STUDENT'S T-TEST

in statistics, method of testing hypotheses about the mean of a small sample drawn from a normally distributed population when the population standard deviation is unknown. An Englishman, William Sealy Gosset, publishing under the pseudonym Student, developed the t-test and t distribution. The t distribution is a family of curves in which the number of degrees of freedom (the number of independent observations in the sample minus one) specifies a particular curve. As the sample size (and thus the degrees of freedom) increases, the t distribution approaches the bell shape of the standard normal distribution. Hence, for tests involving the mean of a sample of size greater than 30, the normal distribution is usually applied. It is usual first to formulate a null hypothesis, which states that there is no effective difference between the observed sample mean and the hypothesized or stated population meani.e., that any measured difference is due only to chance. An alternative hypothesis is also formulated. This may be either two-sided (also termed two-tailed), stating simply that the means are not equivalent, or one-sided, specifying whether the observed mean is larger or smaller than the hypothesized mean. The test statistic t is then calculated. If the observed t-statistic is more extreme than the critical value determined by the appropriate reference distribution, the null hypothesis is rejected. The appropriate reference distribution for the t-statistic is the t distribution. The critical value depends on the significance level of the test (the probability of erroneously rejecting the null hypothesis). For example, suppose a researcher wishes to test the hypothesis that a sample of size n = 25 with mean x = 79 and standard deviation s = 10 was drawn at random from a population with mean m = 75 and unknown standard deviation. Using the formula for the t-statistic, the calculated t equals 2.00. For a two-sided test at the level of significance a = 0.05, the critical values from the t distribution on 24 degrees of freedom are -2.064 and 2.064. The calculated t does not exceed these values, hence the null hypothesis is accepted with 95 percent confidence. (The confidence level is 1 - a.) A second application of the t distribution tests the hypothesis that two independent random samples have the same mean. The t distribution can also be used to construct confidence intervals for the true mean of a population (the first application) or for the difference between two sample means (the second application). See also interval estimation.