Business Statistics:
A First Course
(3rd Edition)
Chapter 8
Hypothesis Tests for Numerical Data
from Two or More Samples
© 2003 Prentice-Hall, Inc.
Chap 8-1
Chapter Topics

Comparing Two Independent Samples


Independent samples Z Test for the difference in
two means
Pooled-variance t Test for the difference in two
means

F Test for the Difference in Two Variances

Comparing Two Related Samples

Paired-sample Z test for the mean difference

Paired-sample t test for the mean difference
© 2003 Prentice-Hall, Inc.
Chap 8-2
Chapter Topics

(continued)
The Completely Randomized Design:
One-Way Analysis of Variance



ANOVA Assumptions
F Test for Difference in More than Two Means
The Tukey-Kramer Procedure
© 2003 Prentice-Hall, Inc.
Chap 8-3
Comparing Two Independent
Samples

Different Data Sources

Unrelated

Independent

Sample selected from one population has no effect
or bearing on the sample selected from the other
population

Use the Difference between 2 Sample Means

Use Z Test or Pooled-Variance t Test
© 2003 Prentice-Hall, Inc.
Chap 8-4
Independent Sample Z Test
(Variances Known)

Assumptions



Samples are randomly and independently drawn
from normal distributions
Population variances are known
Test Statistic

Z
( X 1  X 2 )  ( 1    )

2

n1
© 2003 Prentice-Hall, Inc.


2
n2
Chap 8-5
Independent Sample (Two
Sample) Z Test in EXCEL

Independent Sample Z Test with Variances
Known

Tools | Data Analysis | z-test: Two Sample for
Means
© 2003 Prentice-Hall, Inc.
Chap 8-6
Pooled-Variance t Test
(Variances Unknown)

Assumptions

Both populations are normally distributed

Samples are randomly and independently drawn


Population variances are unknown but assumed
equal
If both populations are not normal, need large
sample sizes
© 2003 Prentice-Hall, Inc.
Chap 8-7
Developing the Pooled-Variance
t Test

Setting Up the Hypotheses
H0:  1 =  2
H1:  1   2
OR
H0:  1 - 2 = 0
H1:  1 -  2  0
Two
Tail
H0:  1   2
H1:  1 >  2
OR
H0:  1 -  2  0
H1:  1 -  2 > 0
Right
Tail
H0:  1   2
H1:  1 <  2
OR
H0:  1 -  2  
H1:  1 -  2 < 0
Left
Tail
© 2003 Prentice-Hall, Inc.
Chap 8-8
Developing the Pooled-Variance
t Test
(continued)

Calculate the Pooled Sample Variance as an
Estimate of the Common Population Variance
(n1  1) S  (n2  1) S
S 
(n1  1)  (n2  1)
2
p
2
1
S p2 : Pooled sample variance
2
1
S : Variance of sample 1
2
2
n1 : Size of sample 1
n2 : Size of sample 2
2
2
S : Variance of sample 2
© 2003 Prentice-Hall, Inc.
Chap 8-9
Developing the Pooled-Variance
t Test
(continued)

Compute the Sample Statistic
X

t
df  n1  n2  2
S
© 2003 Prentice-Hall, Inc.
2
p
1
 X 2    1   2 
1 1
S   
 n1 n2 
2
p
Hypothesized
difference
n1  1 S   n2  1 S


 n1  1   n2  1
2
1
2
2
Chap 8-10
Pooled-Variance t Test: Example
You’re a financial analyst for Charles Schwab. Is there a
difference in dividend yield between stocks listed on the
NYSE & NASDAQ? You collect the following data:
NYSE
NASDAQ
Number
21
25
Sample Mean
3.27
2.53
Sample Std Dev 1.30
1.16
Assuming equal variances, is
there a difference in average
yield (a = 0.05)?
© 2003 Prentice-Hall, Inc.
© 1984-1994 T/Maker Co.
Chap 8-11
Calculating the Test Statistic
X

