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Module 7: Independent Samples t Procedures
Objectives: In this module you will learn an important statistical technique that will allow you to
compare two populations with respect to their means, that is, learn about 1  2 . The objective is to
help you understand the ideas behind confidence intervals, tests of significance, and the statistical
language involved in the comparison of two population means.
Overview: The two independent samples t procedures (sometimes called Student's t procedures) are
used when you want to compare the means of two populations that are not related or matched in any
way. For example you might want to test if the G.P.A. of English majors is different than that of Math
majors.
The idea is to use the two sample means to estimate the corresponding population means. We may
want to construct a confidence interval to estimate the difference between the two population means.
Or we may wish to test if the difference in the two population means is equal to a value specified in a
null hypothesis, the hypothesized value generally being 0. There are 2 versions of independent samples
t procedures: pooled and unpooled (also known as general). For a pooled independent t procedure, we
assume equal population variances for the two populations. Details on determining which version to
use are below.
We will assume that we have a random sample from each of the two populations and that the variable
being measured has a normal model for each population, with possible different means. In addition, the
two random samples are assumed to be independent of each other. Whether or not we can assume the
two populations have equal variances will determine which of the two possible independent samples t
procedures we use.
There are several ways to check the equal population variances assumption required in order to perform
the pooled t test instead of the general version. You can construct side-by-side boxplots and see if the
IQRs are comparable or compare the two sample standard deviations. A third method is to perform a
Levene’s test for homogeneity of variances. Levene’s test is, in fact, a hypothesis test, but it is a bit
different from hypothesis tests you have seen previously since it is a test about variances rather than
means. The null hypothesis is that the populations of interest do have equal variances and the
alternative is that they do not have equal variances. So if the null hypothesis from the Levene’s test is
NOT rejected, it would be reasonable to perform the pooled t-test for two population means. We will
use an alpha of 10% for Levene’s tests. If the null hypothesis from Levene’s test IS rejected, then the
assumption of equal population variances is unreasonable, and the unpooled (general) version should
be used instead of the pooled. Be careful not to confuse the Levene’s test p-value with the p-value from
the two independent samples t-test, since your output will contain both.
Note: A summary of inference procedures for population means (one sample, paired, and independent
samples) and the corresponding assumption checks can be found in Supplement 7. Supplement 6 is also
quite helpful, giving examples of common interpretations.
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Formula Card:
Activity:
Do Men and Women differ in their SSHA scores?
Background: A total of 38 college freshmen were administered the Survey of Study Habits and
Attitudes (SSHA), a psychological test designed to measure motivation and attitude towards study habits
in college students, at a private college. The 38 students were a simple random sample, of which 18
were female and 20 were male. It is known that scores on the SSHA may explain success in college.
Scores on the test range from a low of 0 to a high of 200.
School administrators are interested in whether or not there is a difference between the mean scores
for males and females. The scores for females (Group 1) and males (Group 2) are listed in the SSHA.sav
data set (Source: Moore and McCabe (1999), pg 563).
Task: Perform a two-independent samples t-test to assess if there is a difference between the mean
score for women on the test and the mean score for men on the test.
Recall: Write out the Five Steps for conducting a test of hypotheses (Reference page 51).
1.
2.
3.
4.
5.
69
Before conducting any test, here are a set of questions to ask yourself:
 How many populations are there?
One
Two
More than two
 How many variables are there?
One
Two
 What is the response variable?
 What type of variable is the response?
Categorical
Quantitative
 What type of parameter would be useful for summarizing this response?
Proportion
Mean
Other (see Supplement 3)
Based on the answers to these questions, you should be able to identify the appropriate inference
procedure. You may refer back to Supplement 3 – Name that Scenario for assistance.
The appropriate inference procedure for this scenario is ______________________________
and the specific parameter of interest is ___________________ .
Note: Why is this not a paired t procedure?
1. State the hypotheses: H0: __________ = __________ Ha: _______________________
where _____ represents
Your parameter definition should always be a statement about the population(s) under study.
