size Determination

Formulas for determining adequate sample sizes

One-Sample Mean (Known σ)

n = \left(\frac{(u_{\alpha/2} + u_\beta)\sigma}{\delta}\right)^2

$\delta$ = | $\mu_1$ - $\mu_0$ | $\text{ (effect size)}$ , $\beta$ $\text{ (Type II error)}$

Two-Sample Mean Comparison

n = \frac{2(u_{\alpha/2} + u_\beta)^2\sigma^2}{(\mu_1 - \mu_2)^2}

$\text{Equal sample sizes per group}$

One-Sample Proportion

n = \frac{(u_{\alpha/2} + u_\beta)^2 p_0(1-p_0)}{(p_1 - p_0)^2}

p_0 $\text{ (null proportion)}$ , p_1 $\text{ (alternative proportion)}$

Multiple Testing Corrections

Adjustments for multiple hypothesis tests

Bonferroni Correction

\alpha_{\text{adj}} = \frac{\alpha}{m}

m $\text{ = number of tests}$

Holm-Bonferroni Method

\alpha_i = \frac{\alpha}{m-i+1}

$\text{Applied to ordered p-values }$ p_1 $\leq$ p_2 $\leq$ $\cdots$ $\leq$ p_m

False Discovery Rate (FDR)

\text{FDR} = E\left[\frac{V}{R}\right]

V $\text{ = false discoveries}$ , R $\text{ = total discoveries}$

📊 How to Use These Formulas Effectively

Master hypothesis testing with these formula application strategies and best practices

Choose the Right Test

Select test based on data type, sample size, and whether population parameters are known.

Check Assumptions

Verify normality, independence, and other assumptions before applying formulas.

Interpret Results

Always interpret statistical results in context and consider practical significance.

Apply these formulas

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