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Null distribution

In statistical hypothesis testing, the null distribution is the probability distribution of a test statistic under the assumption that the null hypothesis is true.^[1] It provides the theoretical foundation for evaluating how extreme an observed test statistic is, thereby enabling inferences about the validity of the null hypothesis.^[2] The null distribution is used to compute p-values, which represent the probability of obtaining a test statistic at least as extreme as the observed one assuming the null hypothesis holds, or to define rejection regions based on a chosen significance level (e.g., α = 0.05).^[3] For instance, in a one-sample t-test for a population mean, the test statistic follows a t-distribution with n-1 degrees of freedom under the null, allowing rejection of the null if the observed t-value falls in the tail beyond the critical value.^[1] Similarly, for tests of independence in contingency tables, the null distribution is often a chi-squared distribution, derived from the assumption of no association between variables.^[2] Null distributions can be derived analytically under parametric assumptions, such as normality via the central limit theorem for large samples, or approximated through simulation methods like permutation tests or bootstrapping when exact forms are unavailable.^[3] This flexibility ensures applicability across diverse statistical models, from simple means comparisons to complex regression analyses, while controlling the Type I error rate—the probability of falsely rejecting a true null hypothesis.^[4]

Fundamentals

Definition

In statistics, the null distribution refers to the probability distribution of a test statistic under the assumption that the null hypothesis is true.^[2] It describes the expected sampling variability of the test statistic when there is no effect, no difference, or no association in the population, as specified by the null hypothesis H_0.^[5] The null hypothesis H_0 is a statement asserting the absence of an effect or relationship, such as equality of population means or independence of variables.^[6] A test statistic, which is a function derived from the sample data to summarize evidence against H_0, follows this null distribution when H_0 holds.^[7] In contrast, under the alternative hypothesis H_A, the test statistic would follow a different distribution, potentially shifting the probability mass to more extreme values.^[2] Mathematically, the null distribution is often expressed through the cumulative distribution function P(T \leq t \mid H_0), where T is the test statistic and t is an observed value.^[2] The concept of the null distribution was introduced by R.A. Fisher in the 1920s and further developed within the Neyman-Pearson framework for hypothesis testing, as outlined in their seminal 1933 paper on efficient tests of statistical hypotheses.^[8]^[9] This foundational work formalized the role of distributions under both null and alternative hypotheses to construct optimal decision rules. The null distribution underpins p-value calculations by providing the baseline probability of extreme outcomes if H_0 is correct.^[2]

Role in Hypothesis Testing

In hypothesis testing, the null distribution serves as the foundational probability distribution of the test statistic under the assumption that the null hypothesis H_0 is true, enabling researchers to quantify the evidence against H_0 based on observed data. It is used to compute the p-value, defined as the probability of obtaining a test statistic at least as extreme as the observed value given H_0, such as P(T \geq |t_{\text{obs}}| \mid H_0) for a two-sided test, where T is the test statistic. This p-value measures the compatibility of the data with the null hypothesis, with smaller values indicating stronger evidence against H_0. Additionally, the null distribution defines rejection regions, which are the tails of the distribution where the test statistic would lead to rejecting H_0 at a specified significance level \alpha, such as the upper \alpha-quantile for a one-sided test.^[1]^[10] The null distribution directly controls the Type I error rate, or the probability of falsely rejecting H_0 when it is true, denoted as \alpha = P(\text{reject } H_0 \mid H_0 \text{ true}). By setting \alpha (commonly 0.05 or 0.01), the rejection region is calibrated so that the probability of a Type I error does not exceed this level under the null distribution, ensuring a controlled risk of false positives in the testing procedure. This framework, formalized in the Neyman-Pearson approach, emphasizes error rates over direct probability statements about hypotheses.^[10]^[9] The decision rule in hypothesis testing involves comparing the observed test statistic to critical values derived from the null distribution's quantiles; for instance, in a right-tailed test, reject H_0 if the observed statistic exceeds the $1 - \alpha quantile of the null distribution. Equivalently, rejection occurs if the p-value is less than or equal to \alpha. This rule provides a systematic way to make inferences, balancing the risks of errors while relying on the null distribution for calibration.^[1]^[10] The validity of the null distribution in hypothesis testing depends on key assumptions, including random sampling from the population and the specified probabilistic model holding true under H_0, such as normality or independence of observations. Violations of these assumptions can distort the null distribution, leading to invalid p-values or error rates, underscoring the need for careful verification of preconditions before applying the testing framework.^[1]^[3]

