T-TESTS  

I. t-Test Analysis

a.  Introduction

1)  Many research projects are designed to test the differences between two groups.  Differences that involve interval or ratio data, requires an evaluation of means and distributions of each group.

b. Student's t-test

1) Named after William Gosset, who published under the pseudonym of Student.  Gosset invented the t-test as a more precise method of comparing groups. He described a set of distribution of means of randomly drawn samples from a normally distributed population. Distributions are the t distributions.

2) All t distributions have a normal distribution with a mean equal to the mean of the population. T distributions are used to compare differences between two means. These distributions are described by the sample of differences between means obtained from drawing pairs of random samples from a population. If we drew an infinite number of pairs of samples and plot the differences between the means of all possible pairs of samples, we would find a particular distribution with a mean of zero and a shape similar to the normal curve.

c. The Research Question

1) When compare two groups on a particular characteristic, asking whether or not the groups are different. 

1. The statistical question asks how different the groups are; that is, the difference we find greater than that which could occur by chance alone. 

Symbolically, the null hypothesis is written:

Ho:  µ1 =  µ2

2. The null hypothesis states that any difference that occurs between the means of two groups is a difference in the sampling distribution. The means are different not because the groups are drawn from two theoretical populations, but rather because of different random distributions of the samples from such a population.

3. When we use the t-test to interpret the significance of the difference between groups, we are asking the statistical question, what is the probability of getting a difference of this magnitude in this size if we were comparing random samples drawn from the same population? What is the probability of getting a difference this large by chance alone?

4. Sample size (n), and standard deviation (variability), contributes to significance of the t-test.

d. Type of Data Required

1)  Independent Variable: Nominal-level variable, with two levels (two groups).

2) Dependent Variable: Interval or ratio level, ordinal level data can often be treated as interval-level data.

e.  Assumptions

1)  Interval-level data for the dependent measure.

2) Assumption of independence: subject can belong to one and only one of the two groups and contributes one and only one score.

3) Distribution of the dependent measure is normal.

4) Homogeneity of variance: groups that are compared are similar in their variances.

 Formula:                                   larger variance  

F =     -----------------

smaller variance

                          df for this F ratio are based on the ns for the groups and equal n-1.
                               
*Meeting this assumption protects against Type II errors-incorrectly accepting the null hypothesis.

f. T-Test Formulas

1)  Independent or Pooled Formula: Used to compare two groups when the assumptions for the t-test (including homogeneity of variance) are met.

2) Correlated t-test (Dependent) or Paired Comparisons: Used if there is correlation between the data taken from the two groups, adjustment must be made for that relationship.  Comparing a group of subjects on their pre- and post scores is an example. Because these are not two independent groups, but rather one group measured twice, the scores will most likely be correlated.  Another example is when the two groups consist of matched pairs.

g.  Calculations

1) Basic (Pooled) t-test

 

t =

(x1 - x2) - ( µ1-  µ2)
-----------------------
       s(x1 - x2)

s(x1 -x2) =

standard error

calculated by:

 
s(x1-x2) =

  ___________________
√  (Σx21 +  Σx22 ) (1  + 1)
   ---------------   ---   ---
   (n1 + n2 –2)  (n1   n2)

Where
Σx21  = sum of squares of group 1
Σx22  = sum of squares of group 2
n1     = the number of scores in group 1
n2     = the number of scores in group 2

when two groups have equal ns, the formula is simplified to:

 
s(x1-x2) =

    _____________
 √ (Σx21 +  Σx22 )
   ---------------  
    n (n – 1)

To find the sum of squares for each group, use:

 

Group 1:

                            (ΣX1)2
      (Σx1)2 = ΣX21-    --------
                              n1

Group 2:

                          (ΣX2)2
      (Σx2)2 = ΣX22-    --------
                              n2

T-Test Example (Independent Groups or pooled)

Group I

Group II

9

81

6

36

6

36

7

49

8

64

7

49

8

64

9

81

9

81

8

64

∑X1 = 40

∑X12 = 326

∑X2 = 37

∑X22 = 279

 Set up your computation table.


Group I

Group II

                       _
X1 = 8

                        _
X2= 7.4

n1 = 5

n2 = 5

∑X1 = 40

∑X2 = 37

∑X12 =326

∑X22 = 279

 

Now calculate the sum of squares

                         (40)2
x12  = 326  –-  ---------
                            5

 

x12  = 6 

                         (37)2
x22  =  279 –  --------
                            5

 

x22  = 5.2

 

Now calculate t.

 

t =

  _      _  
 (X1 – X2) – (µ1 -  µ2)
  ___________________________________
√(∑x12  + ∑x22 ) / (n1 + n2 – 2)  (1/n1 + 1/ n2)

 

 

 

 

 

t =

   
      (8 – 7.4) – (0)
  ___________________________
√(6 + 5.2) / (5 + 5 – 2)  (1/5 + 1/5)
    

 

     .6
=   ___
   √(1.4) (.4)

 

       .6
=   ___
   √ .56

 

    .6
=  __
   .75

  

t = .8
(not significant with p < .05)

Now calculate the degrees of freedom.


df = (n1 + n2) – 2 or df = n - 2   

df = (5 + 5) – 2 or 8

Use a critical values table to determine t-value. Use a one-tail distribution if hypothesize a difference in a particular direction. If no directional hypothesis, base interpretation on the two-tailed distribution.