chapter 9 two-sample tests part ii: introduction to hypothesis testing renee r. ha, ph.d. james c....
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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