Hypothesis Testing with
Two Populations
Week 9
GT00303
Comparing 2 Populations
Previously we looked at techniques to estimate and test
parameters for one population:
Population Mean (µ)
Population Variance (σ2)
We will still consider these parameters when we are
looking at two populations, however our interest will now
be:
 The difference between two means (µ1- µ2).
 The ratio of two variances (σ12 / σ22).
9-2
Difference between 2 Means
(Independent Samples)
In order to test and estimate the difference between
two population means, we draw random samples
from each of two populations.
Initially, we will consider independent samples, that
is, samples that are completely unrelated to one
another.
9-3
Population 1
Sample, size: n1
Statistics:
Parameters:
Population 2
Sample, size: n2
Parameters: 2
&
2
2
Statistics:
x2 & s22
9-4
The sampling distribution of x1  x2 .
(1) If populations are normal (approximately normal):
(2) If populations are non-normal:
The expected value of x1  x2 :
The standard error of x1  x2 :
1  2
 12
n1

 22
n2
9-5
Statistic - Parameter
Test statistic 
Standard error
z
 x1  x2  -  1  2 
 12
n1

 22
n2
??
In
practice,
the
population
variances
(σ2) are unknown.
So, z statistic is rarely
used
In this case, we have to replace them with sample
variances (s), and use t-statistic!
9-6
However, the application of t-test depends on 2
conditions:
(1) When we believe the population variances are
equal (equal-variances t-test)
(2) When we believe the population variances are not
equal (unequal-variances t-test)
9-7
(1) Equal-variance t-test for (μ1 - μ2)
Test Statistic: t 
 x1  x2    1  2 
s 2p
n1
CI Estimator:
 x1  x2  
t

s 2p
d.f. = n1  n2  2
n2
s 2p
2
,
n1

s 2p
n2
,
d.f. = n1  n2  2
Pooled variance estimator
9-8
(2) Unequal-variance t-test for (μ1 - μ2)
Test Statistic: t 
s12 s22

n1 n2
 x1  x2  
CI Estimator:
d.f. 
 x1  x2    1  2 
s
2
1
s
2
1
t
n1  s
2
2
n1 
n1  1
2
s


2
n2 
2
2
s12 s22

n1 n2
When the two
population
variances are
unequal, we cannot
pool the data and
produce a common
estimator.
2
n2 
2
n2  1
9-9
So, which t-test to use? Equal-variances or
unequal- variances?
We have to first test the hypothesis of equal variances!
1st Hypothesis
H0: σ12 / σ22 = 1
H1: σ12 / σ22 ≠ 1
If reject H0 and conclude unequal
variances, USE unequal-variances ttest for 2nd hypothesis.
If do not reject H0 and conclude
insufficient evidence that variances are
unequal, USE equal-variances t-test
for 2nd hypothesis
2nd Hypothesis
H0: µ1- µ2 = 0
H1: µ1- µ2 > 0
9-10
How to test 1st Hypothesis?
1st Hypothesis
H0: σ12 / σ22 = 1
H1: σ12 / σ22 ≠ 1
s12
Test Statistic: F  2
s2
d.f.: v1  n1  1 v2  n2  1
The rejection region and critical value can be
obtained from the F-table.
9-11
Illustration 1:
Millions of investors buy mutual funds choosing from thousands of
possibilities. Some funds can be purchased directly from banks or
other financial institutions while others must be purchased through
brokers, who charge a fee for this service. This raises the question,
can investors do better by buying mutual funds directly than by
purchasing mutual funds through brokers.
To help answer this question a group of researchers randomly
sampled the annual returns from mutual funds that can be acquired
directly and mutual funds that are bought through brokers and
recorded the net annual returns, which are the returns on investment
after deducting all relevant fees.
Can we conclude at the 5% significance level that directly-purchased
mutual funds outperform mutual funds bought through brokers?
9-12
Population 1
Net annual return from directly-purchased
mutual funds
Population 2
Net annual return from broker-purchased funds
µ1 = mean net annual return for population 1
µ2 = mean net annual return for population 2
9-13
From the data Xm13-01 (click here), we can compute
sample mean and sample variance using Excel:
n1  50
x1  6.63
s12  37.49
n2  50
x 2  3.72
s22  43.34
Preliminary Test
Since population variances (σ2) are unknown, we will use tdistribution.
But which t-test to use? Equal-variances or unequalvariances?
9-14
To decide, we apply the F-test on the following hypothesis:
H0: σ12 / σ22 = 1
H1: σ12 / σ22 ≠ 1
s12
37.49
F 2 =
 0.86
s2
43.34
We will compare this test statistic with the critical value (or
rejection region).
For F-table, we use α = 5%, and degree of freedom v1 =
50-1 = 49, v2 = 50 – 1 =49. It is a two-tail test.
9-15
Right-tail Critical Value:
F0.025,49,49 » F0.025,50,50 = 1.75
Left-tail Critical Value:
The F-table gives critical value for right-tail test. Because the F
distribution is not symmetric, and there are no negative values, you
CANNOT simply take the opposite of the right critical value to find
the left critical value.
The way to find a left critical value is to reverse the degrees of
freedom, look up the right critical value, and then take the
reciprocal of this value
F0.975,49,49 =
1
F0.025,49,49
»
1
F0.025,50,50
1
=
= 0.57
1.75
9-16
The Rejection Region is
F < 0.57
F > 1.75
The test statistic of 0.86 does not fall into the Rejection
Region.
Do not reject H0 and conclude that there is insufficient
evidence to infer the population variances are unequal.
So, for this illustration, we will conduct the hypothesis
testing using equal-variances t-test.
9-17
Step 1:
The hypothesis to be tested is that the mean net annual
return from directly-purchased mutual funds (µ1) is larger
than (outperform) the mean of broker-purchased funds
(µ2).
H0:
µ1- µ2 = 0
H1:
µ1-µ2 > 0
H0 is presumed to be true
This is what we want to prove!
9-18
Step 2:
Since population variances (σ2) are unknown, we will use tdistribution.
Our F-test earlier suggests the use of equal-variances ttest.
Test Statistic: t 
 x1  x2    1  2 
s
2
p
n1