Obtaining the Null Distribution

Analytical Derivation

Analytical derivation of the null distribution in parametric hypothesis testing follows a systematic process: first, specify the underlying statistical model, such as assuming independent and identically distributed observations from a known parametric family; second, state the null hypothesis H_0 that imposes restrictions on the parameters; third, construct a test statistic as a function of the data that captures deviations from H_0; and fourth, transform the statistic to a pivotal form whose distribution under H_0 does not depend on nuisance parameters, often by standardization or ratio formation, leading to a known distribution—exact in cases like the t or F under normality, or asymptotic like the chi-squared for goodness-of-fit tests. This approach relies on exact distributional properties under model assumptions, such as normality, to obtain closed-form expressions for the null distribution. In the parametric case of testing a population mean, consider independent observations X_1, \dots, X_n \sim N(\mu, \sigma^2) with \sigma^2 unknown. Under H_0: \mu = \mu_0, the sample mean \bar{X} satisfies \sqrt{n} (\bar{X} - \mu_0)/\sigma \sim N(0,1), while the sample variance s^2 = \frac{1}{n-1} \sum (X_i - \bar{X})^2 yields (n-1) s^2 / \sigma^2 \sim \chi^2_{n-1}, and these two quantities are independent. The test statistic is then formed as the ratio

t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}} = \frac{Z}{\sqrt{\chi^2_{n-1} / (n-1)}},

where Z \sim N(0,1). This ratio follows the Student's t-distribution with n-1 degrees of freedom, providing the exact null distribution for computing critical values or p-values. This derivation was originally developed by William Sealy Gosset under the pseudonym "Student" to address small-sample inference in quality control settings.^[11] For the chi-squared goodness-of-fit test in parametric settings, suppose categorical data arise from multinomial probabilities specified under H_0, or more fundamentally, from independent normal variables. Under H_0, standardized residuals (X_i - \mu)/\sigma \sim N(0,1) for i=1,\dots,k, and the test statistic \sum_{i=1}^k [(X_i - \mu)/\sigma]^2 is the sum of squares of independent standard normals, which follows a \chi^2_k distribution. For the goodness-of-fit case with k categories and estimated parameters reducing the degrees of freedom, the statistic \sum (O_i - E_i)^2 / E_i \sim \chi^2_{k-1-p} asymptotically, where p is the number of estimated parameters, derived analogously by recognizing it as a quadratic form in normal deviates under the null model of specified probabilities. Karl Pearson introduced this distribution in 1900 as a criterion for assessing deviations in correlated normal systems, establishing the chi-squared null for exact inference when the model holds.^[12] In analysis of variance (ANOVA) for testing equality of means across g groups, assume group samples from normal distributions with common variance \sigma^2. Under H_0: \mu_1 = \dots = \mu_g, the between-group mean square MS_B estimates \sigma^2 via (MS_B / \sigma^2) \sim \chi^2_{g-1} / (g-1), while the within-group mean square MS_W follows (MS_W / \sigma^2) \sim \chi^2_{n-g} / (n-g), with independence between them. The test statistic is the ratio F = MS_B / MS_W = [( \chi^2_{g-1} / (g-1) )] / [ \chi^2_{n-g} / (n-g) ], which follows an F-distribution with g-1 and n-g degrees of freedom under H_0. This exact null distribution arises from the properties of independent chi-squared variates and was formalized by Ronald Fisher in the context of experimental design, with George Snedecor providing the nomenclature and tables in 1934 to facilitate its use in variance ratio tests.

