Chapter 9 Two-Sample Tests

Part II: Introduction to Hypothesis Testing

Renee R. Ha, Ph.D.James C. Ha, Ph.DIntegrative Statistics for the Social & Behavioral Sciences

Paired t test

Also called Correlated Groups t test

Paired t test

Use when:

You have two measures on the same subjects (“before” and “after” measures are common).

You have two separate samples but the subjects in each are individually matched so that there are similar subjects in each group (but not the same subjects in each group).

Tables 9.1 & 9.2

Table 9.1: An example of “before” and “after” pairing using the same subjects in each “paired” sample

Table 9.2: An example of a “paired” design in which the actual subjects in each sample are different but are “matched” for characteristics which they have in common (in this example, genetics)

Formulas

Working Formula: Paired t test

t obtained = Ds

D =

Ds

n

D =

)1(

D

nn

SSD

t-Test: Paired Two Sample for Means

• Results if you use Microsoft Excel to calculate the t-test

Paired Samples Test

•Results if you use SPSS to calculate the t-test

t-Test: Paired Two Sample for Means

• Results if you use Microsoft Excel to calculate the t-test

Paired Samples Test

•Results if you use SPSS to calculate the t-test

When to use Paired t test

1. You have two samples and a within-groups design.

2. The sampling distribution is normally distributed.

3. The dependent variable is on an interval or ratio scale.

Independent t test

Two completely different (independent) groups of subjects that you want to compare to determine if they are significantly different from one another: a between-groups design.

E.g. Experimental and control conditions.

Compare the means of the two conditions/groups.

Independent t test

Sampling Distribution of the Difference between sample means.

Determining variance when you have TWO estimates.

Weighted Variance

Formula for Weighted Variance

sw2 =

21

222

211

dfdf

sdfsdf

Substituting the equation for degrees of freedom and for variance:

= )1()1(

)1)(1()1)(1(

21

222111

nn

nSSnnSSn

Rearranging to simplify:

= 221

21

nn

SSSS

Formula Independent t test

Conceptual Formula for an Independent t-test:

tobtained =

2

22

1

21

X21

21 µ)XX(

nn

X

=

)11(

µ)XX(

212

X21

21

nn

X

Calculation (practical) Formula to use for an Independent t-test:

tobtained = )11(

XX

212

21

nnsw

Variations of the Independent t-test formula All purpose formulas:

tobtained = )11(

XX

212

21

nnsw

=

)11(2

XX

2121

21

21

nnnn

SSSS

To create the second variation of the formula, we simply substituted the formula

for sw2 directly into the independent t-test formula.

Formula to be used ONLY when n1 = n2:

tobtained =

)1(

XX

21

21

nn

SSSS

Independent t test Formulas

t-Test: Two-Sample Assuming Equal Variances

• Results if you use Microsoft Excel to calculate the t-test

Independent Samples Test

• Results if you use SPSS to calculate the t-test

• Results if you use Microsoft Excel to calculate the t-test

t-Test: Two-Sample Assuming Equal Variances

Independent Samples Test

• Results if you use SPSS to calculate the t-test

When to use Independent t test

1. Two samples and a between-groups design.

2. The sampling distribution is normally distributed.

3. The dependent variable is on an interval or ratio scale

4. The variances of the two groups are the same or are homogeneous.

HOV

The homogeneity of variance assumption (HOV) requires that the variances of the underlying populations are equal or, in practical terms, not significantly different from one another.

Effect Sizes and Power

If testing a sample with no known population parameters, you can estimate the effect size by using Hedges’ G:

Hedge’s G pS

XX 21

Where Sp (when sample sizes are equal) = 2

22yx ss

Where Sp (when sample sizes are unequal) = )1()1(

)1()1( 22

yx

yyxx

nn

snsn