s
2
p
,
d.f. = n1  n2  2
n2
Pooled variance estimator
9-19
2
2
n

1
s

n

1
s
 1 1  2  2
s 2p 
n1  n2  2
(50  1)  37.49   (50  1)  43.34 
=
50  50  2
 40.42
Test Statistic: t 
 x1  x2    1  2 
s 2p
n1


s 2p
n2
d.f. = n1 + n2 -2
= 50 + 50 – 2
= 98
 6.63  3.72    0 
40.42 40.42

50
50
= 2.29
9-20
Step 3:
With α= 0.05 and it is a one-tail test (right tail), the critical
value and rejection region is as follows:
0.05
t0.05,98=???
9-21
t0.05, 98 = 1.661
9-22
Step 4:
Reject H0 if the computed test statistic (from Step 2, t =
2.29) falls into the shaded Rejection Region, or
t > 1.661
Step 5:
Reject H0 at the 5% level of significance and conclude
there is sufficient evidence to infer that on average
directly-purchased mutual funds outperform brokerpurchased mutual funds.
9-23
Can we estimate the 95% confidence
interval for μ1-μ2?
CI Estimator:
 x1  x2  
t
s 2p
2
n1

s 2p
n2
,
d.f. = n1  n2  2
1 
 1
 (6.63  3.72)  1.984 40.42   
 50 50 
 2.91  2.52
  0.39,5.43
It is estimated that the return on directly purchased mutual funds is
on average between 0.39 and 5.43 percentage points larger than
broker-purchased mutual funds.
9-24
Illustration 2:
What happens to the family-run business when the boss’s son or
daughter takes over? Does the business do better after the
change if the new boss is the offspring of the owner or does the
business do better when an outsider is made chief executive
officer (CEO)?
In pursuit of an answer researchers randomly selected 140 firms
between 1994 and 2002, 30% of which passed ownership to an
offspring and 70% appointed an outsider as CEO. For each
company the researchers calculated the operating income as a
proportion of assets in the year before and the year after the new
CEO took over.
Do these data allow us to infer at the 5% level of significance that
the effect of making an offspring CEO is different from the effect
of hiring an outsider as CEO?
9-25
Population 1: Operating income of companies whose CEO is
an offspring of the previous CEO
Population 2: Operating income of companies whose CEO is
an outsider
µ1 = mean operating income for population 1
µ2 = mean operating income for population 2
From the data Xm13-02 (click here), we can compute sample
mean and sample variance using Excel:
n1  42
x1   0.10
s12  3.79
n2  98
x2  1.24
s22  8.03
9-26
Preliminary Test
Since population variances (σ2) are unknown, we will use tdistribution.
But which t-test to use? Equal-variances or unequalvariances?
To decide, we apply the F-test on the following hypothesis:
H0: σ12 / σ22 = 1
H1: σ12 / σ22 ≠ 1
s12
3.79
F 2 =
 0.47
s2
8.03
9-27
We will compare this test statistic with the critical value (or
rejection region).
To use the F-table, we use α = 5%, and degree of freedom
v1 = 42-1 = 41, v2 = 98 – 1 =97. It is a two-tail test.
Right-tail Critical Value:
F0.025,41,97 » F0.025,40,100 = 1.64
Left-tail Critical Value:
F0.975,41,97 =
1
F0.025,97,41
»
1
F0.025,100,40
1
=
= 0.57
1.74
9-28
The Rejection Region is
F < 0.57
F > 1.64
The test statistic of 0.47 falls into the Rejection Region.
Reject H0 and conclude that there is sufficient evidence to
infer the population variances are unequal.
So, for this illustration, we will conduct the hypothesis
testing using unequal-variances t-test.
9-29
Step 1:
The hypothesis to be tested is that the mean operating
income for companies whose CEO is an offspring of the
previous CEO (µ1) is different from the mean operating
income of companies whose CEO is an outsider (µ2).
H0:
µ1- µ2 = 0
H1:
µ1-µ2 ≠ 0
H0 is presumed to be true
This is what we want to prove!
9-30
Step 2:
Since population variances (σ2) are unknown, we will use tdistribution.
Our F-test earlier suggests the use of unequal-variances
t-test.
Test Statistic: t 
d.f. 
s
2
1
s
2
1
 x1  x2    1  2 
s12 s22

