1. Hypothesis Testing Using the Two-Sample t-Test
Chapter 9
Hypothesis Testing Using the Two-Sample t-Test

2. Going Forward Your goals in this chapter are to learn:
The logic of a two-sample experiment
The difference between independent samples and related samples
When and how to perform the independent-samples t-tests
When and how to perform the related-samples t-test
What effect size is and how it is measured using Cohen’s d or

3. Understanding the Two-Sample Experiment

4. Two-Sample Experiment
Participants’ scores are measured under two conditions of the independent variable
Condition 1 produces sample mean representing
Condition 2 produces sample mean representing

5. Two-Sample t-Test
The parametric statistical procedure for determining whether the results of a two-sample experiment are significant is the two-sample t-test
The two versions of the two-sample t-test are
The independent-samples t-test
The related-samples t-test

6. Relationship in the Population in a Two-sample Experiment

7. The Independent Samples t-Test

8. Independent Samples t-Test
The parametric procedure used for testing two sample means from independent samples
Independent samples result when we randomly select participants for a condition without regard to who else has been selected for either condition

9. Assumptions of the Independent Samples t-Test
The dependent scores are normally distributed interval or ratio scores.
The populations have homogeneous variance. Homogeneity of variance means the variance of the populations being represented are equal.

10. Statistical Hypotheses
For a two-tailed test, the statistical hypotheses are
H0 implies both samples represent the same population of scores
Ha implies the means from our conditions each represent a different population of scores

11. Sampling Distribution
The sampling distribution of differences between the means is the distribution of all possible differences between two means when both samples are drawn from the one raw score population that H0 says we are representing.

12. Performing the Independent Samples t-Test
Compute the mean and estimated population variance for each condition
Remember: The formula for the estimated variance in each condition is

13. Performing the Independent Samples t-Test
Compute the pooled variance using the formula

14. Performing the Independent Samples t-Test
Compute the standard error of the difference. This is the standard deviation of the sampling distribution of differences between means. The formula is

15. Performing the Independent Samples t-Test
Compute tobt for two independent samples using the formula

16. One-Tailed Tests
The statistical hypotheses for a one-tailed test of independent samples are OR
If m1 is expected to If m2 is expected to
be larger than m be larger than m1

17. One-Tailed Tests
Conduct one-tailed tests only when you can confidently predict the direction the dependent scores will change.

18. One-Tailed Tests
Decide which and corresponding m is expected to be larger
Arbitrarily decide which condition to subtract from the other
Decide whether the difference will be positive or negative
Create Ha and H0 to match this prediction
Locate the region of rejection
Complete the t-test as described previously

19. Critical Values
Critical values for the independent samples t-test (tcrit) are determined based on
degrees of freedom df = (n1 – 1) + (n2 – 1),
the selected a, and
whether a one-tailed or two-tailed test is used

20. Interpreting the Independent-Samples t-Test
In a two-tailed t-test of independent samples, reject H0 if tobt is greater than (beyond) +tcrit or if tobt is less than (beyond) –tcrit
Otherwise, fail to reject H0

21. The Related Samples t-Test

22. Related-Samples t-Test
The related-samples t-test is used when we have two sample means from two related samples
Related samples occur when we pair each score in one sample with a particular score in the other sample
Two types of research designs producing related samples are the matched-samples design and the repeated-measures design

23. Matched-Samples Design
The researcher matches each participant in one condition with a particular participant in the other condition
We do this so we have more comparable samples

24. Repeated-Measures Design
Each participant is tested under both conditions of the independent variable
That is, each participant is measured under condition 1 and again under condition 2

25. Transforming the Raw Scores
In a related samples t-test, the raw scores are transformed by finding each difference score
The difference score is the difference between the two raw scores in a pair
The symbol for a difference score is D

26. Statistical Hypotheses
The statistical hypotheses for a two-tailed related-samples t-test are

27. Sampling Distribution
The sampling distribution of mean differences shows all possible values of the population mean of the difference scores ( ) that occur when samples are drawn from the population of difference scores that H0 says we are representing.

28. Performing the Related-Samples t-Test
Compute the estimated variance of the difference scores ( ) using the formula
where N equals the number of difference scores

29. Performing the Related-Samples t-Test
Compute the standard error of the mean difference ( ) using the formula

30. Performing the Related-Samples t-Test
Find tobt using the formula

31. One-Tailed Tests
The statistical hypotheses for a one-tailed t-test of related samples are
If we expect the If we expect the
difference to be difference to be
larger than less than 0

32. Critical Values The critical value (tcrit) is determined based on
degrees of freedom df = N – 1 where N is the number of difference scores
the selected a, and
whether a one-tailed or two-tailed test is used

33. Interpreting the Related-Samples t-Test
In a two-tailed test of related samples, reject H0 if tobt is greater than (beyond) +tcrit or if tobt is less than (beyond) –tcrit
Otherwise, fail to reject H0

34. Describing Effect Size

35. Effect Size
Effect size indicates the amount of influence changing the conditions of the independent variable had on dependent scores
The larger the effect size, the more scientifically important the independent variable is

36. Computing Effect Size Cohen’s d is used to compute effect size
Independent Samples
t-Test
Related Samples
t-test

37. Interpreting Effect Size
We interpret the Cohen’s d using a small, medium, or large effect size classification
d = 0.2 is a small effect
d = 0.5 is a medium effect
d = 0.8 is a large effect

38. Proportion of Variance Accounted For
The proportion of variance accounted for is the proportion of the differences in scores that can be attributed to changing the conditions in the independent variable
We use the formula for the squared point-biserial correlation coefficient

39. Example 1
Using the following data set, conduct an independent-samples t-test. Use a = 0.05 and a two-tailed test.
Sample 1
Sample 2 14 13 15 11 10 12 17

40. Example 1

41. Example 1
The standard error of the difference is

42. Example 1

43. Example 1
tcrit for df = (9 – 1) + (9 – 1) = 16 with a = .05 and a two-tailed test is
Reject H0 if tobt is greater than or if tobt is less than –2.120.
Because tobt of – is not beyond the –tcrit of –2.120, it does not lie within the rejection region. We fail to reject H0.

44. Example 2
Using the following data set, conduct a related-samples t-test. Use a = 0.05 and a two-tailed test.
Sample 1
Sample 2 14 13 15 16 10 12 17 18 19

45. Example 2 First, we find the differences between the matched scores
Sample 1
Sample 2
Differences 14 13 15 16 -1 -2 10 12 -3 -4 17 18 19

46. Example 2

47. Example 2 Using a = 0.05 and df = 8, tcrit = 2.306.
Reject H0 if tobt is greater than or if tobt is less than –2.306.
Because tobt of –6.260 is beyond the –tcrit value of –2.306, it lies within the rejection region. We reject H0.

48. Example 2
Effect size

49. Example 2
Proportion of variance accounted for