Statistical Hypothesis

A Statistical Hypothesis is a formal statement about population parameters or probability distributions that can be tested using sample data in a statistical hypothesis testing task.

• AKA: Testable Hypothesis, Statistical Claim, Parametric Hypothesis, Distributional Hypothesis.
• Context:
  • It can typically specify relationships, values, or properties of population parameters such as means, variances, or proportions.
  • It can typically be formulated before data collection to avoid data dredging and p-hacking.
  • It can typically be tested using test statistics calculated from sample data.
  • It can typically be expressed mathematically using equality or inequality relationships.
  • It can often specify the probability distribution underlying observable random phenomena.
  • It can often determine the appropriate statistical test and test statistic distribution.
  • It can often be part of a hypothesis pair consisting of null hypothesis and alternative hypothesis.
  • It can often guide sample size determination and power analysis.
  • It can range from being a Simple Statistical Hypothesis to being a Composite Statistical Hypothesis, depending on its parameter specification completeness.
  • It can range from being a Null Hypothesis to being an Alternative Hypothesis, depending on its hypothesis role.
  • It can range from being a Point Hypothesis to being an Interval Hypothesis, depending on its parameter value constraint.
  • It can range from being a One-Tailed Hypothesis to being a Two-Tailed Hypothesis, depending on its directional specification.
  • It can range from being a Parametric Hypothesis to being a Non-Parametric Hypothesis, depending on its distributional assumption.
  • It can range from being an Accepted Statistical Hypothesis to being a Rejected Statistical Hypothesis, depending on its test outcome.
  • It can range from being a True Statistical Hypothesis to being a False Statistical Hypothesis, depending on its actual validity.
  • ...
• Example(s):
  • Population Mean Hypotheses, such as:
    • H: μ < 0.3 (population mean is less than 0.3).
    • H: μ = 0.3 (population mean equals 0.3).
    • H: μ₁ = μ₂ (two population means are equal).
    • H: μ₁ - μ₂ > 10 (mean difference exceeds 10).
  • Population Variance Hypotheses, such as:
    • H: σ² = 25 (population variance equals 25).
    • H: σ₁² = σ₂² (population variances are equal).
    • H: σ² < 100 (variance is less than 100).
  • Distribution Hypotheses, such as:
    • H: Data follows Normal Distribution.
    • H: Data follows Poisson Distribution with λ = 5.
    • H: Samples come from same distribution.
  • Proportion Hypotheses, such as:
    • H: p = 0.5 (population proportion equals 50%).
    • H: p₁ ≠ p₂ (proportions differ between groups).
    • H: p > 0.7 (success rate exceeds 70%).
  • Correlation Hypotheses, such as:
    • H: ρ = 0 (no population correlation).
    • H: ρ > 0.5 (strong positive correlation).
    • H: β₁ = 0 (no regression effect).
  • Independence Hypotheses, such as:
    • H: Variables X and Y are independent.
    • H: Treatment and outcome are associated.
    • H: Categories have no relationship.
  • ...
• Counter-Example(s):
  • Point Estimate, which is a single value rather than a testable statement.
  • Confidence Interval, which provides a range estimate rather than hypothesis.
  • Significance Level, which is a decision threshold rather than hypothesis.
  • Test Statistic, which is a calculated value rather than statement.
  • P-Value, which is a probability rather than hypothesis.
  • Research Question, which is broader than a statistical hypothesis.
• See: Statistical Hypothesis Testing Task, Null Hypothesis, Alternative Hypothesis, Simple Hypothesis, Composite Hypothesis, Neyman-Pearson Lemma, Test Statistic, Statistical Power, Type I Error, Type II Error, Hypothesis Specification Type, Statistical Inference, Frequentist Inference, Bayesian Inference.

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References

2016

• (Encycplopedia of Mathematics, 2016) ⇒ Statistical hypothesis. © 2011, Encyclopedia of Mathematics Retrieved October 11, 2016, from http://www.encyclopediaofmath.org/index.php?title=Statistical_hypothesis&oldid=13484
  • QUOTE: A specific assumption on the properties of a probability distribution that underlies observable random phenomena. The results of observations are usually represented as the realization of a number of random variables, whether finite or infinite. The joint distribution of these random variables is thus not completely known, and it is assumed in a statistical hypothesis that it belongs to a certain specific class of distributions. The problem of statistical hypotheses testing (cf. Statistical hypotheses, verification of) arises in this type of situation.