t
1
 X 2    1  2 
1 1
S   
 n1 n2 
2
p

 3.27  2.53  0
1 
 1
1.510   
 21 25 
2
2
n

1
S

n

1
S




1
2
2
S p2  1
 n1  1   n2  1
2
2
21

1
1.30

25

1
1.16





 21  1   25  1
© 2003 Prentice-Hall, Inc.
 2.03
 1.502
Chap 8-12
Solution
H0: 1 - 2 = 0 i.e. (1 = 2)
H1: 1 - 2  0 i.e. (1  2)
a = 0.05
df = 21 + 25 - 2 = 44
Critical Value(s):
Reject H0
Reject H0
.025
.025
-2.0154 0 2.0154
© 2003 Prentice-Hall, Inc.
2.03
t
Test Statistic:
t
3.27  2.53
1 
 1
1.502   
 21 25 
 2.03
Decision:
Reject at a = 0.05
Conclusion:
There is evidence of a
difference in means.
Chap 8-13
p -Value Solution
(p-Value is between .02 and .05) < (a = 0.05).
Reject.
p-Value
is between .01 and .025
2
Reject
Reject
a

-2.0154
0
2.0154
2.03
=.025
Z
Test Statistic 2.03 is in the Reject Region
© 2003 Prentice-Hall, Inc.
Chap 8-14
Pooled-Variance t Test in
PHStat and Excel

If the Raw Data are Available


Tools | Data Analysis | t-Test: Two Sample
Assuming Equal Variances
If only Summary Statistics are Available

PHStat | Two-Sample Tests | t Test for Differences
in Two Means...
© 2003 Prentice-Hall, Inc.
Chap 8-15
Solution in EXCEL

Excel Workbook that Performs the PooledVariance t Test
© 2003 Prentice-Hall, Inc.
Chap 8-16
Confidence Interval Estimate for
1  2 of Two Independent Groups

Assumptions

Both populations are normally distributed

Samples are randomly and independently drawn

Population variances are unknown but assumed equal


If both populations are not normal, need large
sample sizes
100 1  a  % Confidence Interval Estimate:
X
© 2003 Prentice-Hall, Inc.
1
 X 2   ta / 2,n1  n2  2
1 1
S   
 n1 n2 
2
p
Chap 8-17
Example
You’re a financial analyst for Charles Schwab. You
collect the following data:
NYSE
NASDAQ
Number
21
25
Sample Mean
3.27
2.53
Sample Std Dev 1.30
1.16
You want to construct a 95%
confidence interval for the
difference in population average
yields of the stocks listed on
NYSE and NASDAQ.
© 2003 Prentice-Hall, Inc.
© 1984-1994 T/Maker Co.
Chap 8-18
Example: Solution
2
2
n

1
S

n

1
S




1
2
2
S p2  1
 n1  1   n2  1
21  11.302   25  11.16 2


 1.502
 21  1   25  1
X
1
 X 2   ta / 2,n1  n2  2
1 1
S   
 n1 n2 
2
p
 1 1
 3.27  2.53  2.0154 1.502   
 21 25 
0.0088  1  2  1.4712
© 2003 Prentice-Hall, Inc.
Chap 8-19
Solution in Excel

An Excel Spreadsheet with the Solution:

© 2003 Prentice-Hall, Inc.
Chap 8-20
F Test for Difference in Two
Population Variances

Test for the Difference in 2 Independent
Populations

Parametric Test Procedure

Assumptions

Both populations are normally distributed


Test is not robust to this violation
Samples are randomly and independently drawn
© 2003 Prentice-Hall, Inc.
Chap 8-21
The F Test Statistic
2
1
2
2
S
F
S
2
1 = Variance of Sample 1
S
n1 - 1 = degrees of freedom
S
2
2= Variance of Sample 2
n2 - 1 = degrees of freedom
0
© 2003 Prentice-Hall, Inc.
F
Chap 8-22
Developing the F Test


