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-TESTA 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 EXPLANATIONOne 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 in situations in which the subjects from the two groups are paired or matched in some ways.
A common example of this case is the same group of subjects tested twice as in a pretest-posttest study. Whether the same or different subjects are in each group as long as there is a systematic relationship between the groups, it is necessary to use the paired t-test to calculate the probability of rejecting the null hypothesis. A more concrete explanation of using the t-test is the following example. Suppose a researcher is interested in finding out whether there is a significant difference between black haired and white haired grade twelve learners with respect to reading achievement.
The research question would be: Is there a difference in the reading achievement (the dependent variable) of the black haired grade 12 learners compared with white haired grade 12 learners (the independent variable)? The null hypothesis would be: There is no difference between black haired and white haired grade 12 learners in reading achievement. To test this hypothesis, the researcher would randomly select a sample of white haired and black haired grade 12 learners from the population of all grade 12 learners.
Let us say that the sample mean of black haired grade 12 learners’ reading achievement is 54 and the sample mean for white haired grade 12 learners is 48. Because we assume the null hypothesis-that the population means are equal. We use the t-test to show how often the difference of scores in the sample would occur if the population means are equal.
4.o WHAT DOES THIS TEST MEASURE?A t-Test determines whether the means of two groups are statistically different from each other. It measures whether there is any statistical difference in the mean of the two groups.It can also be used to determine if two sets of data are significantly different from each other and 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.
Additionally, a t-test is used in statistics to measure the correlation values between two or more samples to determine the validity of the null hypothesis.5.0 WHAT IS THE T-TEST USED FOR?The t-test is used for comparing the means of two samples. In simple terms, the t-test compares the actual difference between two means in relation to the variation in the data expressed as the standard deviation of the difference between the means.
6.0 ADDITIONAL INFORMATIONThe one-sample t-test compares the mean score of a sample to a known value usually the population mean (the average for the outcome of some population of interest). The basic idea of the test is a comparison of the average of the sample (observed average) and the population (expected average) with an adjustment for the number of cases in the sample and the standard deviation of the average.
For example one of the best indicators of the health of a baby is his or her weight at birth. Birth weight is an outcome that is sensitive to the conditions in which mothers experienced pregnancy particularly to issues of deprivation and poor diet which are tied to lower birth weight. In Africa, mothers who live in poverty generally have babies with lower birth weight than those who do not live in poverty.
While the average birth weight for babies born is approximately 3300 grams, the average birth weight for women living in poverty is 2800 grams. In the first year, 25 mothers, all of whom live in poverty, participated in this program. Data drawn from hospital records reveals that the babies born to these women had a birth weight of 3075 grams, with a standard deviation of 500 grams. The question posed to the researcher, is whether this program has been effective at improving the birth weights of babies born to poor women. 6.1 Establish Hypotheses
The first step here is to establish the specific hypotheses. For this example, what is the null hypothesis? What is the alternative hypothesis? In this case:* The Null hypothesis: the difference between the birth weights of babies born to mothers who participated in the program and those born to other poor mothers is 0. * Alternative hypothesis: the difference between the observed mean of birth weight for program babies and the expected mean of birth weight for poor women is not zero. 6.2 Calculate Test Statistic
Calculation of the test statistic requires four components:1. The average of the sample (observed average)2. The population average or other known value (expected average) 3. The standard deviation (SD) of the sample average4. The number of observations.
With this example, the components are as follows:1. Sample average = 3075 grams2. Population average (poor women – remember we’re interested in whether this program improves birth outcomes relative to those of poor women) = 2800 grams 3. SD of the sample average = 300 grams
4. Number of observations = 25
With these four pieces of information, we calculate the following statistic, t:
In the case of our example,
6.3 Use This Value To Determine P-ValueHaving calculated the t-statistic, compare the t-value with a standard table of t-values to determine whether the t-statistic reaches the threshold of statistical significance. Plugging in the values of t (.898) and n (number of cases = 25) yields a p-value of .378. Generally speaking, we require p-values of .05 or less in order to reject the null hypothesis. With a value of .378, we cannot reject the null. Therefore, we conclude that the intervention did not successfully improve birth weight.
6.4 Two sample t-test
Again we often want to know whether the means of two populations on some outcome differ. For example, there are many questions in which we want to compare two categories of some categorical variable (e.g., compare males and females) or two populations receiving different treatments in context of an experiment.
The two-sample t-test is a hypothesis test for answering questions about the mean where the data are collected from two random samples of independent observations each from an underlying normal distribution. In this example rather than comparing the birth weight of a group of infant to some national average, we will examine a program’s effect by comparing the birth weights of babies born to women who participated in an intervention with the birth weights of a group that did not. To evaluate the effects of some intervention, program, or treatment, a group of subjects is divided into two groups.
The group receiving the treatment to be evaluated is referred to as the treatment group, while those who do not are referred to as the control or comparison group. In this example, mothers who are part of the prenatal care program to reduce the likelihood of low birth weight is the treatment group while the control group comprise of women who do not take part in the program. For the two-sample t-test, the steps to conduct the test are similar to those of the one-sample test. 7.0 CONCLUSION
To calculate a one-sample t-test, use the following steps:1. Establish HypothesesNull hypothesis: The difference between observed and expected is 0 Alternative hypothesis: The difference between observed and expected is not 0. 2. Calculate Test StatisticCalculation of the test statistic requires four components:1. The average of the sample (observed average)2. The population average or other known value (expected average) 3. The standard deviation of the average4. The number of observations.With these four pieces of information, we calculate the following statistic, t:
3. Use This Value to Determine P-ValueHaving calculated the t-statistic, compare the t-value with a standard table of t-values to determine whether the t-statistic reaches the threshold of statistical significance. For the two-sample t-test, the steps to conduct the test are similar to those of the one-sample test. Establish Hypotheses:
In this case:* The null hypothesis is that the difference between the mean of the treatment group of birth weight for program babies and the mean of the control group of birth weight for poor women is zero. * Alternative hypothesis: the difference between the observed mean of birth weight for program babies and the expected mean of birth weight for poor women is notzero. Calculate Test Statistic
Calculation of the test statistic requires three components: 1. The average of both sample (observed averages)Statistically, we represent these as
2. The standard deviation (SD) of both averagesStatistically, we represent these as
3. The number of observations in both populations, represented as From hospital records, we obtain the following values for these components:
| Treatment| Control|Average Weight| 3100 g| 2750 g|SD| 420| 425|N| 75| 75|With these pieces of information, we calculate the following statistic, t:
Use This Value To Determine P-ValueHaving calculated the t-statistic, compare the t-value with a standard table of t-values to determine whether the t-statistic reaches the threshold of statistical significance. With a t-score so high, the p-value is 0.001, a score that forms our basis to reject the null hypothesis and conclude that the prenatal care program made a difference.