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| {{DISPLAYTITLE:Student's ''t''-test}}
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| A '''''t''-test''' is any [[statistical hypothesis testing|statistical hypothesis test]] in which the [[test statistic]] follows a [[Student's t-distribution|Student's ''t'' distribution]] if the [[null hypothesis]] is supported. It can be used to determine if two sets of data are significantly different from each other, and is most commonly applied when the test statistic would follow a [[normal distribution]] if the value of a scaling term in the test statistic were known. When the [[Scale parameter|scaling term]] is unknown and is replaced by an estimate based on the [[data]], the test statistic (under certain conditions) follows a Student's ''t'' distribution.
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| ==History==
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| The [[t-statistic]] was introduced in 1908 by [[William Sealy Gosset]], a chemist working for the [[Guinness]] [[brewery]] in [[Dublin, Ireland]] ("Student" was his [[pen name]]).<ref>Richard Mankiewicz, ''The Story of Mathematics'' (Princeton University Press), p.158.</ref><ref name="Gossett">{{MacTutor Biography|id=Gosset}}</ref><ref>{{cite journal | last=Fisher Box | first=Joan | journal=Statistical Science | year=1987 | pages=45–52 | volume=2 | jstor=2245613 | title=Guinness, Gosset, Fisher, and Small Samples | doi=10.1214/ss/1177013437 | issue=1 }}</ref><ref>http://www.aliquote.org/cours/2012_biomed/biblio/Student1908.pdf</ref> Gosset had been hired due to [[Claude Guinness]]'s policy of recruiting the best graduates from [[Oxford University|Oxford]] and [[Cambridge University|Cambridge]] to apply [[biochemistry]] and [[statistics]] to Guinness's industrial processes.<ref name="Gossett" /> Gosset devised the ''t''-test as a cheap way to monitor the quality of [[stout]]. The student t-test work was submitted to and accepted in the journal ''[[Biometrika]]'', the journal that [[Karl Pearson]] had co-founded and was the Editor-in-Chief; the article was published in 1908. Company policy at Guinness forbade its chemists from publishing their findings, so Gosset published his mathematical work under the pseudonym "Student". Actually, Guinness had a policy of allowing technical staff leave for study (so-called study leave), which Gosset used during the first two terms of the 1906-1907 academic year in Professor Karl Pearson's Biometric Laboratory at University College London.<ref>{{cite journal |author=Raju TN |title=William Sealy Gosset and William A. Silverman: two "students" of science |journal=Pediatrics |volume=116 |issue=3 |pages=732–5 |year=2005 |pmid=16140715 |doi=10.1542/peds.2005-1134}}</ref> Gosset's identity was then known to fellow statisticians and the Editor-in-Chief Karl Pearson.
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| It is not clear how much of the work Gosset performed while he was at Guinness and how much was done when he was on study leave at University College London.
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| ==Uses==
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| Among the most frequently used ''t''-tests are:
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| * A one-sample [[location test]] of whether the mean of a population has a value specified in a [[null hypothesis]].
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| * A two-sample location test of the null hypothesis 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 |author=Fadem, Barbara |title=High-Yield Behavioral Science (High-Yield Series) |publisher=Lippincott Williams & Wilkins |location=Hagerstwon, MD |year=2008 |pages= |isbn=0-7817-8258-9 |oclc= |doi=}}</ref>
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| * A test of the null hypothesis that the difference between two responses measured on the same statistical unit has a mean value of zero. For example, suppose we measure the size of a cancer patient's tumor before and after a treatment. If the treatment is effective, we expect the tumor size for many of the patients to be smaller following the treatment. This is often referred to as the "paired" or "repeated measures" ''t''-test:<ref name=fadem/><ref>{{cite journal | last=Zimmerman | first=Donald W. | journal=Journal of Educational and Behavioral Statistics | year=1997 | volume=22 | pages=349–360 | title=A Note on Interpretation of the Paired-Samples t Test | jstor=1165289 | issue=3}}</ref> see [[paired difference test]].
