Data Analysis with Microsoft Excel: Updated for Office 2007

256 Fundamentals of Statistics

Comparing the Pooled and Unpooled Test Statistics

There are important differences between the two test statistics. The unpooled

statistic, although we refer to it as a t test, does not strictly follow a t distri-

bution. However, we can closely approximate the correct p values for this

statistic by assuming it does and then compare the test statistic to a t distri-

bution with degrees of freedom equal to

df 5

a

S^21

n 1

1

S^22

n 2

b

2

£

a

S^21

n 1

b

2

n 121

1

a

S^22

n 2

b

2

n 221

§

Here s 1 and s 2 are the standard deviations of the values in the fi rst and

second samples. The degrees of freedom for this statistic generally result

in a fractional value. In actual practice, you’ll probably never have to make

this calculation yourself; your statistics package will do it for you.

For the pooled statistic, the situation is much easier. The pooled t statis-

tic does follow a t distribution with degrees of freedom equal to

df 5 n 11 n 222

If the standard deviations are different and you apply the pooled t statis-

tic to the data, you run the risk of reporting an erroneous p value. To guard

against this problem, it may be best to perform both a pooled and an un-

pooled test and then compare the results. If they agree, report the pooled t,

because this test statistic is more widely known. Use the unpooled t if the

two tests disagree. You should also examine the standard deviations of the

two samples and determine whether they’re close in value.

Working with the Two-Sample t Statistic

To see how the two-sample t test works, let’s consider two groups of students:

One group has learned to write using a standard teaching approach, and the

other has learned using a new teaching method. There are 25 students in

each group. At the end of the session, each student writes an essay that is

graded on a 100-point scale. The average grade for the group 1 students is

75 with a standard deviation of 8. The average for the group 2 students is 80

with a standard deviation of 6. Could the difference in sample averages be

attributed to differences between the teaching methods? We assume that the

distribution of the data in both groups is normal. Our hypotheses are

H 0 : m 12 m 250

Ha: m 12 m 220