Hypotheses
 H0: 12 = 22
 H1: 12  22
Test Statistic
 F = S12 /S22

Reject H0
Reject H0
a/2
0
Do Not
Reject
FL
a/2
FU
F
Two Sets of Degrees of Freedom
 df1
= n1 - 1; df2 = n2 - 1
 Critical Values: FL(
n1 -1, n2 -1) and FU( n1 -1 , n2 -1)
FL = 1/FU*
© 2003 Prentice-Hall, Inc.
(*degrees of freedom switched)
Chap 8-23
F Test: An Example
Assume you are a financial analyst for Charles Schwab.
You want to compare dividend yields between stocks listed
on the NYSE & NASDAQ. You collect the following data:
NYSE
NASDAQ
Number
21
25
Mean
3.27
2.53
Std Dev
1.30
1.16
Is there a difference in the
variances between the NYSE
& NASDAQ at the a  0.05 level?
© 2003 Prentice-Hall, Inc.
© 1984-1994 T/Maker Co.
Chap 8-24
F Test: Example Solution

Finding the Critical Values for a = .05

df1  n1  1  21  1  20
df 2  n2  1  25  1  24

FL 20,24  1/ FU  24,20  1/ 2.41  .415
FU  20,24  2.33
© 2003 Prentice-Hall, Inc.
Chap 8-25
F Test: Example Solution
H0 :  1 =  2
H1 :  1 2   2 2
a  .05
df1  20 df2  24
2
2
Critical Value(s):
Reject
.025
Reject
.025
0 0.415
© 2003 Prentice-Hall, Inc.
2.33
1.25
Test Statistic:
2
1
2
2
2
S
1.30
F

 1.25
2
S
1.16
Decision:
Do not reject at a = 0.05
Conclusion:
There is insufficient
F
evidence to prove a
difference in variances.
Chap 8-26
F Test in PHStat


PHStat | Two-Sample Tests | F Test for
Differences in Two Variances
Example in Excel Spreadsheet
© 2003 Prentice-Hall, Inc.
Chap 8-27
F Test: One-Tail
H0: 12  22
H1: 12 < 22
or
a = .05
FL n1 1,n2 1 
Reject
H0: 12  22
H1: 12 > 22
1
FU  n2 1,n1 1
Reject
a  .05
a  .05
0
Degrees of
freedom
switched
F
FL n1 1,n2 1
© 2003 Prentice-Hall, Inc.
0
FU  n1 1,n2 1
F
Chap 8-28
Comparing Two Related Samples

Test the Means of Two Related Samples




Paired or matched
Repeated measures (before and after)
Use difference between pairs
Di  X1i  X 2i
Eliminates Variation between Subjects
© 2003 Prentice-Hall, Inc.
Chap 8-29
Z Test for Mean Difference
(Variance Known)

Assumptions




Both populations are normally distributed
Observations are paired or matched
Variance Known
Test Statistic

Z
D  D
D
n
© 2003 Prentice-Hall, Inc.
n
D
D
i 1
i
n
Chap 8-30
t Test for Mean Difference
(Variance Unknown)

Assumptions





Both populations are normally distributed
Observations are matched or paired
Variance unknown
If population not normal, need large samples
Test Statistic
D  D
t
SD
n
© 2003 Prentice-Hall, Inc.
n
D
 Di
i 1
n
n
SD 
2
(
D

D
)
 i
i 1
n 1
Chap 8-31
Paired-Sample t Test: Example
Assume you work in the finance department. Is the new
financial package faster (a=0.05 level)? You collect the
following processing times:
Existing System (1)
9.98 Seconds
9.88
9.84
9.99
9.94
9.84
9.86
10.12
9.90
9.91
© 2003 Prentice-Hall, Inc.
New Software (2) Difference Di
9.88 Seconds
.10
9.86
.02
9.75
.09
9.80
.19
9.87
.07
9.84
.00
9.87
- .01
9.98
.14
9.83
.07
9.86
.05
D

D
i
n
SD 
 .072
D  D
2
i
n 1
 .06215
Chap 8-32
Paired-Sample t Test: Example
Solution
Is the new financial package faster (0.05 level)?
H0: D  
H1: D > 
a .5 D = .072
Critical Value=1.8331
df = n - 1 = 9
Test Statistic
D  D
.072  0
t