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| * A test of whether the slope of a [[linear regression|regression line]] differs [[statistical significance|significantly]] from 0.
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| ==Assumptions==
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| Most ''t''-test statistics have the form ''t'' = ''Z''/''s'', where ''Z'' and ''s'' are functions of the data. Typically, ''Z'' is designed to be sensitive to the alternative hypothesis (i.e., its magnitude tends to be larger when the alternative hypothesis is true), whereas ''s'' is a [[Scale parameter|scaling parameter]] that allows the distribution of ''t'' to be determined.
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| As an example, in the one-sample ''t''-test ''Z'' = <math>\bar{X}/(\hat{\sigma}/\sqrt{n})</math>, where <math>\bar{X}</math> is the [[average|sample mean]] of the data, <math>n</math> is the sample size, and <math>\hat{\sigma}</math> is the [[population variance|population standard deviation]] of the data. s is the sample standard deviation.
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| The assumptions underlying a ''t''-test are that
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| * ''Z'' follows a standard normal distribution under the null hypothesis
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| * ''s''<sup>2</sup> follows a [[chi-squared distribution|χ<sup>2</sup> distribution]] with ''p'' [[Degrees of freedom (statistics)|degrees of freedom]] under the null hypothesis, where ''p'' is a positive constant
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| * ''Z'' and ''s'' are [[statistical independence|independent]].
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| In a specific type of ''t''-test, these conditions are consequences of the [[statistical population|population]] being studied, and of the way in which the data are [[sampling (statistics)|sampled]]. For example, in the ''t''-test comparing the means of two independent samples, the following assumptions should be met:
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| * Each of the two populations being compared should follow a [[normal distribution]]. This can be tested using a [[normality test]], such as the [[Shapiro–Wilk test|Shapiro–Wilk]] or [[Kolmogorov–Smirnov]] test, or it can be assessed graphically using a [[Q–Q plot|normal quantile plot]].
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| * If using Student's original definition of the ''t''-test, the two populations being compared should have the same variance (testable using [[F-test of equality of variances|F test]], [[Levene's test]], [[Bartlett's test]], or the [[Brown–Forsythe test]]; or assessable graphically using a [[Q–Q plot]]). If the sample sizes in the two groups being compared are equal, Student's original ''t''-test is highly robust to the presence of unequal variances.<ref>{{cite journal | last=Markowski | first=Carol A | coauthors=Markowski, Edward P. | year = 1990 | title=Conditions for the Effectiveness of a Preliminary Test of Variance | journal=The American Statistician | pages=322–326 | volume=44 | jstor=2684360 | doi=10.2307/2684360 | issue=4}}</ref> [[Welch's t-test|Welch's ''t''-test]] is insensitive to equality of the variances regardless of whether the sample sizes are similar.
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| * The data used to carry out the test should be sampled independently from the two populations being compared. This is in general not testable from the data, but if the data are known to be dependently sampled (i.e. if they were sampled in clusters), then the classical ''t''-tests discussed here may give misleading results.
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| ==Unpaired and paired two-sample ''t''-tests==
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| Two-sample ''t''-tests for a difference in mean involve independent samples, paired samples and overlapping samples. Paired ''t''-tests are a form of [[blocking (statistics)|blocking]], and have greater [[statistical power|power]] than unpaired tests when the paired units are similar with respect to "noise factors" that are independent of membership in the two groups being compared.<ref>John A. Rice (2006), Mathematical Statistics and Data Analysis, Third Edition, Duxbury Advanced.</ref> In a different context, paired ''t''-tests can be used to reduce the effects of [[confounders|confounding factors]] in an [[observational study]].
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| === Independent (unpaired) samples ===
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| 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, one from each of the two populations being compared. For example, suppose we are evaluating the effect of a medical treatment, and we enroll 100 subjects into our study, then randomly assign 50 subjects to the treatment group and 50 subjects to the control group. In this case, we have two independent samples and would use the unpaired form of the ''t''-test. The randomization is not essential here – if we contacted 100 people by phone and obtained each person's age and gender, and then used a two-sample ''t''-test to see whether the mean ages differ by gender, this would also be an independent samples ''t''-test, even though the data are observational.
