STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
COMPARING TWO POPULATION MEANS: INDEPENDENT SAMPLES
In Chapter 6, we considered inference for a single population mean. Then, in section 7.1
we extended these ideas so that comparisons could be made between two groups. Such
comparisons were made using differences because the observations in the two groups
were related, or dependent.
In this section, we will consider making comparisons between two independent groups.
The methodologies considered here are a bit more complicated because it no longer
makes sense to simply consider differences. Consider the following example.
Example 7.3: In NC Birth Weight study one potential question of interest is the
relationship between smoking and birth weights of babies. The data can be found in
the file NCBirth3.JMP, a portion of which is shown below.
Comment: The first observation for a nonsmoker is in NO WAY related to the first
observation for a smoker. Therefore, these groups are independent.
Next, let’s examine birth weight across the two groups: smokers and nonsmokers. In
JMP, select Analyze > Fit Y by X. Place the categorical variable (Smoker) in the X,
Factor box, and place the numerical variable of interest (Birth Weight (g)) in the Y,
Response box.
162
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Questions:
1. Identify the following quantities from the JMP output on the previous page:
Population Parameters
Sample Statistics
Group 1
Group 2
(Smokers) (Nonsmokers)
Group 1
(Smokers)
Group 2
(Nonsmokers)
Mean
μ1
μ2
x1 
x2 
Standard
Deviation
σ1
σ2
s1 =
s2 =
Variance
𝜎12
𝜎22
𝑠12 =
𝑠22 =
n1 =
n2 =
# of Observations
163
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
2. We are interested in the true population difference between these two means,
μ 1  μ 2 . Based on the data, what is your best guess for this difference, and what
does it indicate about birth weights for smokers compared to nonsmokers?
3. If you were to take another sample, is your “best guess” from the previous
question likely to change? Explain.
Since the difference will change from sample to sample, it is considered a random
variable and therefore has a sampling distribution. In order to make valid inferences
about the true population difference, we must use the sampling distribution so that we
can understand how the difference in sample means is expected to change from sample
to sample.
Characteristics of the Sampling Distribution for the Difference in Means
As long as the samples are INDEPENDENT, the sampling distribution for the difference
in means can be described as follows.
1. The sampling distribution is centered around μ1 – μ2.
2. The standard deviation of our sampling distribution (i.e., standard error)
depends on whether the variances of the two groups are equal or unequal.
If the variances are unequal,
If the variances are equal, we pool
we estimate each of them
the variances and estimate them
individually from our sample
with a common value
2
data.
(called s pooled
).
Standard
error
s 12 s 22

n1 n 2
s 2pooled
n1

s 2pooled
n2
164
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
2
Note that the pooled variance, s pooled
, is calculated as follows:
s 2pooled
(n 1  1)s12  (n 2  1)s 22
.

(n 1  1)  (n 2  1)
3. The shape of the sampling distribution is approximately normal if both sample
sizes are “sufficiently large” OR if both original populations are normally
distributed.
Given these characteristics, the test statistic we will use when testing for differences
in two population means for INDEPENDENT samples is as follows:
Assuming Unequal Variances Assuming Equal (Pooled) Variances
t
(x 1  x 2 )  (μ 1  μ 2 )
s 12 s 22

n1 n 2
t
(x 1  x 2 )  (μ 1  μ 2 )
s 2pooled
n1

s 2pooled
n2
These test statistics follow a t-distribution, and the degrees of freedom for each case
are given in the outline of the formal hypothesis test procedure on the next page.
165
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
The Procedure for Testing for a Difference in Two Population Means
(Assuming Independent Samples)
Step 0:
Check the assumptions behind the test to be sure that the test is valid (and that the
right test is used).
1. Are the two groups independent?
2. Are both sample sizes sufficiently large? If not, is it reasonable to assume
that both populations are normally distributed?
3. Check whether you can assume the variances are equal. Here is one
widely used rule of thumb: If one group’s variance is more than twice as
big as the other group’s variance, then assume the variances are unequal.
Step 1:
Convert the research question into H0 and Ha.
1. Upper-Tailed Test:
H0:
Ha:
2. Lower-Tailed Test:
H0:
Ha:
3. Two-Tailed Test:
H0:
Ha:
Step 2:
Determine α, the significance level.
166
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Step 3:
Calculate a test statistic from your data.
If you assume the variances are equal:
t
(x 1  x 2 )  (μ 1  μ 2 )
s 2pooled
n1