 3.66
SD / n .06215/ 10
© 2003 Prentice-Hall, Inc.
Reject
a .5
1.8331
t
3.66
Decision: Reject H0
t Stat. in the rejection
zone.
Conclusion: The new
software package is faster.
Chap 8-33
Paired-Sample t Test in
EXCEL


Tools | Data Analysis… | t-test: Paired Two
Sample for Means
Example in Excel Spreadsheet
© 2003 Prentice-Hall, Inc.
Chap 8-34
General Experimental Setting

Investigator Controls One or More
Independent Variables



Observe Effects on Dependent Variable


Called treatment variables or factors
Each treatment factor contains two or more groups
(or levels)
Response to groups (or levels) of independent
variable
Experimental Design: The Plan Used to Test
Hypothesis
© 2003 Prentice-Hall, Inc.
Chap 8-35
Completely Randomized Design

Experimental Units (Subjects) are Assigned
Randomly to Groups


Only One Factor or Independent Variable


Subjects are assumed homogeneous
With 2 or more groups (or levels)
Analyzed by One-way Analysis of Variance
(ANOVA)
© 2003 Prentice-Hall, Inc.
Chap 8-36
Randomized Design Example
Factor (Training Method)
Factor Levels
(Groups)
Randomly
Assigned
Units
Dependent
Variable
(Response)
© 2003 Prentice-Hall, Inc.



21 hrs
17 hrs
31 hrs
27 hrs
25 hrs
28 hrs
29 hrs
20 hrs
22 hrs
Chap 8-37
One-way Analysis of Variance
F Test

Evaluate the Difference among the Mean
Responses of 2 or More (c ) Populations


E.g. Several types of tires, oven temperature
settings
Assumptions

Samples are randomly and independently drawn


This condition must be met
Populations are normally distributed
 F Test is robust to moderate departure from
normality

Populations have equal variances

© 2003 Prentice-Hall, Inc.
Less sensitive to this requirement when samples
are of equal size from each population
Chap 8-38
Why ANOVA?



Could Compare the Means One by One using
Z or t Tests for Difference of Means
Each Z or t Test Contains Type I Error
The Total Type I Error with k Pairs of Means
is 1- (1 - a) k

E.g. If there are 5 means and use a = .05



© 2003 Prentice-Hall, Inc.
Must perform 10 comparisons
Type I Error is 1 – (.95) 10 = .40
40% of the time you will reject the null hypothesis
of equal means in favor of the alternative when the
null is true!
Chap 8-39
Hypotheses of One-Way ANOVA

H 0 : 1  2 



 c
All population means are equal
No treatment effect (no variation in means among
groups)
H1 : Not all i are the same



© 2003 Prentice-Hall, Inc.
At least one population mean is different (others
may be the same!)
There is a treatment effect
Does not mean that all population means are
different
Chap 8-40
One-way ANOVA
(No Treatment Effect)
H 0 : 1  2   c
H1 : Not all i are the same
The Null
Hypothesis is
True
1   2  3
© 2003 Prentice-Hall, Inc.
Chap 8-41
One-way ANOVA
(Treatment Effect Present)
H 0 : 1  2 
 c
H1 : Not all i are the same
1   2  3
© 2003 Prentice-Hall, Inc.
The Null
Hypothesis is
NOT True
1  2  3
Chap 8-42
One-way ANOVA
(Partition of Total Variation)
Total Variation SST
=






Variation Due to
Treatment SSA
Commonly referred to as:
Among Group Variation
Sum of Squares Among
Sum of Squares Between
Sum of Squares Model
Sum of Squares Explained
Sum of Squares Treatment
© 2003 Prentice-Hall, Inc.
+
Variation Due to Random
Sampling SSW




Commonly referred to as:
Within Group Variation
Sum of Squares Within
Sum of Squares Error
Sum of Squares Unexplained
Chap 8-43
Total Variation
nj
c
SST   ( X ij  X )
2
j 1 i 1
X ij : the i -th observation in group j
n j : the number of observations in group j
n : the total number of observations in all groups
c : the number of groups
c
X 
© 2003 Prentice-Hall, Inc.
nj
 X
j 1 i 1
n
ij
the overall or grand mean
Chap 8-44
Total Variation