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| === Paired samples ===
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| {{Main|Paired difference test}}
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| Paired samples ''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).
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| 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 between-patient variation has now been eliminated. Note however that 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 'n/2 - 1' degrees of freedom (with '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.
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| 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 | last=David | first=HA | coauthors=Gunnink, 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.
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| Paired samples ''t''-tests are often referred to as "dependent samples ''t''-tests" (as are ''t''-tests on overlapping samples).
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| === Overlapping samples ===
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| An overlapping samples ''t''-test is used when there are paired samples with data missing in one or the other samples (e.g., due to selection of "Don't know" options in questionnaires or because respondents are randomly assigned to a subset question). These tests are widely used in commercial survey research (e.g., by polling companies) and are available in many standard [[crosstab]] software packages.
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| ==Calculations==
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| Explicit expressions that can be used to carry out various ''t''-tests are given below. In each case, the formula for a test statistic that either exactly follows or closely approximates a ''t''-distribution under the null hypothesis is given. Also, the appropriate [[degrees of freedom (statistics)|degrees of freedom]] are given in each case. Each of these statistics can be used to carry out either a [[One-tailed test|one-tailed test or a two-tailed test]].
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| Once a ''t'' value is determined, a [[p-value]] can be found using a table of values from Student's t-distribution. If the calculated p-value is below the threshold chosen for [[statistical significance]] (usually the 0.10, the 0.05, or 0.01 level), then the null hypothesis is rejected in favor of the alternative hypothesis.
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| ===One-sample ''t''-test===
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| In testing the null hypothesis that the population mean is equal to a specified value ''μ''<sub>0</sub>, one uses the statistic
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| :<math> t = \frac{\overline{x} - \mu_0}{s/\sqrt{n}} </math>
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| where <math>\overline{x}</math> is the sample mean, ''s'' is the [[Standard deviation#Estimation|sample standard deviation]] of the sample and ''n'' is the sample size. The degrees of freedom used in this test are ''n'' − 1. Although the parent population does not need to be normally distributed, the distribution of the population of sample means, <math> \overline {x} </math>, is assumed to be normal. By the [[central limit theorem]], if the sampling of the parent population is random then the sample means will be approximately normal.<ref>George Box, William Hunter, and J. Stuart Hunter, "Statistics for Experimenters", ISBN 978-0471093152, pp. 66-67.</ref> (The degree of approximation will depend on how close the parent population is to a normal distribution and the sample size, n.)
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| ===Slope of a regression line===
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| Suppose one is fitting the model
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| : <math> Y_i = \alpha + \beta x_i + \varepsilon_i, </math>
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| where ''x''<sub>''i''</sub>, ''i'' = 1, ..., ''n'' are known, ''α'' and ''β'' are unknown, and ''ε''<sub>''i''</sub> are independent identically normally distributed [[errors and residuals in statistics|random errors]] with expected value 0 and unknown variance ''σ''<sup>2</sup>, and ''Y''<sub>''i''</sub>, ''i'' = 1, ..., ''n'' are observed. It is desired to test the null hypothesis that the slope ''β'' is equal to some specified value ''β''<sub>0</sub> (often taken to be 0, in which case the hypothesis is that ''x'' and ''y'' are unrelated).
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| Let
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| : <math>
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| \begin{align}
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| \widehat\alpha, \widehat\beta & = \text{least-squares estimators}, \\
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| SE_{\widehat\alpha}, SE_{\widehat\beta} & = \text{the standard errors of least-squares estimators}.