Simulation and Monte Carlo Methods

When analytical forms of the null distribution are unavailable or computationally intractable, simulation-based approaches such as Monte Carlo methods provide a robust alternative for approximation. In a Monte Carlo simulation, numerous independent samples are generated directly from the probability distribution specified under the null hypothesis H_0, the test statistic is computed for each sample, and the resulting empirical distribution of these statistics serves as an approximation to the true null distribution. This technique, pioneered by Dwass in 1957 and further developed by Barnard in 1963, enables the estimation of p-values and critical values without relying on asymptotic assumptions.^[13] A related approach is the bootstrap method adapted under the null hypothesis, which leverages the observed data to resample while enforcing H_0 constraints, such as randomly permuting group labels in randomized experiments to simulate exchangeability. For each bootstrap replicate, the test statistic is recalculated, and the null distribution is estimated using the percentile method, where quantiles of the bootstrap statistics define critical regions. Introduced by Efron in 1979, this resampling strategy is particularly effective for finite-sample inference in complex settings. The implementation follows a structured algorithm: first, set a random seed to ensure reproducibility; second, generate B replicates (typically B = 10,000 or more for precision) from the null model or via constrained resampling; third, compute the test statistic for each replicate; and finally, approximate the null distribution through a histogram of the simulated values or kernel density estimation for smoother representations. These steps allow for flexible handling of multifaceted models, such as those in spatial statistics or high-dimensional data.^[14] Monte Carlo and bootstrap methods excel in scenarios involving non-standard distributions or intricate dependencies where exact derivations fail, offering exact control over Type I error rates in finite samples while remaining computationally feasible with modern resources.^[15]

Applications

Parametric Examples

In parametric hypothesis testing, the null distribution plays a central role in determining whether observed data provide sufficient evidence against the null hypothesis H_0. One common example is the Z-test for a population mean when the variance is known. Consider testing H_0: \mu = \mu_0 against H_a: \mu \neq \mu_0, where the sample mean \bar{X} from n observations yields a test statistic Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}, and under H_0, Z follows a standard normal distribution N(0,1).^[16] For an observed Z = 2.5, the two-tailed p-value is $2 \times (1 - \Phi(2.5)) \approx 0.0124, where \Phi is the cumulative distribution function of the standard normal; since this is below a typical significance level of \alpha = 0.05, H_0 is rejected.^[17] Another parametric example is the one-sample t-test, used when the population variance is unknown and estimated from the sample. For testing H_0: \mu = \mu_0 with a sample of size n = 10, the test statistic is t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}, where s is the sample standard deviation, and under H_0, t follows a Student's t-distribution with df = n-1 = 9 degrees of freedom.^[18] If the observed t = 1.8 for a two-tailed test at \alpha = 0.05, the critical value from the t-table is approximately \pm 2.262; since |1.8| < 2.262, the p-value (around 0.11) exceeds \alpha, so H_0 is not rejected. The likelihood ratio test (LRT) provides another parametric framework, particularly for comparing nested models. For a simple binomial proportion test of H_0: p = p_0 (e.g., p_0 = 0.5) with n = 100 trials and k = 60 successes, the likelihood ratio statistic is \Lambda = \frac{L(p_0 | k)}{L(\hat{p} | k)}, where \hat{p} = k/n = 0.6 is the maximum likelihood estimate, and -2 \log \Lambda under H_0 follows a \chi^2 distribution with 1 degree of freedom.^[19] Computing -2 \log \Lambda \approx 4.03, the p-value is $1 - F_{\chi^2_1}(4.03) \approx 0.045, which is below \alpha = 0.05, so H_0 is rejected; the critical value for rejection is 3.84.^[20] Interpreting outputs from these tests typically involves examining the test statistic, p-value, and critical values from distribution tables or software like R or Python's SciPy. For the Z-test and t-test, software reports the statistic alongside the p-value, which indicates the probability of observing data as extreme under H_0; values below \alpha suggest rejection.^[21] In LRT outputs, the \chi^2 statistic and its degrees of freedom are key, with p-values derived from the chi-square cumulative distribution; critical values can be referenced from standard tables for manual verification.^[19]

