# Statistical Hypothesis

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A Statistical Hypothesis is a statistical statement about population parameter(s) and/or sampling distribution that is subject to a Statistical Hypothesis Testing Task.

**AKA:**Research Hypothesis.**Context:**- It can (often) be an input to a Statistical Hypothesis Testing Task.
- It can range from being a Simple Hypothesis to being a Composite Hypothesis.
- It can range from being a Null Hypothesis to being an Alternate Statistical Hypothesis.
- It can range from being an Accepted Statistical Hypothesis to being a Rejected Statistical Hypothesis.
- It can range from being a True Statistical Hypothesis to being a Negative Statistical Hypothesis.

**Example(s):***the population mean is less than 0.3*.*the population mean is equal to 0.3*.*the sample data are normally distributed**the population means are equal**the population variances are equal*.- …

**Counter-Example(s):****See:**Neyman–Pearson Lemma, Test Statistic, Statistical Hypothesis Test Sensitivity, Bayesian Inference, Frequentist Inference, Population Sample.

## 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 (H**_{0})**: A simple hypothesis associated with a contradiction to a theory one would like to prove.****Alternative hypothesis (H**_{1})**: 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.

- QUOTE: The following definitions are mainly based on the exposition in the book by Lehmann and Romano: