Statistical Significance

The z and t-Tests of Statistical Significance

The t-test is an analogue of the z-test where the degrees of freedom replace the n and the t-distribution replaces the normal distribution. Thus the z-test of statistical significance

 

 

 

changes into the t-test of statistical significance

 

 

by replacing the n with the degrees of freedom  equal to n-2.

The t-Test when the Population Mean is Known

The mean temperature of the healthy human body is 98.6 degrees Fahrenheit with very small, negligible variance. This is the population value, true of all healthy human beings. Suppose we measure the body temperature of three patients diagnosed as suffering from an illness. These temperatures are presented in the following table.

 

The framework and results of the regression analysis for this type of experimental design are outlined in the following table.

 

The coefficient of determination of the above regression design equals .4909 / .8241 which is .5958. The coefficient of alienation equals .3333 / .8241 which is .4042. The t-square ratio is defined as

 

 

Substituting these values to the above formula as ( .5958 / .4042 ) 4 returns the t-square equal to 5.896; the t equals 2.43.

Effect Size of the t Ratio

The strength of the relationship, often called the effect size, underlying the critical ratio, can be obtained by algebraically reversing the formula

 

 

 

Multiplying both sides of the above equation by the denominator of the fraction on the right-hand side and at the same time simplifying the left hand expression,

 

 

 

Rearranging the terms of the above equation

 

 

 

Factoring r on the left-hand side of the above equation,

 

 

 

and by dividing the above equation by the expression in the brackets, the formula for the effect size of the t-ratio can be obtained as

 

 

 

For the example of the elevated temperature study, the strength of the relationship can be obtained from the t-square ratio as 5.893 / 5.893 + 4 = .5958.

Independent Measures Design

Five students are preparing for the final examination that they have to take the next day. At midnight, two of them drink a cup of decaffeinated coffee (X₀), and three of them a cup of a regular coffee (X₁). The parent vector X identifies students with respect who drank which type of coffee. The dependent variable Y is the number of hours each student continued to study past midnight.

 

 

 

The coefficient of determination for variables X and Y is .75, the coefficient of alienation is .25, the degrees of freedom are 5 - 2. The t-square

 

is computed as ( .75 / .25) 3 which equals 9. The t equals 3 and the probability associated with the t-ratio is .0562.

Repeated Measures Design

As an example of a repeated-measures t-test, therapy outcomes of five clients suffering from arachnophobia were evaluated by measuring their physiological reactions of fear to a picture of a spider before (Y₁) or after (Y₂) therapy. The conceptual framework of the correlated t-test, for the arachnophobia example, is

 

 

 

The relevant formulae are

 

 

where

 

 

 

and

 

 

 

For the example, the variance of the variable [11 16 20 17 10 8 11 15 11 11] is 12.80 and the correction term is thus 4 (12.80) which equals 51.20. The coefficient of determination is (14.80 - 11.20)² / 51.20 which equals .253. The denominator of the t-square formula is 1.00 - .253 - .625 which equals .122. The degrees of freedom are 4. The t-square ratio equals ( .253 / .122 ) 3 which is 8.31 with the t equal to 2.88.

Summary

The key formulae of the t-test are summarized in the table below. For the experimental design where the population values are known, as well as for the independent measures design, the t-square ratio equals

 

 

For the repeated measures design, the t-square ratio equals

 

 

 

The t-ratio can be associated with a probability, estimated from the areas under the t-distribution.