Editing Student's t-test (section)

==Uses==
[[File:One sample t-test.png|thumb|right]]
[[File:2 Sample Test.png|thumb|right]]

===One-sample ''t''-test===
A '''one-sample Student's ''t''-test''' is a [[location test]] of whether the mean of a population has a value specified in a [[null hypothesis]]. In testing the null hypothesis that the population mean is equal to a specified value {{math|''μ''<sub>0</sub>}}, one uses the statistic

: <math> t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}, </math>

where <math>\bar x</math> is the sample mean, {{math|''s''}} is the [[Standard deviation#Estimation|sample standard deviation]] and {{math|''n''}} is the sample size. The [[Degrees of freedom (statistics)|degrees of freedom]] used in this test are {{math|''n''&nbsp;−&nbsp;1}}. Although the parent population does not need to be normally distributed, the distribution of the population of sample means <math>\bar x</math> is assumed to be normal.

By the [[central limit theorem]], if the observations are independent and the second moment exists, then <math>t</math> will be approximately normal <math display="inline">\mathcal{N}(0, 1)</math>.

===Two-sample ''t''-tests===

[[File:Type 1 error.png|thumb|Type I error of unpaired and paired two-sample ''t''-tests as a function of the correlation. The simulated random numbers originate from a bivariate normal distribution with a variance of 1. The significance level is 5% and the number of cases is 60.]]
[[File:Power of t-tests.png|thumb|Power of unpaired and paired two-sample ''t''-tests as a function of the correlation. The simulated random numbers originate from a bivariate normal distribution with a variance of 1 and a deviation of the expected value of 0.4. The significance level is 5% and the number of cases is 60.]]

A '''two-sample''' location test of the null hypothesis such that the [[expected value|mean]]s of two populations are equal. All such tests are usually called '''Student's ''t''-tests''', though strictly speaking that name should only be used if the [[variance]]s of the two populations are also assumed to be equal; the form of the test used when this assumption is dropped is sometimes called [[Welch's t test|Welch's ''t''-test]]. These tests are often referred to as '''unpaired''' or ''independent samples'' ''t''-tests, as they are typically applied when the [[unit (statistics)|statistical units]] underlying the two samples being compared are non-overlapping.<ref name=fadem>{{cite book |last=Fadem |first=Barbara |title=High-Yield Behavioral Science |series=High-Yield Series |publisher=Lippincott Williams & Wilkins |location=Hagerstown, MD |year=2008 |isbn=9781451130300 }}</ref>

Two-sample ''t''-tests for a difference in means involve independent samples (unpaired samples) or [[paired sample]]s.  Paired ''t''-tests are a form of [[blocking (statistics)|blocking]], and have greater [[statistical power|power]] (probability of avoiding a type II error, also known as a false negative) than unpaired tests when the paired units are similar with respect to "noise factors" (see [[confounder]]) that are independent of membership in the two groups being compared.<ref>{{cite book|first=John A. |last=Rice |date=2006 |title=Mathematical Statistics and Data Analysis |edition= 3rd |publisher=Duxbury Advanced }}{{ISBN missing}}</ref> In a different context, paired ''t''-tests can be used to reduce the effects of [[confounders|confounding factors]] in an [[observational study]].

====Independent (unpaired) samples====
The independent samples ''t''-test is used when two separate sets of [[Independent and identically-distributed random variables|independent and identically distributed]] samples are obtained, and one variable from each of the two populations is compared. For example, suppose we are evaluating the effect of a medical treatment, and we enroll 100&nbsp;subjects into our study, then randomly assign 50&nbsp;subjects to the treatment group and 50&nbsp;subjects to the control group. In this case, we have two independent samples and would use the unpaired form of the ''t''-test.

====Paired samples====
{{Main|Paired difference test}}

[[Paired sample]]s ''t''-tests typically consist of a sample of matched pairs of similar [[unit (statistics)|units]], or one group of units that has been tested twice (a "repeated measures" ''t''-test).

A typical example of the repeated measures ''t''-test would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure-lowering medication. By comparing the same patient's numbers before and after treatment, we are effectively using each patient as their own control. That way the correct rejection of the null hypothesis (here: of no difference made by the treatment) can become much more likely, with statistical power increasing simply because the random interpatient variation has now been eliminated. However, an increase of statistical power comes at a price: more tests are required, each subject having to be tested twice. Because half of the sample now depends on the other half, the paired version of Student's ''t''-test has only {{math|{{sfrac|''n''|2}} − 1}} degrees of freedom (with {{math|''n''}} being the total number of observations). Pairs become individual test units, and the sample has to be doubled to achieve the same number of degrees of freedom. Normally, there are {{math|''n'' − 1}} degrees of freedom (with {{math|''n''}} being the total number of observations).<ref>{{cite web|last1=Weisstein|first1=Eric|title=Student's ''t''-Distribution|url=http://mathworld.wolfram.com/Studentst-Distribution.html|website=mathworld.wolfram.com}}</ref>

A paired samples ''t''-test based on a "matched-pairs sample" results from an unpaired sample that is subsequently used to form a paired sample, by using additional variables that were measured along with the variable of interest.<ref>{{cite journal |last1=David |first1=H.&nbsp;A. |last2=Gunnink |first2=Jason L. |year=1997 |title=The Paired ''t'' Test Under Artificial Pairing |journal=The American Statistician |volume=51 |pages=9–12 |jstor=2684684 |doi=10.2307/2684684 |issue=1}}</ref> The matching is carried out by identifying pairs of values consisting of one observation from each of the two samples, where the pair is similar in terms of other measured variables. This approach is sometimes used in observational studies to reduce or eliminate the effects of confounding factors.

Paired samples ''t''-tests are often referred to as "dependent samples ''t''-tests".