Non-Parametric Examples

Non-parametric tests construct null distributions based on the ranks or permutations of the observed data, avoiding assumptions about the underlying population distribution such as normality. These distributions are typically discrete and exact for finite samples, enabling hypothesis testing in scenarios where parametric assumptions fail. Common examples include rank-based tests and goodness-of-fit procedures, where the null posits no difference in location, shape, or overall form between samples or against a reference distribution. The Wilcoxon signed-rank test assesses whether the median of paired differences is zero, without assuming symmetry in the differences beyond the null. Under the null hypothesis, the test statistic—defined as the sum of ranks assigned to the absolute differences, considering only positive ranks—is symmetrically distributed around its mean of n(n+1)/4, where n is the number of non-zero differences. For small samples, the exact null distribution is derived by considering all possible assignments of signs to the ranked differences, yielding a permutation-based reference that accounts for the discrete nature of ranks. This approach ensures the test's validity even with tied or zero differences, though adjustments for ties reduce the effective sample size in the distribution calculation.^[22]^[23] In the Mann-Whitney U test, applied to two independent samples, the null hypothesis states that the distributions are identical, implying stochastic equality. The test statistic U represents the number of times a value from one sample exceeds a value from the other, computed via ranks of the combined sample. Under the null, the distribution of U arises from all possible permutations of the ranks across the two groups, forming a hypergeometric-like discrete distribution that is symmetric when sample sizes are equal. Ties in the data are handled by assigning average ranks to tied values, which modifies the variance of the null distribution but preserves the uniformity over permutations; for small samples, exact tables or enumeration provide the critical values, while larger samples approximate via normal distribution.^[24] The Kolmogorov-Smirnov test evaluates goodness-of-fit for a single sample against a fully specified continuous distribution or compares empirical cumulative distributions between two samples. Under the null that the sample follows the reference distribution (or the samples share the same distribution), the empirical distribution function converges uniformly to the true one. The test statistic, the supremum of the absolute differences between the empirical and reference (or two empirical) distribution functions, has a known null distribution that is independent of the specific continuous form under test. For the one-sample case, this distribution was originally tabulated for exact inference, with critical values derived from asymptotic theory but applicable exactly via simulation for finite samples; the two-sample version similarly relies on tabulated or permutation-generated null distributions to assess the maximum deviation.^[25]^[26] Permutation tests provide a general non-parametric framework for hypothesis testing across diverse statistics, applicable when exchangeability holds under the null. The null distribution is constructed by uniformly randomizing the observed data over all possible permutations consistent with the null hypothesis—such as reshuffling labels in randomized experiments—yielding an exact discrete distribution for the test statistic. This uniformity ensures each permutation is equally likely, with the p-value computed as the proportion of permuted statistics at least as extreme as the observed one; for computational feasibility with large datasets, Monte Carlo approximations sample from this uniform distribution. The method's foundation traces to randomized experimental designs, where it validates inferences without parametric assumptions.^[27]