2. Assumption Checks and Computing the Test Statistic:
Assumptions:
NOTE: There are TWO versions of this test – pooled and unpooled (general). Both versions are on
your formula card. The pooled test has an additional assumption that the unpooled test does not
have; the pooled test also assumes that the POPULATIONS have equal variances.
a. For this scenario, we need to assume that the two samples are ________________
from each other.
b. We need to assume that each sample is a ___________ sample.
To check this assumption, we would make a __________ plot (if there was time order) for
each sample and look for ________________________________.
c. Each sample needs to come from a normally distributed _________________ .
To check this assumption, we would make a _______ plot for each __________.
d. Finally, for the pooled test, we also need to assume that both populations have
equal ____________________.
70
Some guidelines:
Determining General vs. Pooled (i.e. checking the assumption of equal population variances)
There are three ways to check this assumption depending on the information provided.
o Examine the sample standard deviations. If they are similar, then the assumption is valid.
(This is because variance is standard deviation squared). ‘Similar’ here roughly means one
should not be more than two times the other.
o
Examine side-by-side boxplots of the sample data. If the IQRs are similar, then the assumption is
valid (i.e. the lengths or sizes of the BOXES should be similar; but the two boxes do not need to
line up right next to each other).
o
Use Levene’s test. Levene’s test is a test of assumptions done BEFORE the hypothesis test. SPSS
will provide this output. The Levene’s test null hypothesis is that the populations have equal
variances. Hence, this is a hypothesis test where you WANT to keep the
null hypothesis if you want to do a POOLED test. The Levene’s test statistic is an F,
and the p-value can be found under Sig. in the Levene’s section of the output.
Using Levene’s test:
o If the Levene’s test p-value is greater than 0.10 (or the specified significance level), the
assumption of equal population variances appears to hold, and you can use a POOLED test.
o
If the Levene’s test p-value is less than or equal to 0.10 (or the specified significance level), the
assumption of equal population variances does not appear to be met, and you would use the
GENERAL test.
e. Do the Assumptions Appear Valid?
Comment on each assumption below, using graphs and output when appropriate.
Are the two samples independent?
Are the samples random samples?
Note there is no time order for this data. If there was time order, since you need EACH sample
to be a random sample, how many time plots would you need to make to check this
assumption? ________ time plot(s)
71
Does it appear that the assumption that each sample comes from a normally distributed
population is met? Why?
Note: The equal population variances assumption will be considered after the t-test output is
generated. The t-test output will include results from Levene’s test and allow us to decide
between the pooled versus general two independent samples t procedure.
72
Test-statistic:
e. Generate the t-test output.
Use Analyze>Compare Means>Independent-Samples T-Test.
f.
The test value is _____ (this is the null value from the null hypothesis).
g. The output provides both the pooled (equal variances assumed) and unpooled (equal variances
not assumed) versions of the test. Use the output and the provided side-by-side boxplots to
determine which version of the test can be used.
i. The standard deviation for females is ______
and the standard deviation for males is _______.
The standard deviations are: similar not similar.
ii. Side-by-side boxplots show that the IQRS are:
similar not similar.
iii. Write the null and alternate hypothesis for
Levene’s test using appropriate notation
H0:__________________________
Ha:__________________________
The Levene’s test statistic is ____________ and the Levene’s p-value is __________.
Therefore, we can cannot reject the null hypothesis that the population variances are equal.
o
Hence, for the above reasons, we can say the assumption of equal population variances
is is not valid.
o
What is the symbol of the estimate of the common population standard deviation AND its
value?
Symbol:
Estimate:
h. Based on which version (pooled or general) of the test you decided to conduct in part g, what is
the value of the test statistic? Note that there are two lines of output in your table. The first line
corresponds to using a pooled procedure (equal variances assumed), whereas the second line
corresponds to what you report if you were using the unpooled (equal variances not assumed).
i.
