georges.tesh

t-Tests

Statistical testing was originally designed to use large sample statistics that used the normal distribution as a measure of random chance for the findings. However, that approach does not work for small samples where the N (sample size) is less than 30. The t-Test, based on the t-distribution, is designed to deal with the problem of small sample sizes. The t-test also measures probability of the results being caused by random chance and it does this by using degrees of freedom. Degrees of freedom is a concept of theoretical statistics and is outside the scope of this module, but it is necessary to know how to calculate it for a particular sample. Simply, the degrees of freedom is N-1. In order to get the critical value, select your alpha (p< 0.05, for example) and your degrees of freedom (N-1) on the t-test table found on page 357 of your textbook. However, if you use Excel to do your calculations, it will calculate the alpha and critical value for you. t-Tests also require the use of interval or ratio data.

t-Test For One Sample

t-Tests can be used to answer a variety of research questions. Suppose you were looking at a large research study that reported mean results on a given outcome variable. You wish to see if the results on subjects in your facility match up to the larger study mean. A one-sample t-Test can do this using the following formula:

            

M represents the mean of your sample, u represents the mean of the larger population. S represents the standard deviation of the larger population, and N is the sample size.

As an example, assume that you are looking at the relationship between intelligence as measured on the Wechsler Adult Intelligence Scale (IQ), and the impact of a brain disorder that affects cognitive processing. You want to determine whether the subjects with the disorder have a lower IQ than those in the larger population.

Using the one-sample t-test with 100 subjects testing for p<0.005 with degrees of freedom at 99 (N-1), the table indicates a critical cutoff value for your subjects at t < -2.66. The M (sample mean) of your subjects is 94.3 and the standard deviation (S) is 15.

First calculate Sm = S/Sq Root of N = 15/Sq root of 100 = 1.5

Then calculate t = M-u/Sm = (94.3-100)/1.5 = -3.80

Your critical cutoff value is -2.66, and your finding is -3.80, indicating that the answer to your research question is Yes, with less than a 0.005% chance that the finding was caused by random chance.

Note that the sample size in the example was 100, not 30. The t-test can be used for larger samples, although the ANOVA is sometimes better. However, only the t-test can be used for small sized samples. Its use is also limited to two groups. If you have more than two groups, the ANOVA is the required statistic.

t-Test for Independent Samples

This test can be used in the traditional experimental/control groups design. It requires interval or ratio data, and is designed to answer the research question "Is there a significant difference on measures of the dependent variable between the two groups?"

For example, assume that an experiment is conducted to determine the impact of a new relaxation technique on patients' blood pressures. Subjects were randomly divided into two groups, and Group B received the technique while Group A was given placebo instructions. Mean arterial pressures were measured at the end of the experiment.

The following data were obtained:

      Mean of Group A (control)                       53

      Sum of Squares of Group A                      1011

      Mean of Group B (Exp)                             59

      Sum of Squares of Group B                       879

      N = 30, each group  = 15

The formula for a t-test for two independent samples is:

Translated into English

      M1-M2                   Mean of first group − Mean of second group (53-59)

     

      SS1+SS2                   Sum of squares of first group + sum of squares of second group

 

      n1+n2 -2                 Sample size of first group + sample size of second group -2

 

      (1/n1 + 1/n2)          Divide 1 by sample size of first group; divide 1 by sample size of second group,

                                     Add together.

 

So what we have so far is:

      M1-M2                    53-59= -6

      SS1 + SS2                 1011 + 879 = 1890

      n1 + n2 −     2          15 + 15-2 =28

      (1/n1 + 1/n2)           0.67 + 0.67 = 0.13

When you put it into the formula, it looks like this:

 

 

 

                        -6       t = -----------------------------------------             Sq Rt of 1890/28 *0.13

 

                        -6       t=---------------------------------------             Sq Rt of 67.5 * 0.13

 

                        -6       t=-------------------------------------             Sq Rt of 8.775

 

            -6       t=---------------             2.96

 

      t= -2.02

So what is the critical value for this t-test at an alpha level of p<.025? The degrees of freedom for a two sample t-test are calculated by n1 + n2 − 2, which equals 28. When we look at the table for one-tailed t-tests at p<0.05 and df of 28, the critical value is −2.048. In this case, the question is whether -2.02 is greater on the negative side than -2.048, which it is not. Therefore, the differences between the two groups are not statistically significant, and the patients need to look elsewhere for an effective technique to lower blood pressure.

What are One-Tailed and Two-Tailed t-Tests?

The tail status of a t-test is determined by the research question. For example, if you want to know whether Group I isgreater than Group 2, use a one-tailed t-test since you are only interested in whether one group's score is greater than the other group's score. However, if you want to know whether one group was different than the second group, use a two-tailed test since you'd want to know if one group was greater than or lesser than the other group. In the measure of blood pressure above, the researchers are only interested in whether the experimental group's score was lower than the control group, hence the use of the one-tailed table.

Conclusion

The Pearson Product Moment Correlation Coefficient answers the question of how two variables relate to each other using interval or ratio data with a sample size of at least 4. The one-sample t-test answers the question of how a sample matches up to a larger sample or a population mean using interval or ratio data with a sample size of less than 30. The two sample t-test answers the question of whether one group is different from another one or whether it is greater or lesser than another group on a given variable. The t-test can only be used on interval or ratio data with no more than two groups in the measure and a sample size of less than 30.