Asymptotic Behavior

Large Sample Approximations

In large samples, the null distribution of standardized test statistics exhibits consistency, converging to a fixed limiting form as the sample size n approaches infinity, which facilitates the use of standard normal or chi-squared tables for p-value computation and critical values in hypothesis testing.^[28] This asymptotic behavior under the null hypothesis relies on the parameter being interior to the parameter space and the objective function satisfying regularity conditions such as twice continuous differentiability, with the score converging in distribution to a normal and the Hessian to a nonsingular matrix.^[28] For instance, Wald, Lagrange multiplier, and likelihood ratio test statistics in generalized method of moments frameworks converge in distribution to a chi-squared distribution with degrees of freedom equal to the number of restrictions under the null.^[28] A prominent example is the Student's t-test for the mean, where the exact null distribution follows a t-distribution with \nu = n-1 degrees of freedom, but for large n, it approximates the standard normal distribution \mathcal{N}(0,1), allowing substitution of z-critical values.^[29] The approximation error decreases with n, on the order of O(1/n), enabling reliable inference when the sample standard deviation closely estimates the population parameter.^[30] Bounds on the cumulative distribution function differences between the t and normal further quantify this, with upper and lower error terms derived for practical assessment.^[30] Guidelines for applying these approximations often recommend sample sizes n > 30 as a rule of thumb for tests of means, particularly when the underlying data are not severely skewed, as this threshold ensures the central limit theorem provides sufficient normality for the sampling distribution. However, this heuristic originates from pre-computational era Monte Carlo simulations and assumes moderate skewness; it may overestimate accuracy for certain distributions.^[31] Limitations arise in cases of slow convergence, such as with heavy-tailed data, where the null distribution's approach to the normal can require sample sizes exceeding 10,000 for acceptable approximation due to persistent kurtosis effects.^[32] For heavy-tailed families like Pareto or lognormal with L-kurtosis above 0.4, disruptions in hypothesis testing validity occur even at large n, as tail events delay the central limit theorem's normalization.^[32]

Central Limit Theorem Connections

The Central Limit Theorem (CLT) provides a foundational theoretical justification for the asymptotic normality of null distributions in hypothesis testing, particularly for large samples drawn from independent and identically distributed (i.i.d.) populations with finite variance. Specifically, if X_1, X_2, \dots, X_n are i.i.d. random variables with mean \mu and variance \sigma^2 < \infty, then under the null hypothesis H_0: \mu = \mu_0, the standardized sample mean \sqrt{n} (\bar{X}_n - \mu_0)/\sigma converges in distribution to a standard normal random variable Z \sim N(0,1) as n \to \infty. This convergence implies that, for sufficiently large n, the null distribution of test statistics based on the sample mean can be approximated by the standard normal distribution, enabling the construction of critical regions and p-values without exact knowledge of the underlying population distribution.^[33] Extensions of the CLT broaden its applicability to null distributions beyond simple means, accommodating sums, ratios, and more general functions of estimators, even when observations are independent but not identically distributed. The Lindeberg-Feller theorem generalizes the classical i.i.d. CLT by requiring the Lindeberg condition—that for every \epsilon > 0, the average contribution of individual terms to the variance becomes negligible as n grows—along with a uniform asymptotic negligibility condition on the variances. This allows the normalized sum \sum_{i=1}^n (X_i - \mu_i)/\sqrt{\sum_{i=1}^n \sigma_i^2} to converge to N(0,1) under the null, supporting asymptotic normality for test statistics in heterogeneous data settings, such as regression residuals or weighted averages. These CLT results underpin the asymptotic null distributions of various hypothesis tests, including those yielding chi-squared or Wald statistics. For instance, the scaled sample variance in a single sample, (n-1)s^2 / \sigma^2, follows a chi-squared distribution under normality, and more generally, quadratic forms of asymptotically normal estimators converge to chi-squared via the CLT. Similarly, the Wald test statistic, which measures the squared standardized deviation of a maximum likelihood estimator from its null value, asymptotically follows a chi-squared distribution under the null, as the estimator itself is asymptotically normal by the CLT applied to score functions.^[34] To quantify the accuracy of these normal approximations for finite samples, the Berry-Esseen theorem provides uniform bounds on the deviation between the cumulative distribution function (CDF) of the standardized sum and the standard normal CDF \Phi. For i.i.d. random variables with finite third absolute moment \rho = E[|X_1 - \mu|^3], the bound is \sup_x |F_n(x) - \Phi(x)| \leq C \rho / (\sigma^3 \sqrt{n}), where C is a universal constant (originally around 7.59, refined to approximately 0.4748 in modern estimates). This error rate of O(1/\sqrt{n}) establishes the practical reliability of normal-based null distributions even for moderate sample sizes, provided moments exist.

