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Stat 250 Gunderson Lecture Notes
Learning about the Difference in Population Means
Part 3: Testing about a Difference in Population Means
Chapter 13: Section 4, HT Module 5; Section 13.6
13.4 HT Module 5: Testing Hypotheses about the
Difference in Two Population Means
 We have two populations or groups from which independent samples are available,
(or one population for which two groups can be formed using a categorical variable).
 The response variable is quantitative and we are interested in testing hypotheses
about the means for the two populations.
A Typical Summary of the Responses for a Two Independent Samples Problem:
Population
Sample Size
Sample Mean
Sample Standard Deviation
1
n1
x1
s1
2
n2
x2
s2
Let 1 be the mean for the first population and  2 be the mean for the second population.
Parameter of interest: the difference in the population means 1   2 .
Sample estimate: the difference in the sample means x1  x2 .
Standard error:
s.e. x1  x 2  
s12 s 22

n1 n 2
where s1 and s2 are the two sample standard deviations
Pooled standard error: pooled s.e.x1  x 2   s p
1
1

where s p 
n1 n2
(n1  1)s12  (n2  1)s 22
n1  n2  2
Recall there are two methods for conducting inference for the difference between two
population means for independent samples – the General (Unpooled) Case and the Pooled
Case. Both require we have independent random samples from normal populations (but if the
sample sizes are large, the assumption of normality is not so crucial). Both will result in a t-test
159
statistic, but the standard error used in the denominator differ as well as the degrees of
freedom used for computing the p-value using a t-distribution.
Here is the summary for these two significance tests:
Possible null and alternative hypotheses.
1. H0:
versus Ha:
2. H0:
versus Ha:
3. H0:
versus Ha:
Test statistic = Sample statistic – Null value
Standard error
General
(Unpooled) Two-Sample t-Test
t
x1  x 2  0
x  x2
 1
s.e.( x1  x 2 )
s12 s 22

n1 n 2
where df =
min( n1  1, n 2  1)
Pooled
Pooled Two-Sample t-Test
t
x1  x 2  0

pooled s.e.( x1  x 2 )
x1  x 2
sp
1
1

n1 n 2
(n1  1) s12  (n 2  1) s 22
where s p 
n1  n 2  2
and df = n1  n 2  2
Recall the guidelines to assess which version to use: (also discussed on pages 518 - 519)
 If the sample standard deviations are similar, the assumption of common population
variance is reasonable and the pooled procedure can be used. If the sample sizes
happen to be the same, the pooled and unpooled standard errors are equal anyway.
The advantage for the pooled version is that finding the df is simpler.
 If the larger standard deviation is from the group with the larger sample size, the
pooled procedure is acceptable because it will be conservative (produce a wider
interval).
 If the smaller standard deviation is from the group with the larger sample size, the
pooled procedure can produce a misleading narrower interval.
160
Bottom-line: Pool if reasonable; but if the sample standard deviations are not similar, we have
the unpooled procedure that can be used.
Try It! Effect of Beta-blockers on pulse rate
Do beta-blockers reduce the pulse rate? In a study of heart surgery, 60 subjects were randomly
divided into two groups of 30. One group received a beta-blocker while the other group was
given a placebo. The pulse rate at a particular time during the surgery was measured. The
results are given below.
Group
1=beta-blockers
2=placebo
Sample size
30
30
Sample mean
65.2
70.3
Sample standard deviation
7.8
8.4
a. State the hypotheses to assess if beta-blockers reduce pulse rate on average.
H0: ____________________
versus Ha: _______________________
b. Which test will you perform – the pooled or unpooled test? Explain.
c. Perform the t-test. Show all steps. Are the results significant at a 5% level?
161
Try It! Does the Drug Speed Learning?
In an animal-learning experiment, a researcher wanted to assess if a particular drug speeds
learning. One group of 5 rats (Group 1 = control) is required to learn to run a maze without use
of the drug, whereas a second independent group of 8 rats (Group 2 = experimental) 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
Std.
Deviation
3.42
4.78
Mean
46.80
38.38
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 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: _______________________
Test statistic: _____________________________
p-value: _________________________________
Decision: (circle one)
Fail to reject H0
Reject H0
Thus …
162
Try It! Eat that Dark Chocolate
An Ann Arbor News article entitled: Dark Chocolate may help blood flow, reported the results
of a study in which researchers fed a small 1.6-ounce bar of dark chocolate to each of 22
volunteers daily for two weeks. Half of the subjects were randomly selected and assigned to
receive bars containing dark chocolate’s typically high levels of flavonoids, and the other half
received placebo bars with just trace amounts of flavonoids. The ability of the brachial artery to
dilate significantly improved for those in the high-flavonoid group compared to those in the
placebo group.
Let 1 represent the population average improvement in blood flow for those on the highflavonoid diet and 2 represent the population average improvement in blood flow for those on
the placebo diet. The researchers tested that the high-flavonoid group would have a higher
average improvement in blood flow.
a. State the null and alternate hypotheses
H0: ____________________
versus Ha: _______________________
b. The researchers conducted a pooled two sample t-test. The two assumptions about the
data are that the two samples are independent random samples.
i. Clearly state one of the remaining two assumptions regarding the populations that are
required for this test to be valid.
ii. Explain how you would use the data to assess if the above assumption in part (i) is
reasonable. (Be specific.)
c. A significance level of 0.05 was used. Based on the statements reported above, what can
you say about the p-value? Clearly circle your answer:
p-value > 0.05
p-value ≤ 0.05
can’t tell
d. The researchers also found that concentrations of the cocoa flavonoid epicatechin soared in
blood samples taken from the group that received the high-flavonoid chocolate, rising from
a baseline of25.6 nmol/L to 204.4 nmol/L. In the group that received the low-flavonoid
chocolate, concentrations of epicatechin decreased slightly, from a baseline of 17.9 nmol/L
to 17.5 nmol/L. The average improvement for the high-flavonoid group of 204.4 – 25.6 =
178.8 nmol/L is a … (circle all correct answers):
parameter statistic
sample mean
population mean
163
sampling distribution
13.6 Choosing an Appropriate Inference Procedure
Now that we have covered the inference techniques for all five of the big five parameters,
review this section (pages 523-524) which provides some guidance on how to identify the
appropriate procedure based on the research question(s) of interest.
Questions to Ask:
 Is the response variable measured quantitative or categorical?
Categorical  Proportions, percentages
p: One population proportion
p1- p2: Difference between two population proportions
Quantitative  Means
 One population mean
d : Paired difference population mean
1 - 2 : Difference between two population means

