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The best online torrent player and downloader! Your browser does not support the video tag. The t statistic to test whether the population means are different is calculated as:.
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.
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.
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. The t statistic is calculated as. The pairs are e. Let A 1 denote a set obtained by drawing a random sample of six measurements:.
We will carry out tests of the null hypothesis that the means of the populations from which the two samples were taken are equal.
The difference between the two sample means, each denoted by X i , which appears in the numerator for all the two-sample testing approaches discussed above, is.
The sample standard deviations for the two samples are approximately 0. 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. The test statistic is approximately 1.
The test statistic is approximately equal to 1. The t -test provides an exact test for the equality of the means of two i.
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 -test is relatively robust to moderate violations of the normality assumption.
If the data are substantially non-normal and the sample size is small, the t -test can give misleading results. See Location test for Gaussian scale mixture distributions for some theory related to one particular family of non-normal distributions.
When the normality assumption does not hold, a non-parametric alternative to the t -test may have better statistical power. However, when data are non-normal with differing variances between groups, a t-test may have better type-1 error control than some non-parametric alternatives.
It will also have power in detecting an alternative by which group B has the same distribution as A but after some shift by a constant in which case there would indeed be a difference in the means of the two groups.
However, there could be cases where group A and B will have different distributions but with the same means such as two distributions, one with positive skewness and the other with a negative one, but shifted so to have the same means.
In such cases, MW could have more than alpha level power in rejecting the Null hypothesis but attributing the interpretation of difference in means to such a result would be incorrect.
In the presence of an outlier , the t-test is not robust. For example, for two independent samples when the data distributions are asymmetric that is, the distributions are 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.
For a discussion on choosing between the t -test and nonparametric alternatives, see Lumley, et al. One-way analysis of variance ANOVA generalizes the two-sample t -test when the data belong to more than two groups.
When both paired observations and independent observations are present in the two sample design, assuming data are missing completely at random MCAR , the paired observations or independent observations may be discarded in order to proceed with the standard tests above.
Alternatively making use of all of the available data, assuming normality and MCAR, the generalized partially overlapping samples t-test could be used.
A generalization of Student's t statistic, called Hotelling's t -squared 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.
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 for combining multiple tests with alpha reduced for positive correlation among tests is one.
Another is Hotelling's T 2 statistic follows a T 2 distribution. However, in practice the distribution is rarely used, since tabulated values for T 2 are hard to find.
Usually, T 2 is converted instead to an F statistic. The test statistic is Hotelling's t 2 :. The test statistic is Hotelling's two-sample t 2 :.
From Wikipedia, the free encyclopedia. Statistical method. Main article: Paired difference test. Main article: Welch's t-test. This article may not properly summarize its corresponding main article.
Please help improve it by rewriting it in an encyclopedic style. Learn how and when to remove this template message.
Main article: Hotelling's T-squared distribution. Mathematics portal. October Zeitschrift für Anatomie und Entwicklungsgeschichte.
Astronomische Nachrichten in German. Bibcode : AN A forerunner of the t-distribution". MR Archive for History of Exact Sciences. Skew Variation in Homogeneous Material".
Myers, H. New Delhi: Pearson. Retrieved 24 July Silverman: Two "students" of science". The Concise Encyclopedia of Statistics.
High-Yield Behavioral Science. High-Yield Series. Annual Review of Public Health. The American Statistician. Statistical Methods in Medical Research.
An Introduction to Medical Statistics. Oxford University Press. Mathematical Statistics and Data Analysis 3rd ed. Duxbury Advanced. Clifford Psychological Bulletin.
January The Journal of Experimental Education. Clifford; Higgins, James J. Journal of Educational Statistics. On assumptions for hypothesis tests and multiple interpretations of decision rules".
Statistics Surveys. The Quantitative Methods for Psychology. O'Mahony, Michael CRC Press. Press, William H.
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