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| \end{align}
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| </math>
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| Then
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| : <math>
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| t_\text{score} = \frac{\widehat\beta - \beta_0}{ SE_{\widehat\beta} }\sim\mathcal{T}_{n-2}
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| </math> | |
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| has a ''t''-distribution with ''n'' − 2 degrees of freedom if the null hypothesis is true. The
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| [[Simple linear regression#Normality assumption|standard error of the slope coefficient]]:
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| : <math>
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| SE_{\widehat\beta} = \frac{\sqrt{\frac{1}{n - 2}\sum_{i=1}^n (Y_i - \widehat y_i)^2}}{\sqrt{ \sum_{i=1}^n (x_i - \overline{x})^2 }}
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| </math>
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| can be written in terms of the residuals. Let
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| : <math>
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| \begin{align}
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| \widehat\varepsilon_i & = Y_i - \widehat y_i = Y_i - (\widehat\alpha + \widehat\beta x_i) = \text{residuals} = \text{estimated errors}, \\
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| \text{SSR} & = \sum_{i=1}^n \widehat\varepsilon_i^{\;2} = \text{sum of squares of residuals}.
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| \end{align}
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| </math>
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| Then <math> t_\text{score} </math> is given by:
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| : <math> t_\text{score} = \frac{(\widehat\beta - \beta_0)\sqrt{n-2}}{ \sqrt{\text{SSR}/\sum_{i=1}^n \left(x_i - \overline{x}\right)^2} }. </math>
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| ===Independent two-sample ''t''-test===
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| ==== Equal sample sizes, equal variance ====
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| This test is only used when both:
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| *the two sample sizes (that is, the number, ''n'', of participants of each group) are equal;
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| *it can be assumed that the two distributions have the same variance.
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| Violations of these assumptions are discussed below.
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| The ''t'' statistic to test whether the means are different can be calculated as follows:
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| : <math> t = \frac{\bar {X}_1 - \bar{X}_2}{s_{X_1X_2} \cdot \sqrt{\frac{2}{n}}} </math>
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| where
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| :<math>\ s_{X_1X_2} = \sqrt{\frac{1}{2}(s_{X_1}^2+s_{X_2}^2)}</math>
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| Here <math>s_{X_1X_2}</math> is the grand [[Standard deviation#Estimating population standard deviation from sample standard deviation|standard deviation]] (or [[pooled standard deviation]]), 1 = group one, 2 = group two. <math>s_{X_1}^2</math> and <math>s_{X_2}^2</math> are the [[unbiased estimator]]s of the [[variance]]s of the two samples. The denominator of ''t'' is the [[Standard error (statistics)|standard error]] of the difference between two means.
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| For significance testing, the [[Degrees of freedom (statistics)|degrees of freedom]] for this test is 2''n'' − 2 where ''n'' is the number of participants in each group.
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| ====Unequal sample sizes, equal variance====
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| This test is used only when it can be assumed that the two distributions have the same variance. (When this assumption is violated, see below.) The ''t'' statistic to test whether the means are different can be calculated as follows:
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| :<math>t = \frac{\bar {X}_1 - \bar{X}_2}{s_{X_1X_2} \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}</math>
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| where
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| :<math> s_{X_1X_2} = \sqrt{\frac{(n_1-1)s_{X_1}^2+(n_2-1)s_{X_2}^2}{n_1+n_2-2}}.</math>
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| Note that the formulae above are generalizations of the case where both samples have equal sizes (substitute ''n'' for ''n''<sub>1</sub> and ''n''<sub>2</sub>).
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| <math>s_{X_1X_2}</math> is an estimator of the common [[standard deviation]] of the two samples: it is defined in this way so that its square is an [[unbiased estimator]] of the common variance whether or not the population means are the same. In these formulae, ''n'' = number of participants, 1 = group one, 2 = group two. ''n'' − 1 is the number of degrees of freedom for either group, and the total sample size minus two (that is, ''n''<sub>1</sub> + ''n''<sub>2</sub> − 2) is the total number of degrees of freedom, which is used in significance testing.
