Hypothesis testing

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Hypothesis testing

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A hypothesis is the general term for a proposed explanation for some observed phenomena, which may or may not be the subject of further scrutiny in order to confirm or reject its veracity. Scientific studies investigate hypotheses by conducting experiments and analyzing data. Within the context of scientific analysis, statistics is utilized to assist this evaluation process by helping to identify how likely particular observations are given the hypothesis in question is true, i.e. do they support the hypothesis or suggest it is unlikely to be true? More specifically, a statistical hypothesis is "an assertion regarding the distribution(s) of one or several random variables. It may concern the parameters of the distribution, or the probability distribution of a population under study." (Dodge,2002, [DOD1]).

Statistical tests are used to determine the statistical significance of an observation or set of observations. Whenever such a test is considered, its applicability should be carefully examined: is a statistical test appropriate? do we have sufficient data? is the data measured in a suitable manner? are we satisfied with the sampling procedures? do the data satisfy the requirements of the proposed test procedure?…

A statistical hypothesis test is used to help evaluate whether some hypothesis, often referred to as the null hypothesis, can or cannot be rejected on the basis of the evidence (data) available. The term null hypothesis was introduced by R A Fisher an is often denoted by H0- this concept refers to a hypothesis which is tested for possible rejection under the assumption that it is true - e.g.

H0: the distribution of trees in a study region is random
H0: the population mean value (lifetime) of a sample of electronic devices is 500 hours

In Fisher's view, the aim of a hypothesis test is to help decide whether to reject or not reject the null hypothesis. In the Neyman-Pearson view (see further, the Neyman-Pearson Lemma) emphasis is placed on evaluating the relative merits of the null hypothesis against an alternative hypothesis, HA (see further, Types of Error). Of course, using a statistical test to help 'prove' (being satisfied or confident) that something is not false is not the same as proving something is true.

The table below lists some of the widely used 'classical' hypothesis tests, together with a summary of the core assumptions, data requirements and hypotheses to be evaluated. Each of these tests, and others, are described in the topic Classical tests later in this Handbook.

Classical hypothesis tests (after Crow et al., 1960, [CRO1])

One sample tests

Two sample tests

Assumptions: data is distributed N(μ,σ2), where σ2 is known

Data: random sample size n yields an estimate of the population mean

Hypotheses: Null hypothesis H0: μ=μ0; Alternative HA: μμ0

Test statistic: z-test (one sample); Rejection of H0: if |z|>zα/2 where α is the critical value of z

Assumptions: two data samples, with data distributed:

N(μ1,σ12) and N(μ2,σ22), where σ12 and σ22 are unknown

Data: random samples sizes n1 and n2 yields estimates of the population mean and variance for each sample

Hypotheses: Null hypothesis H0: μ1-μ2=c; Alternative HA: μ1-μ2c

Test statistic: t-test (two sample); Rejection of H0: if |t|>tα/2 where α is the critical value of t with n1+n2-2 DF

Assumptions: data is distributed N(μ,σ2), where σ2 is unknown

Data: random sample size n yields an estimate of the population mean and variance

Hypotheses: Null hypothesis H0: μ=μ0; Alternative HA: μμ0

Test statistic: t-test (one sample); Rejection of H0: if |t|>tα/2 where α is the critical value of t with n-1 DF

Assumptions: two data samples, with data distributed:

N(μ1,σ12) and N(μ2,σ22), where σ12 and σ22 are unknown

Data: random samples sizes n1and n2 yields estimates of the population mean and variance for each sample

Hypotheses: Null hypothesis H0: σ12=σ22; Alternative HA: σ12>σ22

Test statistic: F-test (two sample); Rejection of H0: if |F|>Fα where α is the critical value of F with n1-1 and n2-1 DF

Assumptions: data is distributed N(μ,σ2), where σ2 is unknown

Data: random sample size n yields an estimate of the population  variance

Hypotheses: Null hypothesis H0: σ2=σ20; Alternative HA: σ2>σ20

Test statistic: chi-square (one sample); Rejection of H0: if |χ2|>χ2α where α is the critical value of χ2 with n-1 DF

 

References

[CRO1] Crow E L, Davis F A, Maxfield M W (1960) Statistics Manual. Dover Publications

[DOD1] Dodge Y (2002) The Concise Encyclopedia of Statistics. Springer, New York

[UKM1] UK Maths, Stats & OR Network. Guides to Statistical Information: Probability and statistics Facts and Formulae. www.mathstore.ac.uk

Wikipedia: Hypothesis: http://en.wikipedia.org/wiki/Hypothesis