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| {{DISPLAYTITLE:Welch's ''t''-test}}
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| In [[statistics]], '''Welch's ''t'' test''' is an adaptation of [[Student's t-test|Student's ''t''-test]] intended for use with two samples having possibly unequal [[variance]]s.<ref> {{Cite journal | last = Welch | first = B. L. | title = The generalization of "Student's" problem when several different population variances are involved | journal = [[Biometrika]] | volume = 34 |issue=1–2 | pages = 28–35 | year = 1947 |doi =10.1093/biomet/34.1-2.28 | mr = 19277 }}</ref> As such, it is an approximate solution to the [[Behrens–Fisher problem]].
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| ==Formulas==
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| Welch's t-test defines the statistic ''t'' by the following formula:
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| :<math>
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| t \quad = \quad {\; \overline{X}_1 - \overline{X}_2 \; \over \sqrt{ \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \quad }}\,</math>
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| where <math>\overline{X}_{i}</math>, <math>s_{i}^{2}</math> and <math>N_{i}</math> are the <math>i</math><sup>th</sup> [[mean|sample mean]], [[variance|sample variance]] and [[sample size]], respectively. Unlike in [[Student's t test|Student's ''t''-test]], the denominator is ''not'' based on a [[pooled variance]] estimate.
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| The [[degrees of freedom]] <math>\nu</math> associated with this variance estimate is approximated using the [[Welch–Satterthwaite equation]]:
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| :<math>
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| \nu \quad \approx \quad
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| {{\left( \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \; \right)^2 } \over
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| { \quad {s_1^4 \over N_1^2 \nu_1} \; + \; {s_2^4 \over N_2^2 \nu_2 } \quad }}
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| </math>
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| Here <math>\nu_i</math> = <math>N_i-1</math>, the degrees of freedom associated with the <math>i</math><sup>th</sup> variance estimate.
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| ==Statistical test==
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| Once ''t'' and ''<math>\nu</math>'' have been computed, these statistics can be used with the [[t-distribution]] to test the [[null hypothesis]] that the two population means are equal (using a [[two-tailed test]]), or the null hypothesis that one of the population means is greater than or equal to the other (using a one-tailed test). In particular, the test will yield a [[p-value]] which might or might not give evidence sufficient to reject the null hypothesis.
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| ==References==
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| {{Reflist}}
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| ;Further reading
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| * Daniel Borcard, ''[http://biol09.biol.umontreal.ca/BIO2041e/Correction_Welch.pdf Lecture Note Appendix: t-test with Welch correction''], excerpt from Legendre, P. and D. Borcard. ''Statistical comparison of univariate tests of homogeneity of variances''.
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| * {{cite journal |last=Sawilowsky |first=Shlomo S. |year=2002 |url=http://education.wayne.edu/jmasm/sawilowsky_behrens_fisher.pdf |title=Fermat, Schubert, Einstein, and Behrens–Fisher: The Probable Difference Between Two Means When σ<sub>1</sub> ≠ σ<sub>2</sub> |journal=Journal of Modern Applied Statistical Methods |volume=1 |number=2 |pages=461–472}}
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| {{statistics-stub}}
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| [[Category:Statistical approximations]]
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| [[Category:Statistical tests]]
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Oscar is how he's known as and he totally enjoys this name. Minnesota is where he's been residing for many years. Supervising is my profession. To collect coins is what her family and her enjoy.
my weblog 1A-pornotube.com