s 2pooled
n2
Assuming the two variances are equal, this test statistic comes from a tdistribution with df = (n1 – 1) + (n2 – 1).
If there is evidence of unequal variances:
t
(x 1  x 2 )  (μ 1  μ 2 )
s 12 s 22

n1 n 2
If we assume the variances are unequal, we must use Satterthwaite’s
approximation to find the degrees of freedom for this test statistic:
df 
 s 12 s 22 



 n1 n 2 


2
  2 2  2 2
  s1 
 s2 
n 
n 
 1    2 
 n1  1
n2 1











167
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Step 4:
Determine the p-value and make a decision regarding H0.
Lower-Tailed Test
(Ha contains <)
Step 5:
Upper-Tailed Test
(Ha contains >)
Two-Tailed Test
(Ha contains ≠)
Write a conclusion in terms of the original research question.
Back to Example 7.3: Carry out the hypothesis test associated with the following
research question: Is the average birth weight lower when mothers are classified as
smokers?
Step 0:
Check the assumptions behind the test to be sure that the test is valid.
1. Are the two groups independent? Explain.
2. Are both sample sizes sufficiently large? If not, is it reasonable to assume that
both populations are normally distributed?
168
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
3. Check whether you can assume the variances are equal. Remember the rule of
thumb: If one group’s sample varaince is more than twice as big as the other group’s
variance, then assume the variances are unequal.
Step 1:
Convert the research question into H0 and Ha.
H0:
Ha:
Step 2:
Determine α, the error rate.
169
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Step 3:
Calculate a test statistic from your data.
We can use the formulae to verify the value of the test statistic from JMP:
s 2pooled 
t
(n 1  1)s12  (n 2  1)s 22
=
(n 1  1)  (n 2  1)
(x 1  x 2 )  (μ 1  μ 2 )
s 2pooled
n1

s 2pooled
=
n2
Recall that this test statistic comes from a t-distribution with
df = (n1 – 1) + (n2 – 1) =
170
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Step 4:
Determine the p-value and make a decision regarding H0.
p-value:
Decision:
Step 5:
Write a conclusion in terms of the original research question.
“We have evidence that the average birth weight is lower when mothers are classified
as smokers (p-value < .0001).”
Constructing a Confidence Interval for the Difference in Means
Lower Endpoint =
Upper Endpoint =
Interpret the meaning of this 95% confidence interval:
171
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Example 7.4: Consider the WSU Student Survey.JMP data and the issue of the
relationship between skipping at least one class per week and cumulative GPA.
Your research question is as follows: Is the average cumulative GPA of students
who skip at least one class per week lower than the cumulative GPA of students who
don’t?
The data are summarized by selecting Analyze > Fit Y by X and placing Skip Class
in the X, Factor box and GPA in the Y, Response box. Then, use the Display
Options menu to add different graphical enhancements.
It is clear that the SAMPLE of students who do not regularly skip classes have a higher
mean GPA than that for the SAMPLE of students who do. However, to determine
whether if the skippers have a smaller POPULATION mean we will carry out a
hypothesis test.
Step 0:
Check the assumptions behind the test to be sure that the test is valid.
1. Are the two groups independent?
2. Are both sample sizes sufficiently large? If not, is it reasonable to assume
that both populations are normally distributed?
3. Check whether you can assume the variances are equal. Your text suggests
using the following rule of thumb: If one group’s variance is more than twice
as big as the other group’s variance, then assume the variances are unequal.
172
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Step 1:
Convert the research question into H0 and Ha.
H0:
Ha:
Step 2:
Determine α, the level of significance.
Step 3:
Calculate a test statistic from your data.
We could verify the value of the test statistic from JMP. The order of the
calculations is shown below. I will make you do this once on your next
assignment.
(n  1)s12  (n 2  1)s 22
s 2pooled  1
=
(n 1  1)  (n 2  1)
t
(x 1  x 2 )  (μ 1  μ 2 )
s 2pooled
n1

