# T Test

**Topics:**Statistics, Statistical hypothesis testing, Normal distribution

**Pages:**7 (2258 words)

**Published:**September 17, 2013

1.0 INTRODUCTION

The t-test was developed by W. S. Gossett, a statistician employed at the Guinness brewery. However, because the brewery did not allow employees to publish their research, Gossett's work on the t-test appears under the name "Student". The t-test is sometimes referred to as "Student's t-test." Gossett was a chemist and was responsible for developing procedures for ensuring the similarity of batches of Guinness. The t-test was developed as a way of measuring how closely the yeast content of a particular batch of beer corresponded to the brewery's standard. And the same statistical methodology that compares a particular batch of beer to a standard can be used to compare how different any two batches are from each other. The test can be used to compare the yeast content of two kegs of beer brewed at separate times. Extending this into the realm of social phenomena, this methodology was used to address questions such as whether SAT preparation courses improve test scores or not and one of the advantages of the t-test is that it can be applied to a relatively small number of cases. It was specifically designed to evaluate statistical differences for samples of 30 or less.

2.0 DEFINITION OF T-TEST

A t-Test is any statistical hypothesis test in which the test statistics follows a Student’s t distribution if the null hypothesis is supported. 3.0 A BRIEF EXPLANATION

One of the most commonly used statistical procedures is the t-test. There are actually three variations of the t-test that we will consider. These are the single-sample, two samples with different groups and the two- sample with the same group. 3.1 SINGLE-SAMPLE T-TEST

Single Sample t-test involves one group. The single sample t-test is used to describe the nature of the population confidence intervals or compare the group mean to a specified value. To establish confidence intervals, the mean and the standard error of the mean are calculated and the confidence intervals are established, typically at 95 or 99 percent. This gives the researcher confidence that the true mean of the population is between the end point of the interval. A single mean can also be compared to a specified value. In this case, the researcher tests the null hypothesis that there is no difference between the sample mean and the fixed numerical value. For example, a researcher could draw a sample of high school learners’ SAT scores, calculate the mean and then compare the sample mean to the national average. In this way, conclusion about whether the sample of learners was significantly above or below the national norm can be determined. 3.2 TWO-SAMPLE T-TEST WITH INDEPENDENT GROUPS

Independent samples t-test involves two groups. This is the most common use of the t-test. It is usually referred to as an independent sample t-test. The purpose of this procedure is to determine if there is a statistically significant difference in the dependent variable between two different populations of subjects. The mean and standard deviation of each sample are calculated and used to determine the t-statistics, which is the difference between the samples means divided by the standard error of the mean (the denominator is calculated from the standard deviations). The formula is t= mean of group 1- mean of group 2 divided by the standard error of mean differences. One way of thinking about this formula is that the difference between the groups is divided by the variation that exists between both between groups and within groups. Researchers may refer to this as simply variation between divided by variation within. As the distance between the groups’ means gets larger and as the standard error gets smaller, the t statistics gets larger 3.3 PAIRED T-TEST WITH DEPENDENT GROUPS

Paired t-Test is one group with two measures. The third form of the t-test can be referred to by several different names including paired, dependent samples, correlated or matched t-test. This t-test is used...

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