n1 n2
n1  s
2
2
n1 
n1  1
2
s


n2 
2
2
2
n2 
2
n2  1
9-31
Test Statistic: t 
 x1  x2    1  2 
s12 s22

n1 n2
 0.10  1.24    0 

3.79 8.03

42
98
=  3.22
d.f. 
s
2
1
s
2
1
n1  s
2
2
n1 
n1  1
2
s


n2 
2
2
2
n2 
n2  1
2

 3.79
 3.79
42  8.03 98 
2
42 
8.03 98 


42  1
98  1
2
2
 111
9-32
Step 3:
With α= 0.05 and it is a two-tail test, the critical value and
rejection region is as follows:
-t0.025,111 » -t0.025,110 = -1.982
t0.025,111 » t0.025,110 = 1.982
0.025
0.025
-t0.025,111=???
t0.025,111=???
9-33
Step 4:
Reject H0 if the computed test statistic (from Step 2, t =
-3.22) falls into the shaded Rejection Region, or
t < -1.982
t > 1.982
Step 5:
Reject H0 at the 5% level of significance and conclude
that there is sufficient evidence to infer that mean
operating income for the two populations are different.
9-34
Difference between 2 Means
(Matched Pairs)
Previously, we consider independent samples, that
is, samples that are completely unrelated to one
another.
When an observation in one sample is matched with
an observation in a second sample, this is called a
matched pairs experiment.
9-35
Illustration 3A:
In the last few years, a number of web-based companies that
offer job placement services have been created. The manager
of one such company wanted to investigate the job offers
recent MBAs were obtaining. In particular, she wanted to know
whether finance majors were being offered higher salaries than
marketing majors.
In a preliminary study she randomly sampled 50 recently
graduated MBAs half of whom majored in finance and half in
marketing. From each she obtained the highest salary
(including benefits) offer.
Can we infer at the 5% level of significance that finance majors
obtain higher salary offers than do marketing majors among
MBAs?
9-36
Population 1
Highest salary offer to finance majors
Population 2
Highest salary offer to marketing majors
µ1 = mean highest salary offer for population 1
µ2 = mean highest salary offer for population 2
9-37
From the data Xm13-04 (click here), we can compute
sample mean and sample variance using Excel:
n1 = 25
n2 = 25
x1 = 65624
x2 = 60423
s12 = 360433294
s22 = 262228559
Preliminary Test
Since population variances (σ2) are unknown, we will use tdistribution.
But which t-test to use? Equal-variances or unequalvariances?
9-38
To decide, we apply the F-test on the following hypothesis:
H0: σ12 / σ22 = 1
H1: σ12 / σ22 ≠ 1
s12
360433294
F= 2 =
= 1.37
262228559
s2
We will compare this test statistic with the critical value (or
rejection region).
To use the F-table, we use α = 5%, and degree of freedom
v1 = 25-1 = 24, v2 = 25 – 1 =24. It is a two-tail test.
9-39
Right-tail Critical Value
F0.025,24,24  2.27
The Rejection Region is
Left-tail Critical Value
F0.975,24,24 
1
F0.025,24,24
1