s 2pooled
=
n2
Recall that this test statistic comes from a t-distribution with
df = (n1 – 1) + (n2 – 1) =
173
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Step 4:
Determine the p-value and make a decision regarding H0.
p-value:
Decision:
Step 5:
Write a conclusion in terms of the original research question.
“We have evidence that the average GPA of students who skip at least one class
per week is lower than the average GPA of student who do not skip class
(p-value < .0001).”
Finally, construct a 95% confidence interval for the difference in means:
Lower Endpoint =
Upper Endpoint =
174
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Interpret the meaning of this confidence interval:
Example 7.5: Now suppose you wish to compare the mean sleep time of female and
male WSU students. The research question is: Is there a significant difference in the
average amount of sleep per day of female and male WSU students?
There does not appear to be a large difference in the SAMPLE average sleep times based
on gender. However, to determine whether the POPULATION mean sleep times differ
between the genders, we will carry out a hypothesis test.
Step 0:
Check the assumptions behind the test to be sure that the test is valid.
1. Are the two groups independent?
2. Are both sample sizes sufficiently large? If not, is it reasonable to assume
that both populations are normally distributed?
175
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
3. Check for unequal variances. If one group’s variance is more than twice as big
as the other group’s standard deviation, then assume the variances are unequal.
Step 1:
Convert the research question into H0 and Ha.
H0:
Ha:
Step 2:
Determine α, the level of significance.
176
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Step 3:
Calculate a test statistic from your data.
We could verify the value of the test statistic from JMP, but I will skip it this time.
Feel free to fill in the calculations on your own.
s 2pooled 
t
(n 1  1)s12  (n 2  1)s 22
=
(n 1  1)  (n 2  1)
(x 1  x 2 )  (μ 1  μ 2 )
s 2pooled
n1

s 2pooled
=
n2
Recall that this test statistic comes from a t-distribution with
df = (n1 – 1) + (n2 – 1) =
Step 4:
Determine the p-value and make a decision regarding H0.
p-value:
177
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Step 5:
Write a conclusion in terms of the original research question.
Finally, we can construct a confidence interval for the difference in means:
Lower Endpoint =
Upper Endpoint =
Interpret the meaning of this 95% confidence interval:
178
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Wilcoxon Rank Sum Test
This test is an alternative to the two-sample t-test for comparing the “average” value of
two populations where the samples from each population are taken independently. This
test is appropriate when the assumption of normality is violated (as it may have been in
the previous example due to the discrete nature of the response sleep time).
To obtain the output for the Wilcoxon Rank Sum test, simply select Nonparametric >
Wilcoxon Test from the drop-down menu in the upper left-hand corner of the boxplots.
JMP returns the following output:
The decision based on Wilcoxon’s test is the same decision that we arrived at using the
standard t-test above. Often, these procedures will agree each other because the above ttest is robust to (i.e. not greatly affected by) departures from normality.
179
STAT 110: Section 7.2 – Comparing Two Population Means Using
Independent Samples
May 2012
Example 7.6: Cell Radii of Malignant vs. Benign Breast Tumors
Research Question: Is there evidence of a difference in the mean cell radius of benign and
malignant breast tumors?
The variance for malignant tumors is over twice that of benign tumors thus we should not
assume the variances are equal. All that means is that we conduct a non-pooled t-Test by
selecting the t-Test option instead as shown below.
The resulting output is the same.
We have extremely strong evidence to suggest that malignant breast tumors have a larger
mean cell radius than benign tumors (p < .0001). We estimate the mean cell radius for
malignant tumors is between 4.84 and 5.79 units larger than that for benign tumors.
180