A t test is a type of statistical test that is used to compare the means of two groups. It is one of the most widely used statisti- cal hypothesis tests. The t-test is used as an example of the basic principles of statistical inference. .. The calculation of the mathematical form (pdf) of the null sampling distribu-. t-tests offer an opportunity to compare two groups on scores such as differences between boys and girls or between children in different school grades. A t-test is .

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PDF | The t distribution is a probability distribution similar to the Normal distribution. It is commonly used to test hypotheses involving numerical data. This paper. A paired t-test is used to compare two population means where you have two Suppose a sample of n students were given a diagnostic test before studying a. Types of t-test. • One sample: – compare the mean of a sample to a predefined value. • Dependent (related) samples: – compare the means of two conditions in .

Oxford University Press. It can also compare average scores of samples of individuals who are paired in some way such as siblings, mothers, daughters, persons who are matched in terms of a particular characteristics. A t-test is an analysis of two populations means through the use of statistical examination; a t-test with two samples is commonly used with small sample sizes, testing the difference between the samples when the variances of two normal distributions are not known. See [8]. One-way analysis of variance ANOVA generalizes the two-sample t -test when the data belong to more than two groups. Z -test normal Student's t -test F -test. O'Mahony, Michael

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. Note that the previous formulae are a special case of the formulae below, one recovers them when both samples are equal in size: This test, also known as 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:. 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. 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: The average X D and standard deviation s D of those differences are used in the equation. 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 normal populations with unknown, but equal, variances.

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. 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. 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 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 can often have better statistical power. 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 Sawilowsky 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 [18].

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 The test statistic is Hotelling's two-sample t From Wikipedia, the free encyclopedia. Main article: Paired difference test.

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. Hotelling's T-squared distribution. Statistics portal. The Story of Mathematics Paperback ed. Princeton, NJ: Princeton University Press. Statistical Science. Retrieved 24 July Two "students" of science". The Concise Encyclopedia of Statistics.

High-Yield Behavioral Science.

High-Yield Series. Hagerstown, MD: The American Statistician. An Introduction to Medical Statistics. Oxford University Press. Mathematical Statistics and Data Analysis 3rd ed. Duxbury Advanced.

Clifford Psychological Bulletin. Clifford; Higgins, James J. Journal of Educational Statistics. On assumptions for hypothesis tests and multiple interpretations of decision rules". Statistics Surveys. Journal of Modern Applied Statistical Methods. Retrieved A companion to Derrick, Russ, Toher and White ".

The Quantitative Methods for Pschology.

O'Mahony, Michael Sensory Evaluation of Food: Statistical Methods and Procedures. CRC Press. Press, William H. When this assumption is violated, see below. Note that the previous formulae are a special case of the formulae below, one recovers them when both samples are equal in size: This test, also known as 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:.

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. 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: The average X D and standard deviation s D of those differences are used in the equation.

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 normal populations with unknown, but equal, variances. 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.

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.

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 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 can often have better statistical power. 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 Sawilowsky 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 [18]. 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 The test statistic is Hotelling's two-sample t From Wikipedia, the free encyclopedia. Main article: Paired difference test. 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. Hotelling's T-squared distribution. Statistics portal. The Story of Mathematics Paperback ed.

Princeton, NJ: Princeton University Press. Statistical Science. Retrieved 24 July Two "students" of science". The Concise Encyclopedia of Statistics. High-Yield Behavioral Science. High-Yield Series. Hagerstown, MD: The American Statistician. An Introduction to Medical Statistics.

Oxford University Press. Mathematical Statistics and Data Analysis 3rd ed. Duxbury Advanced. Clifford Psychological Bulletin. Clifford; Higgins, James J. Journal of Educational Statistics. On assumptions for hypothesis tests and multiple interpretations of decision rules". Statistics Surveys. Journal of Modern Applied Statistical Methods. Retrieved A companion to Derrick, Russ, Toher and White ".

The Quantitative Methods for Pschology. O'Mahony, Michael Sensory Evaluation of Food: Statistical Methods and Procedures. CRC Press. Press, William H.

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