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| ====Equal or Unequal sample sizes, unequal variances====
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| This test, also known as [[Welch's t test|Welch's ''t''-test]], is used only when the two population variances are not assumed to be equal (the two sample sizes may or may not be equal) and hence must be estimated separately. The ''t'' statistic to test whether the population means are different is calculated as:
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| :<math>t = {\overline{X}_1 - \overline{X}_2 \over s_{\overline{X}_1 - \overline{X}_2}}</math>
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| where
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| :<math>s_{\overline{X}_1 - \overline{X}_2} = \sqrt{{s_1^2 \over n_1} + {s_2^2 \over n_2}}.
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| </math>
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| Here ''s''<sup>2</sup> is the [[unbiased estimator]] of the [[variance]] of the two samples, ''n<sub>i</sub>'' = number of participants in group ''i'', ''i''=1 or 2. Note that in this case <math> {s_{\overline{X}_1 - \overline{X}_2}}^2 </math> is not a pooled variance. For use in significance testing, the distribution of the test statistic is approximated as an ordinary Student's t distribution with the degrees of freedom calculated using
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| :<math> \mathrm{d.f.} = \frac{(s_1^2/n_1 + s_2^2/n_2)^2}{(s_1^2/n_1)^2/(n_1-1) + (s_2^2/n_2)^2/(n_2-1)}.
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| </math>
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| This is known as the [[Welch–Satterthwaite equation]]. The true distribution of the test statistic actually depends (slightly) on the two unknown population variances (see [[Behrens–Fisher problem]]).
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| ===Dependent ''t''-test for paired samples===
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| This test is used when the samples are dependent; that is, when there is only one sample that has been tested twice (repeated measures) or when there are two samples that have been matched or "paired". This is an example of a [[paired difference test]].
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| :<math>t = \frac{\overline{X}_D - \mu_0}{s_D/\sqrt{n}}. </math>
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| For this equation, the differences between all pairs must be calculated. The pairs are either one person's pre-test and post-test scores or between pairs of persons matched into meaningful groups (for instance drawn from the same family or age group: see table). The average (''X<sub>D</sub>'') and standard deviation (''s<sub>D</sub>'') of those differences are used in the equation. The constant ''μ''<sub>0</sub> is non-zero if you want to test whether the average of the difference is significantly different from ''μ''<sub>0</sub>. The degree of freedom used is ''n'' − 1.
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| {| border="1" cellspacing="0" cellpadding="5" align="right"
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| | colspan="4" | ''Example of repeated measures''
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| |-
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| ! Number !! Name !! Test 1 !! Test 2
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| |-
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| | 1 || Mike || 35% || 67%
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| |-
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| | 2 || Melanie || 50% || 46%
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| |-
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| | 3 || Melissa|| 90% || 86%
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| |-
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| | 4 || Mitchell || 78% || 91%
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| |}
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| {| border="1" cellspacing="0" cellpadding="5"
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| | colspan="4" | ''Example of matched pairs''
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| |-
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| ! Pair !! Name !! Age !! Test
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| |-
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| | 1 || John || 35 || 250
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| |-
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| | 1 || Jane || 36 || 340
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| |-
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| | 2 || Jimmy || 22 || 460
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| |-
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| | 2 || Jessy || 21 || 200
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| |}
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| ==Worked examples==
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| Let A<sub>1</sub> denote a set obtained by taking 6 random samples out of a larger set:
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| :<math>A_1=\{30.02,\ 29.99,\ 30.11,\ 29.97,\ 30.01,\ 29.99\}</math>
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| and let A<sub>2</sub> denote a second set obtained similarly:
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| :<math>A_2=\{29.89,\ 29.93,\ 29.72,\ 29.98,\ 30.02,\ 29.98\}</math>
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| These could be, for example, the weights of screws that were chosen out of a bucket.
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| We will carry out tests of the null hypothesis that the [[Arithmetic mean|mean]]s of the populations from which the two samples were taken are equal.
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| The difference between the two sample means, each denoted by <math>\overline{X}_i</math>, which appears in the numerator for all the two-sample testing approaches discussed above, is
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| :<math>
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| \overline{X}_1 - \overline{X}_2 = 0.095.
