The significance level is 5% and the number of cases is 60. The simulated random numbers originate from a bivariate normal distribution with a variance of 1. Unpaired and paired two-sample t-tests Type I error of unpaired and paired two-sample t-tests as a function of the correlation. 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.The most frequently used t-tests are one-sample and two-sample tests: 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. 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. 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 the pseudonym 'Student' because his employer preferred staff to use pen names when publishing scientific papers. 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'. History William Sealy Gosset, who developed the ' t-statistic' and published it under the pseudonym of 'Student' In many cases, a Z-test will yield very similar results to a t-test since the latter converges to the former as the size of the dataset increases. The t-test's most common application is to test whether the means of two populations are different. When the scaling term is estimated based on the data, the test statistic-under certain conditions-follows a Student's t distribution. It is 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 (typically, the scaling term is unknown and is therefore a nuisance parameter). It is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis. A t-test is a statistical hypothesis test used to test whether the difference between the response of two groups is statistically significant or not.

Calculate degrees of freedom for t test