Editing Statistical hypothesis test (section)

==Definition of terms==
{{See also|Notation in probability and statistics}}
The following definitions are mainly based on the exposition in the book by Lehmann and Romano:<ref name="LR">{{cite book|title=Testing Statistical Hypotheses|edition=3E|isbn=978-0-387-98864-1|last1=Lehmann|first1=E. L.|first2=Joseph P.|last2=Romano|year=2005|publisher=Springer|location=New York}}</ref>

*'''Statistical hypothesis''': A statement about the parameters describing a [[Statistical population|population]] (not a [[Statistical sample|sample]]).
*Test statistic: A value calculated from a sample without any unknown parameters, often to summarize the sample for comparison purposes.
*{{visible anchor|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<sub>0</sub>) 
*Positive data: Data that enable the investigator to reject a null hypothesis.
*[[Alternative hypothesis]] (H<sub>1</sub>) 
[[File:One tailed critical value with significance level alpha.jpg|thumb|260x260px|Suppose the data can be realized from an N(0,1) distribution. For example, with a chosen significance level α = 0.05, from the Z-table, a one-tailed critical value of approximately 1.645 can be obtained. The one-tailed critical value C<sub>α</sub> ≈ 1.645 corresponds to the chosen significance level. The critical region [C<sub>α</sub>, ∞) is realized as the tail of the standard normal distribution.]]
*'''{{vanchor|Critical value}}s''' of a statistical test are the boundaries of the acceptance region of the test.<ref>{{cite book |first1=Ann J. |last1=Hughes |first2=Dennis E. |last2=Grawoig |title=Statistics: A Foundation for Analysis |location=Reading, Mass. |publisher=Addison-Wesley |year=1971 |isbn=0-201-03021-7 |page=[https://archive.org/details/trent_0116302260611/page/191 191] |url=https://archive.org/details/trent_0116302260611 |url-access=registration }}</ref> The acceptance region is the set of values of the test statistic for which the null hypothesis is not rejected. Depending on the shape of the acceptance region, there can be one or more than one critical value. 
**'''{{vanchor|Region of rejection}}''' / '''{{vanchor|Critical region}}''': The set of values of the test statistic for which the null hypothesis is rejected.
*'''[[statistical power|Power of a test]] (1&nbsp;−&nbsp;''β'')'''
* [[Size (statistics)|'''Size''']]: For simple hypotheses, this is the test's probability of ''incorrectly'' rejecting the null hypothesis. The [[false positive]] rate. For composite hypotheses this is the supremum of the probability of rejecting the null hypothesis over all cases covered by the null hypothesis. The complement of the false positive rate is termed '''specificity''' in [[biostatistics]]. ("This is a specific test. Because the result is positive, we can confidently say that the patient has the condition.") See [[sensitivity and specificity]] and [[type I and type II errors]] for exhaustive definitions.
*[[Significance level]] of a test (''α)''
*'''[[p-value|''p''-value]]''' 
*'''{{vanchor|Statistical significance test}}''': A predecessor to the statistical hypothesis test (see the Origins section). An experimental result was said to be [[statistical significance|statistically significant]] if a sample was sufficiently inconsistent with the (null) hypothesis. This was variously considered common sense, a pragmatic heuristic for identifying meaningful experimental results, a convention establishing a threshold of statistical evidence or a method for drawing conclusions from data. The statistical hypothesis test added mathematical rigor and philosophical consistency to the concept by making the alternative hypothesis explicit. The term is loosely used for the modern version which is now part of statistical hypothesis testing.
*Conservative test: A test is conservative if, when constructed for a given nominal significance level, the true probability of ''incorrectly'' rejecting the null hypothesis is never greater than the nominal level.
*[[Exact test]]

A statistical hypothesis test compares a test statistic (''z'' or ''t'' for examples) to a threshold. The test statistic (the formula found in the table below) is based on optimality. For a fixed level of Type I error rate, use of these statistics minimizes Type II error rates (equivalent to maximizing power). The following terms describe tests in terms of such optimality:

*Most powerful test: For a given ''size'' or ''significance level'', the test with the greatest power (probability of rejection) for a given value of the parameter(s) being tested, contained in the alternative hypothesis.
*[[Uniformly most powerful test]] (UMP)