References

[1]
1.6 - Hypothesis Testing | STAT 462
Under the assumption that the null hypothesis is true, this test statistic will have a particular probability distribution. For testing a univariate population ...
[2]
[PDF] Null Hypothesis Significance Testing p-values, significance level ...
Test statistic: value: x, computed from data. Null distribution: f (x|H0) (assumes null hypothesis is true). Sides: HA determines if the rejection region is one ...
[3]
[PDF] Lecture Three Normal theory null distributions
Null distribution: from the assumptions, if H0 is true then t(Y) ∼ tn−1,. i.e. the population of hypothetical values of t(Y) that might have been sampled is a t ...<|control11|><|separator|>
[4]
What is a test statistic? - Support - Minitab
The sampling distribution of the test statistic under the null hypothesis is called the null distribution. When the data show strong evidence against the ...
[5]
[PDF] 05 Hypothesis testing ljh_pruned - UBC Zoology
look like under the null hypothesis? • Compare the ... • The null distribution is the sampling ... • Test statistic = a quantity calculated from the ...
[6]
Null & Alternative Hypotheses - Statistics Resources
Oct 27, 2025 · “Null” meaning “nothing.” This hypothesis states that there is no difference between groups or no relationship between variables. The null ...
[7]
[PDF] 9 Hypothesis Tests
The test statistic is a function of the sample data that will be used to make a decision about whether the null hypothesis should be rejected or not. Page 12 ...
[8]
IX. On the problem of the most efficient tests of statistical hypotheses
Neyman Jerzy and; Pearson Egon Sharpe. 1933IX. On the problem of the most efficient tests of statistical hypothesesPhilosophical Transactions of the Royal ...
[9]
[PDF] Null Hypothesis Significance Testing I Class 17, 18.05
Null distribution: the probability distribution of X assuming H0. • Rejection region: if X is in the rejection region we reject H0 in favor of HA. • Non- ...
[10]
[PDF] THE PROBABLE ERROR OF A MEAN Introduction - University of York
The usual method of determining the probability that the mean of the pop- ulation lies within a given distance of the mean of the sample is to assume a normal ...
[11]
[PDF] Karl Pearson a - McGill University
This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub ...Missing: original | Show results with:original
[12]
Finite Sample Properties and Asymptotic Efficiency of Monte Carlo ...
Since their introduction by Dwass (1957) and Barnard (1963), Monte Carlo tests have attracted considerable attention. The aim of this paper is to give a ...
[13]
[PDF] Markov chain Monte Carlo Significance tests - arXiv
Jun 19, 2024 · Abstract. Monte Carlo significance tests are a general tool that produce p-values by generating samples from the null distribution.
[14]
Monte Carlo tests with nuisance parameters: A general approach to ...
We give general conditions under which a Monte Carlo test obtained after replacing an unknown nuisance parameter yield an asymptotically valid test in cases ...Missing: seminal | Show results with:seminal
[15]
Z Test: Uses, Formula & Examples - Statistics By Jim
A Z test compares means when you know the population standard deviation. Learn about a Z test vs t test, its formula, and interpret examples.
[16]
10.1 - Z-Test: When Population Variance is Known | STAT 415
As one bit of evidence, n = 25 boys (of the same age) are weighed and found to have a mean weight of x ¯ = 80.94 pounds. It is known that the population ...
[17]
One-Sample t-Test | Introduction to Statistics - JMP