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| </math>
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| The sample [[standard deviations]] for the two samples are approximately 0.05 and 0.11, respectively. For such small samples, a test of equality between the two population variances would not be very powerful. Since the sample sizes are equal, the two forms of the two sample ''t''-test will perform similarly in this example.
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| ===Unequal variances===
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| If the approach for unequal variances (discussed above) is followed, the results are
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| :<math>
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| \sqrt{{s_1^2 \over n_1} + {s_2^2 \over n_2}} \approx 0.0485
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| </math>
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| and | |
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| :<math>
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| \text{df} \approx 7.03. \,
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| </math>
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| The test statistic is approximately 1.959. The [[two-tailed test]] [[p-value]] is approximately 0.091 and the [[one-tailed]] p-value is approximately 0.045. | |
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| ===Equal variances===
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| If the approach for equal variances (discussed above) is followed, the results are
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| :<math>
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| S_{X_1X_2} \approx 0.084 \,
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| </math> | |
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| and
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| :<math>
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| df = 10. \,
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| </math> | |
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| Since the sample sizes are equal (both are 6), the test statistic is again approximately equal to 1.959. Since the degrees of freedom is different from what it is in the unequal variances test, the p-values will differ slightly from what was found above. Here, the [[two-tailed]] p-value is approximately 0.078, and the [[one-tailed]] p-value is approximately 0.039. Thus if there is good reason to believe that the population variances are equal, the results become somewhat more suggestive of a difference in the mean weights for the two populations of screws.
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| ==Alternatives to the ''t''-test for location problems==
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| The ''t''-test provides an exact test for the equality of the means of two normal populations with unknown, but equal, variances. (The [[Welch's t-test|Welch's ''t''-test]] is a nearly exact test for the case where the data are normal but the variances may differ.) For moderately large samples and a one tailed test, the ''t'' is relatively robust to moderate violations of the normality assumption.<ref name="Sawilowsky S., Blair R. C. 1992 353–360">{{cite journal | author = Sawilowsky S., Blair R. C. | year = 1992 | title = A more realistic look at the robustness and type II error properties of the t test to departures from population normality | url = | journal = Psychological Bulletin | volume = 111 | issue = 2 | pages = 353–360 | doi = 10.1037/0033-2909.111.2.352 }}</ref>
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| For exactness, the ''t''-test and Z-test require normality of the sample means, and the ''t''-test additionally requires that the sample variance follows a scaled [[Chi-squared distribution|χ<sup>2</sup> distribution]], and that the sample mean and sample variance be [[independence (probability theory)|statistically independent]]. Normality of the individual data values is not required if these conditions are met. By the [[central limit theorem]], sample means of moderately large samples are often well-approximated by a normal distribution even if the data are not normally distributed. For non-normal data, the distribution of the sample variance may deviate substantially from a χ<sup>2</sup> distribution. However, if the sample size is large, [[Slutsky's theorem]] implies that the distribution of the sample variance has little effect on the distribution of the test statistic. If the data are substantially non-normal and the sample size is small, the ''t''-test can give misleading results. See [[location testing for Gaussian scale mixture distributions|Location test for Gaussian scale mixture distributions]] for some theory related to one particular family of non-normal distributions.
