We begin our exploration of bivariate analysis with the two-sample t-test. In chapter 20, we saw how to use the one-sample t-test to evaluate the probability that the mean of a sample reflects the mean of the population from which the sample was drawn. The two- sample f-test evaluates whether the means of two independent groups differ on some variable. Table 21.1 shows some data that Penn Handwerker collected in 1978 from American and Liberian college students on how many children those students wanted.

Table 21.1 Number of Children Wanted (CW) by College Students in the United States and in Liberia

#

CW

Region

Sex

#

CW

Region

Sex

#

CW

Region

Sex

1

1

USA

M

29

0

USA

F

57

12

WA

M

2

3

USA

M

30

0

USA

F

58

12

WA

M

3

3

USA

M

31

0

USA

F

59

8

WA

M

4

3

USA

M

32

0

USA

F

60

6

WA

M

5

3

USA

M

33

1

USA

F

61

4

WA

M

6

3

USA

M

34

1

USA

F

62

4

WA

M

7

2

USA

M

35

2

USA

F

63

2

WA

M

8

2

USA

M

36

2

USA

F

64

3

WA

M

9

2

USA

M

37

2

USA

F

65

3

WA

M

10

2

USA

M

38

2

USA

F

66

4

WA

M

11

2

USA

M

39

2

USA

F

67

4

WA

M

12

2

USA

M

40

2

USA

F

68

4

WA

M

13

1

USA

M

41

2

USA

F

69

4

WA

M

14

6

USA

M

42

2

USA

F

70

3

WA

F

15

1

USA

M

43

2

USA

F

71

3

WA

F

16

1

USA

M

44

6

WA

M

72

3

WA

F

17

4

USA

M

45

4

WA

M

73

3

WA

F

18

0

USA

M

46

4

WA

M

74

7

WA

F

19

5

USA

F

47

4

WA

M

75

2

WA

F

20

4

USA

F

48

5

WA

M

76

4

WA

F

21

4

USA

F

49

5

WA

M

77

6

WA

F

22

4

USA

F

50

5

WA

M

78

4

WA

F

23

3

USA

F

51

5

WA

M

79

4

WA

F

24

2

USA

F

52

5

WA

M

80

4

WA

F

25

3

USA

F

53

7

WA

M

81

4

WA

F

26

0

USA

F

54

12

WA

M

82

4

WA

F

27

2

USA

F

55

6

WA

M

83

4

WA

F

28

0

USA

F

56

6

WA

M

84

4

WA

F

SOURCE: Data Analysis with MYSTAT by H. R. Bernard and W. P Handerwerker, 1995. McGraw-Hill. Repro- ducedwith permission of The McGraw-Hill Companies.

Table 21.2 shows the relevant statistics for the data in table 21.1. (I generated these with SYSTAT®, but you can use any statistics package.)

There are 43 American students and 41 Liberian students. The Americans wanted, on average, 2.047 children, 1.396 sd, SEM 0.213. The Liberians wanted, on average, 4.951 children, 2.387 sd, SEM 0.373.

Table 21.2 Descriptive Statistics for Data in Table 21.1

CW-USA

CW-Liberia

N of cases

43

41

Minimum

0

2

Maximum

6

12

Mean

2.047

4.951

95% CI Upper

2.476

5.705

95% CI Lower

1.617

4.198

Standard Error

0.213

0.373

Standard Deviation

1.396

2.387

The null hypothesis, H_{0}, is that these two means, 2.047 and 4.951, come from random samples of the same population—that there is no difference, except for sampling error, between the two means. Stated another way, these two means come from random samples of two populations with identical averages. The research hypothesis, H_{1}, is that these two means, 2.047 and 4.951, come from random samples of truly different populations.

The formula for calculating t for two independent samples is:

That is, t is the difference between the means of the samples, divided by the fraction of the standard deviation, ^, of the total population, that comes from each of the two separate populations from which the samples were drawn. (Remember, we use Roman letters, like s, for sample statistics, and Greek letters, like ^, for parameters.) Since the standard deviation is the square root of the variance, we need to know the variance, ^^{2}, of the parent population.

The parent population is the general population from which the two samples were pulled. Our best guess at ^^{2} is to pool the variances from the two samples:

which is very messy, but just a lot of arithmetic. For the data on the two groups of students, the pooled variance is:

Now we can solve for t:

Testing the Value of t

We can evaluate the statistical significance of t using appendix B. Recall from chapter 20 that we need to calculate the degrees of freedom and decide whether we want a one- or a two-tailed test to find the critical region for rejecting the null hypothesis. For a two- sample t-test, the degrees of freedom equals:

so there are 43 + 41 — 2 = 82 degrees of freedom in this particular problem (box 21.1).

We’ll use a two-tailed test for the problem here because we are only interested in the magnitude of the difference between the means, not its direction or sign (plus or minus). We are only interested here, then, in the absolute value of t, 6.844.

Looking at the values in appendix B, we see that any t-value above 3.291 is significant for a two-tailed test at the .001 level. Assuming that our samples represent the populations of American students and Liberian students, we’d expect the observed difference in the means of how many children they want to occur by chance less than once every thousand times we run this survey.

BOX 21.1

ONE- AND TWO-TAILED TESTS

To recap from chapter 20, if you test the possibility that one mean will be higher than another, then you need a one-tailed test. After all, you're only asking whether the mean is likely to fall in one tail of the t-distribution (see figure 6.7). With a one-tailed test, a finding of no difference (the null hypothesis) is equivalent to finding that you predicted the wrong mean to be the one that was higher. If you want to test only whether the two means are different (and not that one will be higher than the other), then you need a two-tailed test. In appendix B, as in appendix A, scores significant at the .10 level for a two-tailed test are significant at the .05 level for a one-tailed test; scores significant at the .05 level for a two-tailed test are significant at the .025 level for a one-tailed test, and so on.