SST  X 11  X
 X
2
21
X
  X
(continued)
2
nc c
X

2
Response, X
X
Group 1
© 2003 Prentice-Hall, Inc.
Group 2
Group 3
Chap 8-45
Among-Group Variation
c
SSA   n j ( X j  X )
j 1
2
SSA
MSA 
c 1
X j : The sample mean of group j
X : The overall or grand mean
i  j
© 2003 Prentice-Hall, Inc.
Variation Due to Differences Among Groups.
Chap 8-46
Among-Group Variation(continued)

SSA  n1 X 1  X
 n X
2
2
2
X

2

 nc X c  X

2
Response, X
X3
X1
Group 1
© 2003 Prentice-Hall, Inc.
Group 2
X2
X
Group 3
Chap 8-47
Within-Group Variation
c
nj
SSW   ( X ij  X j )
2
j 1 i 1
SSW
MSW 
nc
X j : The sample mean of group j
X ij : The i -th observation in group j
Summing the variation
within each group and then
adding over all groups.
© 2003 Prentice-Hall, Inc.
j
Chap 8-48
Within-Group Variation(continued)
SSW   X 11  X 1    X 21  X 1  
2
2

 X nc c  X c

2
Response, X
X3
X1
Group 1
© 2003 Prentice-Hall, Inc.
Group 2
X2
X
Group 3
Chap 8-49
Within-Group Variation(continued)
For c = 2, this is the
SSW
MSW 
pooled-variance in the
nc
t-Test.
2
2
2
(n1  1) S1  (n2  1) S2      (nc  1) Sc

(n1  1)  (n2  1)      (nc  1)
•If more than 2 groups,
use F Test.
•For 2 groups, use t-Test.
F Test more limited.
© 2003 Prentice-Hall, Inc.
j
Chap 8-50
One-way ANOVA
F Test Statistic

Test Statistic

MSA
F
MSW



MSA is mean squares among
MSW is mean squares within
Degrees of Freedom


© 2003 Prentice-Hall, Inc.
df1  c 1
df 2  n  c
Chap 8-51
One-way ANOVA
Summary Table
Degrees
Source of
of
Variation
Freedom
Among
c–1
(Factor)
Within
(Error)
Total
© 2003 Prentice-Hall, Inc.
Sum of
Squares
SSA
n–c
SSW
n–1
SST =
SSA + SSW
Mean
Squares
(Variance)
F
Statistic
MSA =
MSA/MSW
SSA/(c – 1 )
MSW =
SSW/(n – c )
Chap 8-52
Features of One-way ANOVA
F Statistic

The F Statistic is the Ratio of the Among
Estimate of Variance and the Within Estimate
of Variance




The ratio must always be positive
df1 = c -1 will typically be small
df2 = n - c will typically be large
The Ratio Should be Close to 1 if the Null is
True
© 2003 Prentice-Hall, Inc.
Chap 8-53
Features of One-way ANOVA
F Statistic
(continued)

If the Null Hypothesis is False


The numerator should be greater than the
denominator
The ratio should be larger than 1
© 2003 Prentice-Hall, Inc.
Chap 8-54
One-way ANOVA F Test
Example
As production manager, you
want to see if 3 filling
machines have different mean
filling times. You assign 15
similarly trained and
experienced workers, 5 per
machine, to the machines. At
the .05 significance level, is
there a difference in mean
filling times?
© 2003 Prentice-Hall, Inc.
Machine1 Machine2
Machine3
25.40
26.31
24.10
23.74
25.10
23.40
21.80
23.50
22.75
21.60
20.00
22.20
19.75
20.60
20.40
Chap 8-55
One-way ANOVA Example:
Scatter Diagram
Machine1 Machine2
Machine3
25.40
26.31
24.10
23.74
25.10
23.40
21.80
23.50
22.75
21.60
27
20.00
22.20
19.75
20.60
20.40
X 1  24.93
X 2  22.61
X 3  20.59
X  22.71
© 2003 Prentice-Hall, Inc.
26
25
24
23
22
21
20
•
••
•
•
X1
••
•
••
X2
•
••
••
X
X3
19
Chap 8-56
One-way ANOVA Example
Computations
Machine1 Machine2 Machine3
25.40
26.31
24.10
23.74
25.10
23.40
21.80
23.50
22.75
21.60
20.00
22.20
19.75
20.60
20.40
X 1  24.93
nj  5
X 2  22.61
c3
X 3  20.59
n  15
X  22.71
2
2
2