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| When the normality assumption does not hold, a [[non-parametric]] alternative to the ''t''-test can often have better [[statistical power]]. For example, for two independent samples when the data distributions are asymmetric (that is, the distributions are [[skewness|skewed]]) or the distributions have large tails, then the Wilcoxon rank-sum test (also known as the [[Mann–Whitney U test]]) can have three to four times higher power than the ''t''-test.<ref name="Sawilowsky S., Blair R. C. 1992 353–360"/><ref>{{cite journal | last=Blair | first=R. C. | coauthors=Higgins, J.J. | journal=Journal of Educational Statistics |
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| year=1980 | pages=309–334 | volume=5 | title=A comparison of the power of Wilcoxon’s rank-sum statistic to that of Student’s t statistic under various nonnormal distributions. | doi=10.2307/1164905 | jstor=1164905 | issue=4}}</ref><ref>{{cite journal | last=Fay | first=MP | coauthors=Proschan, MA | journal=Statistics Surveys | year=2010 | pages=1–39 | volume=4 | url=http://www.i-journals.org/ss/viewarticle.php?id=51 | title=Wilcoxon–Mann–Whitney or ''t''-test? On assumptions for hypothesis tests and multiple interpretations of decision rules | doi=10.1214/09-SS051 | pmid=20414472 | pmc=2857732 }}</ref> The nonparametric counterpart to the paired samples t test is the [[Wilcoxon signed-rank test]] for paired samples. For a discussion on choosing between the t and nonparametric alternatives, see Sawilowsky.<ref>{{cite journal | author = Sawilowsky S | year = 2005 | title = Misconceptions leading to choosing the t test over the Wilcoxon Mann–Whitney U test for shift in location parameter | url = | journal = Journal of Modern Applied Statistical Methods | volume = 4 | issue = 2| pages = 598–600 }}</ref>
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| One-way [[analysis of variance]] generalizes the two-sample ''t''-test when the data belong to more than two groups.
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| ==Multivariate testing==
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| {{main|Hotelling's T-squared distribution}}
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| A generalization of Student's ''t'' statistic, called [[Hotelling's T-square statistic]], allows for the testing of hypotheses on multiple (often correlated) measures within the same sample. For instance, a researcher might submit a number of subjects to a personality test consisting of multiple personality scales (e.g. the [[Minnesota Multiphasic Personality Inventory]]). Because measures of this type are usually positively correlated, it is not advisable to conduct separate univariate ''t''-tests to test hypotheses, as these would neglect the covariance among measures and inflate the chance of falsely rejecting at least one hypothesis ([[Type I error]]). In this case a single multivariate test is preferable for hypothesis testing. [[Fisher's_Method#Limitations_of_independent_assumption|Fisher's Method]] for combining multiple tests with ''[[Type_I_and_type_II_errors#Type_I_error|alpha]]'' reduced for positive correlation among tests is one. Another is Hotelling's ''T''<sup> 2</sup> statistic follows a ''T''<sup> 2</sup> distribution. However, in practice the distribution is rarely used, since tabulated values for ''T''<sup> 2</sup> are hard to find. Usually, ''T''<sup> 2</sup> is converted instead to an ''F'' statistic.
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| ===One-sample ''T''<sup> 2</sup> test===
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| For a one-sample multivariate test, the hypothesis is that the mean vector (<math>{\mathbf\mu}</math>) is equal to a given vector (<math>{\mathbf\mu_0}</math>). The test statistic is [[Hotelling's_T-squared_distribution#Sum_of_p_squared_t.27s|Hotelling's ''T''<sup> 2</sup>]]:
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| :<math>
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| T^2=n(\overline{\mathbf x}-{\mathbf\mu_0})'{\mathbf S}^{-1}(\overline{\mathbf x}-{\mathbf\mu_0})
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| </math>
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| where ''n'' is the sample size, <math>\overline{\mathbf x}</math> is the vector of column means and <math>{\mathbf S}</math> is a <math>m\times m</math> [[sample covariance matrix]].
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| ===Two-sample ''T''<sup> 2</sup> test===
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| For a two-sample multivariate test, the hypothesis is that the mean vectors (<math>{\mathbf\mu}_1</math>, <math>{\mathbf\mu}_2</math>) of two samples are equal. The test statistic is [[Hotelling's_T-squared_distribution#Hotelling.27s_two-sample_T-squared_statistic|Hotelling's 2-sample''T''<sup> 2</sup>]]:
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| :<math>T^2 = \frac{n_1 n_2}{n_1+n_2}(\overline{\mathbf x}_1-\overline{\mathbf x}_2)'{\mathbf S_\text{pooled}}^{-1}(\overline{\mathbf x}_1-\overline{\mathbf x}_2).</math>
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| ==Software implementations==
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| Many [[spreadsheet]] programs and statistics packages, such as [[QtiPlot]], [[OpenOffice.org Calc]], [[LibreOffice Calc]], [[Microsoft Excel]], [[SAS (software)|SAS]], [[SPSS]], [[Stata]], [[DAP (software)|DAP]], [[gretl]], [[R (programming language)|R]], [[Python (programming language)|Python]] ([http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ttest_ind.html]), [[PSPP]], and [[Minitab]], include implementations of Student's ''t''-test.
