1. Overview Two paired samples: Within-Subject Designs
-Hypothesis test
-Confidence Interval
-Effect Size
Two independent samples: Between-Subject Designs
Hypothesis test
Confidence interval
Effect Size

2. Comparing Two Populations
Until this point, all the inferential statistics we have considered involve using one sample as the basis for drawing conclusion about one population.
Although these single sample techniques are used occasionally in real research, most research studies aim to compare of two (or more) sets of data in order to make inferences about the differences between two (or more) populations.
What do we do when our research question concerns a mean difference between two sets of data?

3. Two kinds of studies
There are two general research strategies that can be used to obtain the two sets of data to be compared:
The two sets of data could come from two independent populations (e.g. women and men, or students from section A and from section B)
The two sets of data could come from related populations (e.g. “before treatment” and “after treatment”)
<- between-subjects design
<- within-subjects design

4. Part I Two paired samples: Within-Subject Designs -Hypothesis test
-Confidence Interval
-Effect Size

5. Paired T-Test for Within-Subjects Designs
Our hypotheses:
Ho: D = 0
HA: D  0
To test the null hypothesis, we’ll again compute a t statistic and look it up in the t table.
Paired Samples t
t = D - D sD = sD

6. Steps for Calculating a Test Statistic
Paired Samples T
Calculate difference scores
Calculate D
Calculate sd
Calculate T and d.f.
Use Table E.6

7. Confidence Intervals for Paired Samples
General formula
X  t (SE)
Paired Samples t
D  t (sD)

8. Effect Size for Dependent Samples
One Sample d
Paired Samples d

9. Exercise
In Everitt’s study (1994), 17 girls being treated for anorexia were weighed before and after treatment. Difference scores were calculated for each participant.
Change in Weight
n = = 7.26 sD = 7.16
Test the null hypothesis that there was no change in weight.
Compute a 95% confidence interval for the mean difference.
Calculate the effect size

10. Change in Weight
n = = 7.26 sD = 7.16
Exercise
T-test

11. Change in Weight
n = = 7.26 sD = 7.16
Exercise
Confidence Interval

12. Change in Weight
n = = 7.26 sD = 7.16
Exercise
Effect Size

13. Part II Two independent samples: Between-Subject Designs
-Hypothesis test
-Confidence Interval
-Effect Size

14. T-Test for Independent Samples
The goal of a between-subjects research study is to evaluate the mean difference between two populations (or between two treatment conditions).
We can’t compute difference scores, so …
Ho: 1 = 2
HA: 1  2

15. T-Test for Independent Samples
We can re-write these hypotheses as follows:
Ho: 1 - 2 = 0
HA: 1 - 2  0
To test the null hypothesis, we’ll again compute a t statistic and look it up in the t table.

16. T-Test for Independent Samples
General t formula
t = sample statistic - hypothesized population parameter estimated standard error
One Sample t
Independent samples t

17. T-Test for Independent Samples
Standard Error for a Difference in Means
The single-sample standard error ( sx ) measures how much error expected between X and .
The independent-samples standard error (sx1-x2) measures how much error is expected when you are using a sample mean difference (X1 – X2) to represent a population mean difference.

18. T-Test for Independent Samples
Standard Error for a Difference in Means
Each of the two sample means represents its own population mean, but in each case there is some error.
The amount of error associated with each sample mean can be measured by computing the standard errors.
To calculate the total amount of error involved in using two sample means to approximate two population means, we will find the error from each sample separately and then add the two errors together.

19. T-Test for Independent Samples
Standard Error for a Difference in Means
But…
This formula only works when n1 = n2. When the two samples are different sizes, this formula is biased.
This comes from the fact that the formula above treats the two sample variances equally. But we know that the statistics obtained from large samples are better estimates, so we need to give larger sample more weight in our estimated standard error.