SSA  5  24.93  22.71   22.61  22.71   20.59  22.71 


 47.164
SSW  4.2592  3.112  3.682  11.0532
MSA  SSA /(c -1)  47.16 / 2  23.5820
MSW  SSW /( n - c)  11.0532 /12  .9211
© 2003 Prentice-Hall, Inc.
Chap 8-57
Summary Table
Degrees
Source of
of
Variation
Freedom
Sum of
Squares
Mean
Squares
(Variance)
F
Statistic
MSA/MSW
=25.60
Among
(Factor)
3-1=2
47.1640
23.5820
Within
(Error)
15-3=12
11.0532
.9211
Total
15-1=14
58.2172
© 2003 Prentice-Hall, Inc.
Chap 8-58
One-way ANOVA Example
Solution
Test Statistic:
H0: 1 = 2 = 3
H1: Not All Equal
a = .05
df1= 2
MSA
23.5820
 25.6
F

MSW
.9211
df2 = 12
Decision:
Reject at a = 0.05
Critical Value(s):
a = 0.05
0
© 2003 Prentice-Hall, Inc.
3.89
F
Conclusion:
There is evidence that at least
one  i differs from the rest.
Chap 8-59
Solution In EXCEL


Use Tools | Data Analysis | ANOVA: Single
Factor
EXCEL Worksheet that Performs the One-way
ANOVA of the example
© 2003 Prentice-Hall, Inc.
Chap 8-60
The Tukey-Kramer Procedure

Tells which Population Means are Significantly
Different



1= 2
Post Hoc (a posteriori) Procedure


e.g., 1 = 2  3
2 groups whose means
may be significantly
different
f(X)
3
X
Done after rejection of equal means in ANOVA
Pairwise Comparisons

Compare absolute mean differences with critical
range
© 2003 Prentice-Hall, Inc.
Chap 8-61
The Tukey-Kramer Procedure:
Example
Machine1 Machine2 Machine3
25.40
23.40
20.00
26.31
21.80
22.20
24.10
23.50
19.75
23.74
22.75
20.60
25.10
21.60
20.40
2. Compute Critical Range:
Critical Range  QU ( c,nc )
1. Compute absolute mean
differences:
X 1  X 2  24.93  22.61  2.32
X 1  X 3  24.93  20.59  4.34
X 2  X 3  22.61  20.59  2.02
MSW
2
1 1 
    1.618
 nj nj' 
3. All of the absolute mean differences are greater than the
critical range. There is a significance difference between
each pair of means at the 5% level of significance.
© 2003 Prentice-Hall, Inc.
Chap 8-62
Solution in PHStat


Use PHStat | c-Sample Tests | Tukey-Kramer
Procedure …
EXCEL Worksheet that Performs the TukeyKramer Procedure for the Previous example
© 2003 Prentice-Hall, Inc.
Chap 8-63
Chapter Summary

Compared Two Independent Samples




Performed Z test for the differences in two means
Performed t test for the differences in two means
Addressed F Test for Difference in two
Variances
Compared Two Related Samples


Performed paired sample Z tests for the mean
difference
Performed paired sample t tests for the mean
difference
© 2003 Prentice-Hall, Inc.
Chap 8-64
Chapter Summary

(continued)
Described The Completely Randomized
Design: One-way Analysis of Variance



ANOVA Assumptions
F Test for Difference in c Means
The Tukey-Kramer Procedure
© 2003 Prentice-Hall, Inc.
Chap 8-65