1. Lecture 8 PY 427 Statistics 1 Fall 2006 Kin Ching Kong, Ph.D
Chicago School of Professional Psychology
Lecture 8
Kin Ching Kong, Ph.D

2. Agenda The t Test for Two Independent Samples Hypothesis Testing
The independent-measures or between-subjects design
The hypotheses for an independent-measures t test
The formula for an independent-measures t test
Pooled Variance
Estimated Standard Error
The Degree of Freedom
Hypothesis Testing
Directional Tests
Assumptions for the independent-measures t test
Measuring Effect Size

3. Two Research Designs
Most research involves comparing two or more sets of data.
Two general research designs used to obtain two or more sets of data:
Independent-measures or between-subjects design: each set of data come from a completely separate sample of individuals. In other words, a separate sample is used for each treatment condition or each level of the independent variable.
Figure 10.1 of your book
Repeated-measures or within-subjects design: two or more sets of data come from the same sample of individuals. In other words, the same sample is repeatedly measured across treatment conditions or levels of the independent variable.
So far we talked about inferential statistics using one sample to make inferences about an unknown population.

4. The hypotheses for an independent-measures t test
The independent-measures t test:
Evaluate the mean difference between two populations (or between two treatment conditions)
The Null Hypothesis:
H0: m1 – m2 = 0
(there is no difference between the population means)
The Alternative Hypothesis:
H1: m1 – m2 = 0 (there is a mean difference)

5. The formula for an independent-measures t test
The basic structure of the t Statistic:
t = sample statistic – hypothesized population parameter
estimated standard error
t = data – hypothesis
error
The single-sample t
t = M – m sM
The independent-measures t
t = (M1 – M2) – (m1 – m2)
s(M1 – M2)

6. The Standard Error
In general, standard error measure how accurately the sample statistic represents the population parameter. In other words, how much difference to expect between statistic and parameter by chance alone.
sstatistic
For the independent-measures t, the standard error measures the amount of error that is expected when you use a sample mean difference (M1 – M2) to represent a population mean difference (m1 – m2).
s(M1 – M2)
For a one sample t test, the statistic is M, so the standard error is sM

7. The Standard Error for independent-measures t
Two sources of error:
M1 approximates m1 with some error
M2 approximates m2 with some error
Standard error for each sample:
sM =
For two samples, the conceptual formula:
s(M1 – M2) =

8. Pooled Variance Problems with the conceptual formula for s(M1 – M2)
When n1 = n2, the formula is biased.
The variance (s2) from the larger sample is a more accurate estimate of s2 and therefore should be given more weight (law of large numbers).
The Pooled Variance
s2p = SS1 + SS2
df1 + df2
Ans to 2: df = 5, t =

9. Estimated Standard Error for independent-measures t
An unbiased measure of the standard error:
s(M1 – M2) =
The pooled variance is used instead of the individual sample variances.

10. Interpreting the estimated standard error in independent-measures t
The meaning of the estimated standard error:
Average discrepancy between a sample statistic (M1 – M2) and the corresponding population parameter (m1 – m2) expect by chance.
Estimated std error = average ((M1 – M2) - (m1 – m2)) expected by chance
When H0 is true, m1 – m2 = 0:
Estimated std error = average (M1 – M2) expected by chance
Then:
t = actual difference between M1 and M2
average difference (expected by chance) between M1 and M2

11. The Final Formula and df
The independent-measures t
t = (M1 – M2) – (m1 – m2)
s(M1 – M2)
s(M1 – M2) = s2p = SS1 + SS2
df1 + df2
The degree of freedom for the independent-measures t:
df = df for the first sample + df for the second sample
df = df1 + df2
= (n1 -1) + (n2 – 1)

12. Single Sample t vs Independent-Measures t
Same conceptual structure for both t formulas:
t = data – hypothesis
error
Same basic elements in both t formulas, but the independent-measures t doubles each aspect of the single-sample t:
Table 10.1 of your book

13. Hypothesis Testing (the experiment)
Research question:
Does using mental images affect memory?
Experiment:
Participants: Two groups of 10 participants (2 separate samples)
Stimuli: a list of 40 pairs of nouns (e.g. dog/bicycle, lamp/piano)
Treatment (IV):
Group 2: given the list for 5 minutes and told to memorize the noun pairs
Group 1: same list for 5 minutes and told to memorize, but also told to form a mental image for each pair of nouns (e.g. dog riding a bicycle).
Dependent Variable:
Each group is given a memory test in which they are given the first word and told to recall the second word of each pair
Number of words correctly recalled was recorded for each participant (DV)

14. Hypothesis Testing (the data)
Group 1 (images)
Group 2 (no images) 19 32 23 12 20 30 22 16 24 27 15 14 31 25 18
n1 = 10
n2 = 10
M1 = 26
M2 = 18
SS = 200
SS = 160

15. Hypothesis Testing (Steps 1 & 2)
Step 1: State the hypotheses (& choose a alpha level)
H0: m1 – m2 = 0 (no difference, imagery has no effect)
H1: m1 – m2 = 0 (there is a difference, imagery has an effect)
a = .05
Step 2: Define Critical Region
df = df1 + df2 = (n1 - 1) + (n2 – 1) =9 + 9 = 18
For a = .05, the critical region consist of the extreme 5% of the distribution beyond t = , i.e. tcritical =
Fig of your book

16. Hypothesis Testing (Step 3)
Step 3: Compute test statistic
t = (M1 – M2) – (m1 – m2)
s(M1 – M2)
s(M1 – M2) = s2p = SS1 + SS2
df1 + df2
s2p = SS1 + SS2 = = = 20
df1 + df
s(M1 – M2) = = = 2

17. Hypothesis Testing (Step 3 continue & Step 4)
Step 3: Compute test statistic
t = (M1 – M2) – (m1 – m2)
s(M1 – M2)
= (26 – 18) – 0 2
= 8/2 = 4.00
Step 4: Make a decision
Since the obtained value (t = 4.00) is in the critical region (i.e. greater than ), we reject H0 and concluded that using mental images produced a significant difference in memory performance.
Another way to think about this: the obtained difference between the group is 4 times greater than would be expected by chance (the standard error). This result is very unlikely if the null hypothesis is true.

18. Effect Size for the independent-measures t
Cohen’s d = mean difference
standard deviation
Cohen’s d = M1 – M2
d = 26 – 18 = 8/4.47 = 1.79
r2: percentage of variance explained by the treatment.
r2 = t = = 16/34 = 0.47
t 2 + df
So 47% of the variance in the data are explained by the treatment effect.

19. Assumptions Three assumptions underlying the independent-measures t:
The observations within each sample must be independent.
The two populations from which the samples are selected must be normally distributed.
The two populations from which the samples are selected must have equal variance (homogeneity of variance).
Test for the homogeneity of variance
If the largest variance is no more than 3 to 4 times the smallest variance, then assume homogeneity of variance
F-max = s2 (largest)/s2(smallest)
Use Table B.3 to find the critical value:
k = number of separate samples
df = n -1 (F-max test is used when all n’s are equal)