This assumption is met when the observations used for estimating s 2 come from a normal distribution (and i.i.d for each group). s 2( n − 1)/ σ 2 follows a χ 2 distribution with n − 1 degrees of freedom.X follows a normal distribution with mean μ and variance σ 2 / n.The assumptions underlying a t-test in the simplest form above are that:
T = Z s = X ¯ − μ σ ^ / n is the estimate of the standard deviation of the population, and μ is the population mean. Z may be sensitive to the alternative hypothesis (i.e., its magnitude tends to be larger when the alternative hypothesis is true), whereas s is a scaling parameter that allows the distribution of t to be determined. Most test statistics have the form t = Z / s, where Z and s are functions of the data. 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. 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. A two-sample location test of the null hypothesis such that the means of two populations are equal.A one-sample location test of whether the mean of a population has a value specified in a null hypothesis.
Uses Īmong the most frequently used t-tests are: Gosset's identity was then known to fellow statisticians and to editor-in-chief Karl Pearson. 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. The t-test work was submitted to and accepted in the journal Biometrika and published in 1908. Gosset devised the t-test as an economical way to monitor the quality of stout. Gosset had been hired owing to Claude Guinness's policy of recruiting the best graduates from Oxford and Cambridge to apply biochemistry and statistics to Guinness's industrial processes. Although it was William Gosset after whom the term "Student" is penned, it was actually through the work of Ronald Fisher that the distribution became well known as "Student's distribution" and "Student's t-test". Hence a second version of the etymology of the term Student is that Guinness did not want their competitors to know that they were using the t-test to determine the quality of raw material (see Student's t-distribution for a detailed history of this pseudonym, which is not to be confused with the literal term student). Gosset worked at the Guinness Brewery in Dublin, Ireland, and was interested in the problems of small samples – for example, the chemical properties of barley with small sample sizes. However, the T-Distribution, also known as Student's t-distribution gets its name from William Sealy Gosset who first published it in English in 1908 in the scientific journal Biometrika using his pseudonym "Student" because his employer preferred staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity. The t-distribution also appeared in a more general form as Pearson Type IV distribution in Karl Pearson's 1895 paper.
In statistics, the t-distribution was first derived as a posterior distribution in 1876 by Helmert and Lüroth. The term " t-statistic" is abbreviated from "hypothesis test statistic". William Sealy Gosset, who developed the " t-statistic" and published it under the pseudonym of "Student"