 0.44
2.27
F < 0.44
F > 2.27
The test statistic of 1.37 does not fall into the Rejection
Region.
Do not reject H0 and conclude that there is insufficient
evidence to infer the population variances are unequal.
So, for this illustration, we will conduct the hypothesis testing
using equal-variances t-test.
9-40
Step 1:
The hypothesis to be tested is that the mean highest salary
offer for finance majors (µ1) is larger than the mean
highest salary offer for marketing majors (µ2).
H0:
µ1- µ2 = 0
H1:
µ1-µ2 > 0
H0 is presumed to be true
This is what we want to prove!
9-41
Step 2:
Since population variances (σ2) are unknown, we will use tdistribution.
Our F-test earlier suggests the use of equal-variances ttest.
Test Statistic: t 
 x1  x2    1  2 
s
2
p
n1

s
2
p
,
d.f. = n1  n2  2
n2
Pooled variance estimator
9-42
s 2p =
=
( n - 1) s + ( n
1
2
1
s
)
2
2
2
1
n1 + n2 - 2
(25 - 1) ( 360433294 ) + (25 - 1) ( 262228559 )
25 + 25 - 2
= 311330926
Test Statistic: t =
(x
1
- x2 ) - ( m1 - m2 )
s 2p
n1
=
+
s 2p
n2
d.f. = n1 + n2 -2
= 25 + 25 – 2
= 48
( 65624 - 60423) - ( 0)
311330926 311330926
+
25
25
= 1.04
9-43
Step 3:
With α= 0.05 and it is a one-tail test (right tail), the critical
value and rejection region is as follows:
t0.05,48  t0.05,50  1.676
0.05
t0.05,48=???
9-44
Step 4:
Reject H0 if the computed test statistic (from Step 2, t =
1.04) falls into the shaded Rejection Region, or
t > 1.676
Step 5:
Do not reject H0 at the 5% level of significance and
conclude there is insufficient evidence to infer that
finance majors receive higher salary offers than
marketing majors.
9-45
Illustration 3B:
Suppose now that we redo the experiment in the following way.
We examine the transcripts of finance and marketing MBA
majors.
We randomly sample a finance and a marketing major whose
grade point average (GPA) falls between 3.92 and 4 (based on
a maximum of 4). We then randomly sample a finance and a
marketing major whose GPA is between 3.84 and 3.92. We
continue this process until the 25th pair of finance and
marketing majors are selected whose GPA fell between 2.0
and 2.08 (The minimum GPA required for graduation is 2.0.)
As in Illustration 3A, we recorded the highest salary offer.
Can we infer at the 5% level of significance that finance majors
obtain higher salary offers than do marketing majors among
MBAs?
9-46
In Illustration 3A, the samples are independent.
Illustration 3B is a matched pairs experiment, i.e., each
observation in one sample is matched with an observation in the
other sample.
The matching is conducted
by selecting finance and
marketing majors with
similar GPAs.
For full data, click here
9-47
For each GPA group (e.g., Group 1 has GPA between 3.92
and 4), we calculate the matched pair difference between
the salary offers for finance and marketing majors.
The difference of the means is equal to the mean of the differences, hence
we will consider the “mean of the paired differences” as our parameter of interest:
9-48
From the data Xm13-05 (click here), we can compute sample
mean and sample variance using Excel:
xD = 5065
sD = 6647
Step 1:
The hypothesis to be tested is that the mean highest salary
offer for finance majors (µ1) is larger than the mean
highest salary offer for marketing majors (µ2).
H0:
µD = 0
H1:
µD > 0
H0 is presumed to be true
This is what we want to prove!
9-49
Step 2:
The test statistic for the mean of the population of
differences (µD) is:
Test Statistic: t =
t=
5605 - 0
6647
25
xD - m D
sD
nD
= 3.81
,
d.f. = nD - 1
d.f. = nD – 1
= 25 – 1
= 24
9-50
Step 3:
With α= 0.05 and it is a one-tail test (right tail), the critical
value and rejection region is as follows:
t0.05,24  1.711
0.05
t0.05,24=???
9-51
Step 4:
Reject H0 if the computed test statistic (from Step 2, t =
3.81) falls into the shaded Rejection Region, or
t > 1.711
Step 5:
Reject H0 at the 5% level of significance and conclude
there is sufficient evidence to infer that finance majors
receive higher salary offers than marketing majors.
9-52