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| {| class="wikitable sortable"
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| |-
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| ! Language/Program !! Function !! Notes
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| |-
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| | [[Microsoft Excel]] pre 2010 || <code>TTEST(''array1'', ''array2'', ''tails'', ''type'')</code> || See [http://office.microsoft.com/en-us/excel-help/ttest-HP005209325.aspx]
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| |-
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| | [[Microsoft Excel]] 2010 and later || <code>T.TEST(''array1'', ''array2'', ''tails'', ''type'')</code> || See [http://office.microsoft.com/en-us/excel-help/t-test-function-HA102753135.aspx]
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| |-
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| | [[OpenOffice.org]] || <code>TTEST(''data1''; ''data2''; ''mode''; ''type'')</code> ||
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| |-
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| | [[Python (programming language)|Python]] || <code>scipy.stats.ttest_ind(''a'', ''b'', ''axis=0'', ''equal_var=True'')</code> || See [http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ttest_ind.html]
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| |-
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| | [[R (programming language)|R]] || <code>t.test(data1, data2)</code> ||
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| |-
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| | [[SAS (software)|SAS]] || <code>PROC TTEST</code> || See [http://www.sas.com/offices/europe/belux/pdf/academic/ttest.pdf]
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| |}
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| ==See also==
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| {{Portal|Statistics}}
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| * [[Conditional change model]]
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| * [[F-test]]
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| * [[Student's t-statistic]]
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| * [[Z-test]]
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| ==Notes==
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| {{Reflist|30em}}
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| ==References==
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| * {{cite book
| |
| | last = O'Mahony
| |
| | first = Michael
| |
| | title = Sensory Evaluation of Food: Statistical Methods and Procedures
| |
| | publisher = [[CRC Press]]
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| | year = 1986
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| | pages = 487
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| | isbn = 0-8247-7337-3
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| }}
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| * {{cite book
| |
| | last = Press
| |
| | first = William H.
| |
| | coauthors = Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery
| |
| | title = Numerical Recipes in C: The Art of Scientific Computing
| |
| | publisher = [[Cambridge University Press]]
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| | year = 1992
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| | pages = [http://www.nrbook.com/a/bookcpdf/c14–2.pdf p. 616]
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| | isbn = 0-521-43108-5
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| | url = http://www.nr.com/
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| }}
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| ==Further reading==
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| *{{Cite journal |year=1960 |author=Boneau, C. Alan |title=The effects of violations of assumptions underlying the ''t'' test |journal=Psychological Bulletin |volume=57 |issue=1 |pages=49–64 |doi=10.1037/h0041412 |postscript=<!--None--> }}
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| *{{Cite journal |year=1984 |author=Edgell, Stephen E., & Noon, Sheila M |title=Effect of violation of normality on the ''t'' test of the correlation coefficient |journal=Psychological Bulletin |volume=95 |issue=3 |pages=576–583 |doi=10.1037/0033-2909.95.3.576}}
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| ==External links==
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| {{wikiversity|t-test}}
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| * {{springer|title=Student test|id=p/s090720}}
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| * [http://www.socialresearchmethods.net/kb/stat_t.php A conceptual article on the Student's ''t''-test]
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| {{Statistics}}
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| {{DEFAULTSORT:Student's T-Test}}
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| [[Category:Statistical tests]]
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| [[Category:Statistical methods]]
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| [[Category:Parametric statistics]]
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