Statistical Null Hypothesis

A Statistical Null Hypothesis is a null hypothesis specifically formulated for statistical hypothesis testing that asserts observed sample data results from chance variation alone, with no systematic effect or relationship present.

• AKA: H₀ Statistical, Chance Hypothesis, Random Variation Hypothesis, No-Effect Statistical Hypothesis.
• Context:
  • It can typically state that any observed difference between sample statistics and hypothesized values is due to sampling error.
  • It can typically specify exact parameter values enabling calculation of test statistic distributions.
  • It can typically serve as the basis for computing p-values in statistical significance tests.
  • It can typically be formulated to be falsifiable through empirical evidence.
  • It can often represent the assumption of no treatment effect in experimental designs.
  • It can often be tested using parametric tests when distributional assumptions are met.
  • It can often determine the null distribution against which test statistics are compared.
  • It can often be rejected at a predetermined significance level when evidence is sufficient.
  • It can range from being a Simple Statistical Null Hypothesis to being a Composite Statistical Null Hypothesis, depending on its parameter specification.
  • It can range from being a Point Statistical Null Hypothesis to being a Interval Statistical Null Hypothesis, depending on its value constraint.
  • It can range from being a Parametric Statistical Null Hypothesis to being a Non-Parametric Statistical Null Hypothesis, depending on its distribution assumption.
  • It can range from being a Frequentist Statistical Null Hypothesis to being a Bayesian Statistical Null Hypothesis, depending on its inference framework.
  • It can range from being a Conservative Statistical Null Hypothesis to being a Liberal Statistical Null Hypothesis, depending on its assumption strength.
  • ...
• Example(s):
  • Mean Statistical Null Hypotheses, such as:
    • H₀: μ = μ₀ (population mean equals hypothesized value).
    • H₀: μ₁ - μ₂ = 0 (no difference between group means).
    • H₀: μd = 0 (mean of paired differences is zero).
  • Proportion Statistical Null Hypotheses, such as:
    • H₀: p = 0.5 (probability equals chance level).
    • H₀: p₁ = p₂ (equal proportions across groups).
    • H₀: π = π₀ (population proportion equals specified value).
  • Correlation Statistical Null Hypotheses, such as:
    • H₀: ρ = 0 (no linear correlation exists).
    • H₀: R² = 0 (no variance explained by model).
    • H₀: β = 0 (regression coefficient is zero).
  • Distribution Statistical Null Hypotheses, such as:
    • H₀: F(x) = F₀(x) (data follows specified distribution).
    • H₀: Samples from same population distribution.
    • H₀: Residuals are normally distributed.
  • Independence Statistical Null Hypotheses, such as:
    • H₀: Variables are statistically independent.
    • H₀: No association between categorical variables.
    • H₀: Treatment assignment independent of outcome.
  • Variance Statistical Null Hypotheses, such as:
    • H₀: σ₁² = σ₂² (equal population variances).
    • H₀: σ² = σ₀² (variance equals specified value).
  • ...
• Counter-Example(s):
  • Legal Null Hypothesis, such as "defendant is innocent" (non-statistical context).
  • Scientific Null Hypothesis, which may be conceptual rather than statistical.
  • Alternative Statistical Hypothesis, which proposes systematic effects exist.
  • Research Hypothesis, which is broader than statistical formulation.
• See: Statistical Alternative Hypothesis, Statistical Hypothesis Testing Task, Null Hypothesis, Type I Error, Type II Error, P-Value, Test Statistic, Significance Level, Statistical Power, Null Distribution, Neyman-Pearson Framework, Fisher's Null Hypothesis Testing.

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References

2009

• http://www.introductorystatistics.com/escout/main/Glossary.htm
  • QUOTE: The presumed model (such as that of a fair coin) in hypothesis testing. The data provide a measure of how weak or strong the evidence for or against this null hypothesis is; it is the model of step 1 if the six-step method is being used.

2008

• (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised." Oxford University Press. ISBN:0199541450
  • QUOTE: The hypothesis, in a hypothesis test, which is used to obtain the probability distribution, and hence the critical region, of the statistic used in the test. The phrase ‘null hypothesis’ was introduced by Sir Ronald Fisher in 1935.

1935

• (Fisher, 1935) ⇒ Ronald A. Fisher. (1935). "The Design of Experiments." Oliver and Boyd.
  • QUOTE: The null hypothesis must be exact, that is, free from vagueness and ambiguity, because it must supply the basis of the 'problem of distribution,' of which the test of significance is the solution. A null hypothesis concerning the value of a parameter can be simple (specifying a single value) or composite (specifying a range).

1933

• (Neyman & Pearson, 1933) ⇒ Jerzy Neyman and Egon Pearson. (1933). "On the Problem of the Most Efficient Tests of Statistical Hypotheses."
  • QUOTE: In testing a statistical hypothesis, we distinguish between the hypothesis under test, which we call the null hypothesis, and alternative hypotheses which represent departures from it. The null hypothesis typically represents a theory that has been put forward, either because it is believed to be true or because it is to be used as a basis for argument.

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