How many samples?
If two sets of measurements – are they paired or independent?

What is the main purpose?
To estimate a numerical value of a parameter?  confidence interval
To make a ‘maybe not’ or ‘maybe yes’ type of conclusion
about a specific hypothesized value?  hypothesis test
Table 13.3 page 524
Variable Type
(Parameter Type)
One Sample
(No Pairing)
Paired Data
Two Independent
Samples
Categorical
(proportions)
p
None (at this time)
p1- p2
Quantitative
(means)

d
1 - 2



From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012.
Used with permission.
164
Hypothesis Testing Procedures
Summary Table from page 534
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012.
Used with permission.
165
Inference Procedures for the Big 5 Parameters
Stats 250 Formula Card Summary
Population Proportion
Two Population Proportions
Population Mean
Parameter
p1  p 2
Parameter
pˆ 1  pˆ 2
Statistic
Standard Error
Parameter
p
p̂
Statistic
Standard Error
pˆ (1  pˆ )
n
s.e.( pˆ ) 
Confidence Interval
x
Statistic
Standard Error
pˆ 1 (1  pˆ 1 ) pˆ 2 (1  pˆ 2 )

n1
n2
s.e.( pˆ 1  pˆ 2 ) 
 pˆ 1  pˆ 2   z
pˆ  z s.e.( pˆ )
*
n
Confidence Interval
s.e. pˆ 1  pˆ 2 
x  t *s.e.( x )
Conservative
Confidence Interval
pˆ 
z
s
s.e.( x ) 
Confidence Interval
*

df = n – 1
Paired Confidence Interval
d  t *s.e.(d )
df = n – 1
*
2 n
Large-Sample z-Test
z
Large-Sample z-Test
pˆ  p0
z
p0 (1  p0 )
n
Sample Size
 z* 

n
 2m 


t
1 1
pˆ (1  pˆ )  
 n1 n2 
Where pˆ 
2
One-Sample t-Test
x  0 x  0
pˆ1  pˆ 2
n1 pˆ1  n2 pˆ 2
n1  n2
s.e.( x )

s
n
df = n – 1
Paired t-Test
d 0
d
t

df = n – 1
s.e.(d ) s d n
Two Population Means
General
1   2
Parameter
Statistic
Standard Error
s.e.x1  x2  
x1  x 2
Pooled
Confidence Interval
1 1

n1 n2
(n1  1)s12  (n 2  1) s 22
n1  n 2  2
Confidence Interval
x1  x2   t s.e.(x1  x2 )
df = min( n1  1, n2  1)
*
Two-Sample t-Test
x1  x2  0

s.e.( x1  x2 )
x1  x 2
pooled s.e.x1  x2   s p
s12 s22

n1 n2
where s p 
t
1   2
Parameter
Statistic
Standard Error
x1  x2   t * pooled s.e.(x1  x2 )
df = n1  n2  2
Pooled Two-Sample t-Test
x1  x2
2
1
2
2
df = min( n1  1, n2  1)
t
s
s

n1 n2
166
x1  x2  0

pooled s.e.( x1  x2 )
x1  x2
sp
1 1

n1 n2
df = n1  n2  2
Additional Notes
A place to … jot down questions you may have and ask during office hours, take a few extra notes, write
out an extra practice problem or summary completed in lecture, create your own short summary about
this chapter.
167
168