Example of paired sample t-test

Let us consider a simple example of what is often termed "pre/post" data or "pretest/posttest" data. Suppose you wish to test the effect of Prozac on the well-being of depressed individuals, using a standardized "well-being scale" that sums Likert-type items to obtain a score that could range from 0 to 20. Higher scores indicate greater well-being (that is, Prozac is having a positive effect). While there are flaws in this design (e.g., lack of a control group) it will serve as an example of how to analyze such data.

The value that we are interested in is the change score, and we obtain it by taking the difference between time two and time one.

Mood Pre / Mood Post / Difference
3 / 5 / 2
0 / 1 / 1
6 / 5 / -1
7 / 7 / 0
4 / 10 / 6
3 / 9 / 6
2 / 7 / 5
1 / 11 / 10
4 / 8 / 4

Notice that I have subtracted the first score away from the second to get a difference score or change score. Person #3's well-being score decreased by one point at the post-test. Person #5 increased their well-being score from 4 point to 10 points. The mean of the Pretest data is 3.33 and the mean of the Post test data is 7.0. The question is, "Is this a significant increase?"

Step 1: Stating the hypotheses:

Ho: md=0

H1: md 0

The alternative is two-tailed and alpha=.05

Step 2: Check assumptions

1. The observations are independent of each other

2. The dependent variable is measured on an interval scale

3. The differences are normally distributed in the population.

The measures are approximately interval scale numbers (self-report scores) and we assume that each person's score has not been influenced by other people's scores. The numbers look to have no major extremes or unusual distribution.

Step 3: Calculate test statistic

Step 4: Evaluate the result

Step 5: Interpret the result

Practice Problem: t-test

A research study was conducted to examine the differences between older and younger adults on perceived life satisfaction. A pilot study was conducted to examine this hypothesis. Ten older adults (over the age of 70) and ten younger adults (between 20 and 30) were given a life satisfaction test (known to have high reliability and validity). Scores on the measure range from 0 to 60 with high scores indicative of high life satisfaction; low scores indicative of low life satisfaction. The data are presented below. Compute the appropriate t-test.

Older Adults / Younger Adults
45 / 34
38 / 22
52 / 15
48 / 27
25 / 37
39 / 41
51 / 24
46 / 19
55 / 26
46 / 36
Mean = / Mean =

1.  What is your computed answer?

2.  What would be the null hypothesis in this study?

3.  What would be the alternate hypothesis?

4.  What probability level did you choose and why?

5.  What is your tcrit?

6.  Is there a significant difference between the two groups?

7.  Interpret your answer.

I read an article in a journal recently that said Canadian researchers studied 56 competitive bike riders and their nutrition habits. The study followed the athletes for 6 months. The athletes were divided into two groups. The first group drank a soy based beverage before and after their workouts. The second group drank skim milk before and after. The researchers were interested in whether those who drank the milk increased their muscle mass more than those drinking soy. The results of the muscle mass gained after the 6 months are below. The question is: Did those who drank skim milk increase their muscle mass by significantly more than those who drank soy?

1=soy group; 2=skim milk group

1 / 5
1 / 4
1 / 5
1 / 6
1 / 5
1 / 4
1 / 5
1 / 6
1 / 4
1 / 2
1 / 5
1 / 6
1 / 2
1 / 1
1 / 5
1 / 3
1 / 6
1 / 5
1 / 3
1 / 2
1 / 4
1 / 2
1 / 5
1 / 6
1 / 3
1 / 2
1 / 4
1 / 1
2 / 4
2 / 5
2 / 7
2 / 5
2 / 2
2 / 4
2 / 7
2 / 8
2 / 4
2 / 8
2 / 6
2 / 5
2 / 7
2 / 5
2 / 4
2 / 2
2 / 3
2 / 5
2 / 8
2 / 7
2 / 9
2 / 5
2 / 4
2 / 5
2 / 6
2 / 5
2 / 4
2 / 8