The one-sample t-test is a statistical hypothesis test used to determine whether an unknown population mean is different from a specific value.Missing: 10 | Show results with:10
[18]
2.2 - Tests and CIs for a Binomial Parameter - STAT ONLINE
Likelihood Ratio Test and CI For large $n$, $G^2$ is approximately chi-square with one degree of freedom, and $\pi_0$ will be rejected if \(G^2\ge \chi^2 ...
[19]
[PDF] STAT 226 Lecture 1 & 2
chi−square curve w/ df = 1. 25. Page 35. Likelihood Ratio Test Statistic for a Binomial Proportion. Recall the likelihood function for a binomial proportion π ...
[20]
Test statistics | Definition, Interpretation, and Examples - Scribbr
Jul 17, 2020 · The smaller the p value, the less likely your test statistic is to have occurred under the null hypothesis of the statistical test. Because the ...Missing: critical | Show results with:critical
[21]
Individual Comparisons by Ranking Methods - jstor
INDIVIDUAL COMPARISONS BY RANKING METHODS. Frank Wilcoxon. American Cyanamid Co. The comparison of two treatments generally falls into one of the following ...
[22]
Wilcoxon Signed‐Rank Test - Woolson - Wiley Online Library
Sep 19, 2008 · The null hypothesis is that the differences, or individual observations in the single-sample case, have a distribution centered about zero.
[23]
On a Test of Whether one of Two Random Variables is ... - jstor
[1] FRANK WILCOXON, "Individual comparisons by ranking methods", Biometrics Bull.,. Vol. 1 (1945), pp. 80>83. [2] A. WALD AND J. WOLFOWITZ, "On a test ...Missing: pdf | Show results with:pdf
[24]
Kolmogorov, A. (1933) Sulla determinazione empirica di una legge ...
Jul 25, 2019 · Kolmogorov, A. (1933) Sulla determinazione empirica di una legge di distribuzione. Giornale dell'Istituto Italiano degli Attuari, 4, 83-91.
[25]
1.3.5.16. Kolmogorov-Smirnov Goodness-of-Fit Test
The Kolmogorov-Smirnov test (Chakravart, Laha, and Roy, 1967) is used to decide if a sample comes from a population with a specific distribution. ... where n(i) ...Missing: reference | Show results with:reference
[26]
[PDF] Design of Experiments - Free
R. A. Fisher (1935). The logic of inductive inference. Journal. Royal ... On the validity of Fisher's 2 test when applied to an actual example of non ...
[27]
[PDF] LARGE SAMPLE ESTIMATION AND HYPOTHESIS TESTING*
Large sample distribution theory is the cornerstone of statistical inference ... it means that under the null hypothesis the test that ignores the first stage ...
[28]
SticiGui Approximate Hypothesis Tests: the z Test and the t Test
Jun 17, 2021 · These tests are approximate: They are based on approximations to the probability distribution of the test statistic when the null hypothesis is ...
[29]
None
### Summary of n ≥ 30 Rule for Normality Approximation in t-Tests
[30]
Errors in Normal Approximations to the $t,\tau,$ and Similar Types of ...
For the cdf's of Student's (t) ( t ) and Thompson's (τ) ( τ ) distributions, upper and lower bounds are obtained in terms of the normal cdf.
[31]
Full article: When Heavy Tails Disrupt Statistical Inference
The term disruption denotes the idea that the convergence guaranteed, for example, by the CLT and LLN become inordinately slow, so that millions of observations ...
[32]
Central limit theorem: the cornerstone of modern statistics - PMC
The central limit theorem is the most fundamental theory in modern statistics. Without this theorem, parametric tests based on the assumption that sample data ...
[33]
Wald test | Formula, explanation, example - StatLect
Asymptotically, the test statistic has a Chi-square distribution. Proposition Under the null hypothesis that [eq11] , the Wald statistic $W_{n}$ converges ...