20. T-Test for Independent Samples
Standard Error for a Difference in Means
We are going to change the formula slightly so that we use the pooled sample variance instead of the individual sample variances.
This pooled variance is going to be a weighted estimate of the variance derived from the two samples.

21. Steps for Calculating a Test Statistic
One-Sample T
Calculate sample mean
Calculate standard error
Calculate T and d.f.
Use Table D

22. Steps for Calculating a Test Statistic
Independent Samples T
Calculate X1-X2
Calculate pooled variance
Calculate standard error
Calculate T and d.f.
Use Table E.6
d.f. = (n1 - 1) + (n2 - 1)

23. Illustration
A developmental psychologist would like to examine the difference in verbal skills for 8-year-old boys versus 8-year-old girls. A sample of 10 boys and 10 girls is obtained, and each child is given a standardized verbal abilities test. The data for this experiment are as follows:
Girls
Boys
n1 = = 37 SS1 = 150
n2 = = 31 SS2 = 210

24. Illustration Girls Boys n1 = 10 = 37 SS1 = 150 n2 = 10 = 31 SS2 = 210
STEP 1: get mean difference

25. Illustration Girls Boys n1 = 10 = 37 SS1 = 150 n2 = 10 = 31 SS2 = 210
STEP 2: Compute Pooled Variance

26. Illustration Girls Boys n1 = 10 = 37 SS1 = 150 n2 = 10 = 31 SS2 = 210
STEP 3: Compute Standard Error

27. Illustration Girls Boys n1 = 10 = 37 SS1 = 150 n2 = 10 = 31 SS2 = 210
STEP 4: Compute T statistic and df
d.f. = (n1 - 1) + (n2 - 1) = (10-1) + (10-1) = 18

28. Illustration Girls Boys n1 = 10 = 37 SS1 = 150 n2 = 10 = 31 SS2 = 210
STEP 5: Use table E.6
T = 3 with 18 degrees of freedom
For alpha = .01, critical value of t is 2.878
Our T is more extreme, so we reject the null
There is a significant difference between boys and girls

29. T-Test for Independent Samples
Sample Data
Hypothesized Population Parameter
Sample Variance
Estimated Standard Error
t-statistic
Single sample t-statistic
Independent samples t-statistic

30. Confidence Intervals for Independent Samples
General formula
X  t (SE)
One Sample t
X  t (sx)
Independent Sample t
(X1-X2)  t (sx1-x2)

31. Effect Size for Independent Samples
One Sample d
Independent Samples d

32. Exercise
Subjects are asked to memorize 40 noun pairs. Ten subjects are given a heuristic to help them memorize the list, the remaining ten subjects serve as the control and are given no help. The ten experimental subjects have a X-bar = 21 and a SS = The ten control subjects have a X-bar = 19 and a SS = 120.
Test the hypothesis that the experimental group differs from the control group.
Give a 95% confidence interval for the difference between groups
Give the effect size

33. Exercise Experimental Control n1 = 10 = 21 SS1 = 100
T-test

34. Exercise Experimental Control n1 = 10 = 21 SS1 = 100
T-test
d.f. = (n1 - 1) + (n2 - 1) = (10-1) + (10-1) = 18

35. Exercise Experimental Control n1 = 10 = 21 SS1 = 100
Confidence Interval

36. Exercise Experimental Control n1 = 10 = 21 SS1 = 100
Effect Size

37. Summary Hypothesis Tests Confidence Intervals Effect Sizes 1 Sample
2 Paired Samples
2 Independent Samples

38. Review

39. Estimated Standard Error t-statistic
Sample Data
Hypothesized Population Parameter
Sample Variance
Estimated Standard Error
t-statistic
One sample t-statistic
Paired samples t-statistic
Independent samples t-statistic

40. Confidence Intervals One Sample t X  t (SE) Paired Samples t
D  t (sD)
Independent Sample t
(X1-X2)  t (sx1-x2)

41. Effect Sizes
One Sample d
Paired Samples d
Independent Samples d