What is the distribution of the test statistic if the null hypothesis is true?
This is the same as asking what model you use to find the p-value.
73
3. Calculate the p-value:
a. What is the SPSS reported p-value? _____________. Is it the p-value we want? _____
b. Draw a picture of the p-value we want.
c. So, our p-value is _____________________
d. Provide an interpretation of the p-value (see supplement 6).
4. Decision:
What is your decision at a 5% significance level? Reject H0 Fail to reject H0
Remember:
Reject H0

Fail to reject H0 
Results statistically significant
Results not statistically significant
5. Conclusion: What is your conclusion in context of the problem?
Conclusions should not be too strong -- say you have sufficient evidence or equivalent, do NOT say
we have proven.
Conclusions should always include a reference to the population parameter of interest.
6. Confidence Intervals (CI):
a. What is the 95% confidence interval given in the output for the difference in the two population
mean scores?
b. Based on the confidence interval, would you reject the null hypothesis at the 5% significance
level?
Circle one: Yes No
Explain.
Did your conclusions on 4 and 6b match?
74
c. Interpret the 95% confidence interval.
d. Provide an interpretation of what the 95% confidence level means.
Check Your Understanding:
1. T F
The test statistic of 2.032 in the activity implies that the sample means differed by 2.032
pooled standard errors.
2. T F
The additional assumption required by the pooled test is that the samples have equal
variances.
3. Consider the following three sets of boxplots of scores between two age groups. Circle the sets that
indicate a pooled test is appropriate.
Set 1
Set 2
Set 3
For which set are you most likely to reject the null hypothesis that the population mean scores are
equal?
Set 1
Set 2
75
Set 3
Check Your Understanding:
Paired or Independent Samples? Consider the following studies in which two sets of quantitative
measurements were obtained. Determine whether the two samples are independent or paired.
a. The effectiveness of a new headache medicine is tested by measuring the intensity of a headache in
patients before and after drug treatment. The data consist of before and after intensities for each
patient.
independent samples
paired samples
b. The effectiveness of a new headache medicine is tested by measuring the amount of time before
the headache is cured for patients who use the new medicine and for another group of patients who
use a placebo drug.
independent samples
paired samples
c. The accuracy of verbal responses is tested in an experiment in which individuals are first asked to
report their height and then their actual height is measured. The data consist of the reported height
and the measured height for each individual.
independent samples
paired samples
76
Example Exam Question on Two-Sample t-Test
In an animal-learning experiment, a researcher wanted to assess if a particular drug speeds learning.
One group of 5 rats (Group 1 = control group) is required to learn to run a maze without use of the drug,
whereas a second independent group of 8 rats (Group 2 = experimental group) is administered the drug.
The running times (time to complete the maze) for the rats in each group were entered into SPSS.
Group Statistics
Time to complete maze
Group
Control
Experimental
N
5
8
Mean
46.80
38.38
Std.
Deviation
3.42
4.78
Std. Error
Mean
1.53
1.69
Independent Samples Test
Levene's Tes t for
Equality of Variances
Run Time
Equal variances as sumed
Equal variances not ass umed
F
1.09
Sig.
.32
t-tes t for Equality of Means
t
3.41
3.70
df
11
10.653
Sig.
(2-tailed)
.006
.004
Mean
Difference
8.42
8.42
Std. Error
Difference
2.47
2.28
Conduct the two independent samples t-test to address the theory of the researcher (state the null and
alternate hypotheses, report the test statistic, p-value, and state your decision and conclusion at the 5%
level of significance).
H0: _______________________
Ha: _______________________
Based on the above output, which results should you report? pooled or un-pooled? Explain.
Test statistic: ______________
Decision: (circle one)
Fail to reject H0
p-value: ____________________
Reject H0
Thus …
Would the corresponding 95% CI for the difference in population means contain 0?
Is it appropriate to use a 95% CI to perform the hypothesis test above? Why or why not?
77