• (Leard Statistics, 2016) ⇒ "Hypothesis Testing - Structure and the Research, Null and Alternative Hypothesis" Laerd Statistics, © 2013 Lund Research Ltd, n.d. Web. Retrieved October 11, 2016, from http://statistics.laerd.com/statistical-guides/hypothesis-testing.php
  • QUOTE: (...) The first step in hypothesis testing is to set a research hypothesis. In Sarah and Mike's study, the aim is to examine the effect that two different teaching methods – providing both lectures and seminar classes (Sarah), and providing lectures by themselves (Mike) – had on the performance of Sarah's 50 students and Mike's 50 students. More specifically, they want to determine whether performance is different between the two different teaching methods. Whilst Mike is skeptical about the effectiveness of seminars, Sarah clearly believes that giving seminars in addition to lectures helps her students do better than those in Mike's class. This leads to the following research hypothesis:

Research Hypothesis: When students attend seminar classes, in addition to lectures, their performance increases.

• (Stat Trek, 2016) ⇒ "Hypothesis Testing Intro", © 2011, Encyclopedia of Mathematics] Retrieved October 11, 2016, from http://stattrek.com/hypothesis-test/hypothesis-testing.aspx
  • QUOTE: A statistical hypothesis is an assumption about a population parameter. This assumption may or may not be true. Hypothesis testing refers to the formal procedures used by statisticians to accept or reject statistical hypotheses.

(...) The best way to determine whether a statistical hypothesis is true would be to examine the entire population. Since that is often impractical, researchers typically examine a random sample from the population. If sample data are not consistent with the statistical hypothesis, the hypothesis is rejected.

There are two types of statistical hypotheses.

• Null hypothesis. The null hypothesis, denoted by H0, is usually the hypothesis that sample observations result purely from chance.
• Alternative hypothesis. The alternative hypothesis, denoted by H1 or Ha, is the hypothesis that sample observations are influenced by some non-random cause.

• (Wikipedia, 2016) ⇒ https://en.wikipedia.org/Statistical_hypothesis_testing#Definition_of_terms Retrieved October 11, 2016
  • QUOTE: The following definitions are mainly based on the exposition in the book by Lehmann and Romano:
    • Statistical hypothesis : A statement about the parameters describing a population (not a sample).
    • Statistic : A value calculated from a sample, often to summarize the sample for comparison purposes.
    • Simple hypothesis : Any hypothesis which specifies the population distribution completely.
    • Composite hypothesis : Any hypothesis which does not specify the population distribution completely.
    • Null hypothesis (H₀) : A simple hypothesis associated with a contradiction to a theory one would like to prove.
    • Alternative hypothesis (H₁) : A hypothesis (often composite) associated with a theory one would like to prove.
    • Statistical test : A procedure whose inputs are samples and whose result is a hypothesis.

2008

• (Lehmann & Romano, 2008) ⇒ E.L. Lehmann and Joseph P. Romano. (2008). "Testing Statistical Hypotheses." Springer.
  • QUOTE: In the classical formulation, a statistical hypothesis concerns the probability distribution of the observations. If the distribution is completely specified, the hypothesis is simple; otherwise it is composite. The specification may concern parameters of the distribution or the functional form itself.

1933

• (Neyman & Pearson, 1933) ⇒ Jerzy Neyman and Egon Pearson. (1933). "On the Problem of the Most Efficient Tests of Statistical Hypotheses." Philosophical Transactions of the Royal Society A.
  • QUOTE: We are inclined to think that as far as a particular hypothesis is concerned, no test based upon the theory of probability can by itself provide any valuable evidence of the truth or falsehood of that hypothesis. But we may look at the purpose of tests from another viewpoint. Without hoping to know whether each separate hypothesis is true or false, we may search for rules to govern our behaviour with regard to them, in following which we insure that, in the long run of experience, we shall not be too often wrong.

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