Can we estimate the 95% confidence
interval for μD?
CI Estimator: x D ± ta
sD
2
nD
= 5065 ± 2.064
d.f. = nD - 1
,
6647
25
= 5065 ± 2744
= ( 2321,7809 )
It is estimated that the mean salary offer to finance majors exceeds
the mean salary offer to marketing majors by an amount that lies
between $2,321 and $7,809.
9-53
Independent Samples versus Matched Pairs
Conclusion
Test Statistic
Independent Samples
Matched Pairs
Do not reject H0 at the 5%
level of significance and
conclude there is insufficient
evidence to infer that finance
majors receive higher salary
offers than marketing majors.
Reject H0 at the 5% level of
significance and conclude there
is sufficient evidence to infer
that finance majors receive
higher salary offers than
marketing majors.
Test Statistic: t =
(x
1
- x2 ) - ( m1 - m2 )
s 2p
n1
+
s 2p
n2
Test Statistic: t =
xD - m D
sD
nD
= 3.81
= 1.04
9-54
Mean
Difference
(Numerator)
Standard
Errors
(Denominator)
Independent Samples
Matched Pairs
x1 - x2 = 5201
xD = 5065
s 2p
n1
+
s2p
n2
= 4991
sD
nD
= 1329
The difference in the test statistic was caused not by the numerator,
but by the denominator. So, the matched pairs experiment reduces
the variation in the data (But it will not always be the case).
9-55
The Ratio of Two Population Variances
When looking at two population variances, we
consider the ratio of the variances, i.e. the parameter
of interest to us is:
s
2
1
s12
Test Statistic: F  2
s2
s
2
2
d.f.: v1  n1  1 v2  n2  1
æ s12 ö
1
CI Estimator: LCL = ç 2 ÷
è s2 ø Fa 2,v1 ,v2
æ s12 ö
UCL = ç 2 ÷ Fa
è s2 ø
2,v2 ,v1
9-56
Illustration 4:
Container-filling machines are used to package a variety of liquids;
including milk, soft drinks, and paint. Ideally, the amount of liquid
should vary only slightly, since large variations will cause some
containers to be under-filled (cheating the customer) and some to be
overfilled (resulting in costly waste).
The president of a company that developed a new type of machine
boasts that this machine can fill 1 liter (1,000 cubic centimeters)
containers consistently. A random sample of 25 l-liter fills was taken
and the results (cubic centimeters) recorded.
Suppose that the statistics practitioner also collected data from
another container-filling machine and recorded the fills of a randomly
selected sample.
Can we infer at the 5% significance level that the second machine is
superior in its consistency?
9-57
Because the information we want is about the
consistency of the two machines, so the parameter of
interest is σ12 / σ22
σ12 = Variance of machine 1
σ22 = Variance of machine 2
From the data Xm13-07 (click here), we can compute
sample mean and sample variance using Excel:
n1 = 25
n2 = 25
x1 = 999.68
x2 = 999.81
s12 = 0.6333
s22 = 0.4528
9-58
Step 1:
We need to determine whether machine 2 is more
consistent than machine 1 (consistent means smaller
variance).
Hence, we are testing whether there is sufficient evidence
to infer that σ12 > σ22
H0: σ12 / σ22 = 1
H1: σ12 / σ22 > 1
H0 is presumed to be true
This is what we want to prove!
9-59
Step 2:
s12
0.6333
F= 2 =
= 1.40
0.4528
s2
Step 3:
To use the F-table, we use α = 5%, and degree of freedom
v1 = 25-1 = 24, v2 = 25 – 1 =24. It is a one-tail (right tail)
test.
F0.05,24,24 = 1.98
9-60
Step 4:
Reject H0 if the computed test statistic (from Step 2, F =
1.40) falls into the shaded Rejection Region, or
F > 1.98
Step 5:
Do not reject H0 at the 5% level of significance and
conclude that there is insufficient evidence to infer the
variance of machine 2 is less than the variance of
machine 1 (or machine 2 is more consistent).
9-61
Can we estimate the 95% confidence
interval for σ12 / σ22 ?
æ s12 ö
1
CI Estimator: LCL = ç 2 ÷
è s2 ø Fa 2,v1 ,v2
æ s12 ö
UCL = ç 2 ÷ Fa
è s2 ø
2,v2 ,v1
d.f.: v1 = n1 -1 v2 = n2 -1
From F table, F0.025,24,24 = 2.27
æ 0.6333 ö 1
LCL = ç
= 0.616
÷
è 0.4528 ø 2.27
æ 0.6333 ö
UCL = ç
2.27 = 3.17
÷
è 0.4528 ø
It is estimated that the ratio of the two population variances lies
between 0.616 and 3.17
9-62