The t-test is any statistical hypothesis test in which the test statistic follows a Student’s t-distribution under the null hypothesis.

t-test is the 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 scaling term is unknown and is replaced by an estimate based on the data, the test statistics (under certain conditions) follow a Student’s t distribution. The t-test can be used, for example, to determine if the means of two sets of data are significantly different from each other.

A two-sample location test of the null hypothesis such that the means 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 variances 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. These tests are often referred to as “unpaired” or “independent samples” t-tests, as they are typically applied when the statistical units underlying the two samples being compared are non-overlapping{\displaystyle t={\frac {Z}{s}}={\frac {{\bar {X}}-\mu }{{\widehat {\sigma }}/{\sqrt {n}}}}}

The means of the two populations being compared should follow normal distributions. Under weak assumptions, this follows in large samples from the central limit theorem, even when the distribution of observations in each group is non-normal. If using Student’s original definition of the t-test, the two populations being compared should have the same variance (testable using F-testLevene’s testBartlett’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. Welch’s t-test is insensitive to equality of the variances regardless of whether the sample sizes are similar.

The data used to carry out the test should either be sampled independently from the two populations being compared or be fully paired. This is in general not testable from the data, but if the data are known to be dependent (e.g. paired by test design), a dependent test has to be applied. For partially paired data, the classical independent t-tests may give invalid results as the test statistic might not follow a t distribution, while the dependent t-test is sub-optimal as it discards the unpaired data.

Independent Samples student t test: Is a test of significance between two samples for each one, the mean and standard deviation are known.

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Comments to: Tests of significance, part 2
• March 25, 2021

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