Lab 14: Related-Samples t-test


Hypothesis tests analyzed with related samples t-tests

 

In the prior lab we examined how to use a t-test to compare a treatment sample against a population (for which s isn't known). Today we'll consider situations where the two samples means come from related samples. The are two ways the samples can be related. In one case, there are two separate but related samples. In the other case, there is a single sample of individuals, each of which gets measured on the dependent variable twice.

 

Consider the following examples:

In the first example, the situation has been decided for you, there is a pre-existing relationship (romantic relationship) between the two samples.

 

In the second and third examples, you, as the experimenter, make a decision to make the two samples related. Why would you ever want to do that? To control for individual differences that might add more noise (error) to your data. In Example 2, each individual acts as their own control.

Okay, so now we know that for repeated-measures and matched-subject designs we need a new t-test. So, what is the t statistic for related samples?

 

Again, the logic of the hypothesis test is pretty much the same as it was for the one-sample cases we've already considered. Once again we'll go through the same steps. However, the nature of the hypothesis, and how the tobs is computed will change from our one-sample case.

 

All of the tests that we've looked at are examining differences. In the previous lab we were interested in comparing a known population with a treatment sample. The t-test for this lab considers differences between scores from a related pair of subjects. Because the two scores for each pair are related, the differences are based on differences between each individual or matched pair.

 

Consider the following example:

The results of the two ratings are presented below. D stands for the difference between the pre- and post-ratings for each individual.


Note:
 = the mean of the differences

 

Student

Pre-test
(first day)

Post-test
(end of semester)

D

D-

(D-)2

1

1

4

3

2

4

2

3

5

2

1

1

3

4

6

2

1

1

4

7

8

1

0

0

5

2

3

1

0

0

6

2

2

0

-1

1

7

4

6

2

1

1

8

3

4

1

0

0

9

6

6

0

-1

1

10

8

6

-2

-3

9

S

40

50

10

0

18

Differences

SSD

Mean difference for the sample = = 10/10 = 1.0

Okay, now let's start our 5 step process of hypothesis testing.

Step 1: State your H0 and H1

Before we can state hypotheses, we need to know if this will be a one-tailed or a two-tailed test. All we are asking in this example is if taking statistics has an impact (any impact - either direction) on the students' feelings about statistics. Since no direction of impact is predicted, this will be a two-tailed test.

 

Now we're ready to state the hypotheses and set our decision criterion. For this example let's assume that a = 0.05. What is our H0? Conceptually it is similar to the one-sample t-test, because we've got a single population of differences to consider that will be represented by a single sample of differences. In other words, the distribution that we're interested in is the distribution of D, the distribution of the pre-test scores subtracted from the post-test scores. So our H0 will be a statement of population comparisons such that taking stats has no effect on a person's preference for statistics. If taking statistics has no effect, we would expect no difference between the pre-test scores and post-test scores, giving us a mean difference in the population of 0. So, we state:

The H1 will state the opposite case, that taking statistics does have an impact. Therefore, our alternative hypothesis is:

    H1: mD ¹ 0

Step 2: Set up the decision criterion

The t Distribution Table

 

Two-tailed test

a = 0.05

With only one sample, our df = n - 1. So df = 10 -1 = 9. Finding tcrit is the same as usual, look at the table. a = 0.05, two-tailed, df = 9, tcrit = \ 2.262

Step 3: Collect the sample.

Our sample is given above in the table with sample mean = 1.0.

Step 4: Compute the tobs for your sample.

Okay, as was the case in last lab, the overall form of the t statistic equation is the same, but the details are a little different. For related samples we'll use:

 

 

 

So we already computed our , and we know mD = 0 (for the H0), so we just need to figure out what   is equal to. This is the estimated standard error of the difference distribution. So first we need to figure out the variance.

 

SSD = S (D - Dbar)2 = [(3-1)2 + (2-1)2 + (2-1)2 + (1-1)2 + (1-1)2 + (0-1)2 + (2-1)2 + (1-1)2 + (0-1)2 + (-2-1)2] = 18

    standard deviation of the differences   

     

     

Now we can figure out the estimated standard error

Now we are read to compute our tobs

     

        =(1-0)/0.45

        = 2.24

Step 5: Compare tobs with tcrit to make a decision about our H0.

  1. A major university would like to improve its tarnished image following a large on-campus scandal. Its marketing department develops a short television commercial and tests it on a sample of n = 7 subjects. People¨s attitudes about the company are measured with a short questionnaire, both before and after viewing the commercial. Was there a difference? Assume a = 0.05 level. Which test should be used? Follow five steps of hypothesis testing to solve the problem. Draw the distribution and indicate the rejection region(s). The data are as follows:

 

Person

X1 (before)

X2 (after)

A

15

15

B

11

13

C

10

18

D

11

12

E

14

16

F

10

10

G

11

19

 

2. An instructor asks his statistics class, on the first day of classes, to rate how much they like statistics, on a scale of 1 to 10 (1 = hate it; 10 = love it). Then at the end of the semester, the instructor asks the same students, the same question. The instructor wants to know if taking the stats course makes any difference on the students¨ feelings about statistics. Assume alpha = 0.05 level. Which test should be used? Follow five steps of hypothesis testing to solve the problem.

 

The data are as follows: n = 30, D-bar = 2, SD = 0.71.

3. An experimenter was interested in dieting and weight losses among men and women. It was believed that in the first 2 weeks of a standard dieting program, women would tend to lost more weight than men. As a check on this notion, random sample of 15 brother-sister pairs were put on the same strenuous diet. Their weight losses after 2 weeks showed the following. Follow five steps of hypothesis testing to solve the problem. Drew the distrition and indicate the rejection region(s). Assume a = 0.01 level.

           
Pair Brother Sister
1 5.0 2.7
2 3.3 4.4
3 4.3 3.5
4 6.1 3.7
5 2.5 5.6
6 1.9 5.1
7 3.2 3.8
8 4.1 3.5
9 4.5 5.6
10 2.7 4.2

 


 

Using SPSS to compute a related samples (paired samples) t-test

 

We can use SPSS to compute paired samples t-tests.

To set up a paired samples t-test you will need two columns of data, one for each sample (related samples) or one for each measurement (repeated measures).


4. Enter the data of Q1 into SPSS. Test your H0 using a paired-samples t-test. Do you get the same result? Explain your SPSS result.  Explain your SPSS result, e.g., what is the standard error? What is the tobs? What is the p-value. Do you reject or fail to reject the null hypothesis? Why? Attach the output with your worksheet.

5. For a study concerned with the reading interests of women and their husbands, a sample of 18 college-educated married couples between the ages of 30 and 40 years was taken. Each individual in the sample was interviewed and asked how many books he or she had finished reading in the year just past. The results were as follows:

Couple Wife Husband
1 1.4 1.1
2 7 2.2
3 8 1.5
4 6.6 8.1
5 4.3 2
6 5.1 3.2
7 3.2 5
8 4 4
9 5.2 7
10 2 0
11 4 1.1
12 6 3
13 8 12
14 5 3
15 8 2
16 6.1 9
17 4 2
18 5.2 6

Are wives and husbands significantly (alpha = .05) different in the average number of books read per year? Write the null hypothesis and the alternative hypothesis. Is this one-tailed or two-tailed test? Using SPSS to run the test, and report the results. What conclusions do you have?