Business Statistics, 4e
by Ken Black
Chapter 10
Discrete Distributions
Statistical
Inferences about
Two Populations
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-1
Learning Objectives
• Test hypotheses and construct confidence
intervals about the difference in two
population means using the Z statistic.
• Test hypotheses and construct confidence
intervals about the difference in two
population means using the t statistic.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-2
Learning Objectives
• Test hypotheses and construct confidence
intervals about the difference in two
related populations.
• Test hypotheses and construct confidence
intervals about the differences in two
population proportions.
• Test hypotheses and construct confidence
intervals about two population variances.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-3
Sampling Distribution of the
Difference Between Two Sample
Means
x1
X
Population 1
1
x
x
n
1
1
x
2

x1  X
x2
X
1
Xx1 xX2
1
2
x
n
2
x2
X
2
Population 2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-4
2
Sampling Distribution of the
Difference between Two Sample
Means

 xXxX
1
1

2
12  22
x  x 


X X
nn1 nn2
1 2
  1  2
2
1
 X
x1  xX2 2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
1

1
2
2
2
1
2
2
1
2
xX1  Xx2
1
2
10-5
Z Formula for the Difference
in Two Sample Means
When 12 and22 are known and
Independent Samples
x  x     
z
 
n n
1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
2
1
2
2
2
1
2
1
2
10-6
Hypothesis Testing for Differences Between
Means: The Wage Example (part 1)
Ho: 1  2  0
Ha: 1  2  0
Rejection
Region
Rejection
Region

2

 .025
2
 .025
Non Rejection Region
 Xx  Xx

11
22
Xx1  Xx2
1
2
Critical Values
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-7
Hypothesis Testing for Differences Between
Means: The Wage Example (part 2)
Rejection
Region
Rejection
Region

2
 .025
If z < - 1.96 or z > 1.96, reject Ho.
If - 1.96  z  1.96, do not reject Ho.

 .025
2
Non Rejection Region
zZc c 
.96
11.96
0
96
zZc c11.96
Critical Values
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-8
Hypothesis Testing for Differences
Between Means: The Wage Example
(part 3)
Advertising Managers
74.256
57.791
71.115
Auditing Managers
n  32
x  70.700
1
69.962
77.136
43.649
55.052
66.035
63.369
57.828
54.335
59.676
63.362
42.494
54.449
37.194
83.849
46.394
96.234
65.145
67.574
89.807
96.767
59.621
93.261
77.242
62.483
103.030
67.056
69.319
74.195
64.276
35.394
99.198
67.160
71.804
75.932
74.194
86.741
61.254
37.386
72.401
80.742
65.360
57.351
73.065
59.505
56.470
39.672
73.904
45.652
54.270
93.083
59.045
63.384
68.508
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
1


n
x
 16.253
1
 264.164
2
1
2
 34
2
 62.187
48.036
72.790
67.814
 12.900
60.053
71.351
71.492
 166.411
66.359
58.653
61.261
63.508


2
2
2
10-9
Hypothesis Testing for Differences between
Means: The Wage Example (part 4)
Rejection
Region
Rejection
Region

2

(xX
x2X
) 2 (1 
 2
1 1 
)

2
 .025
zZ
 .025
Non Rejection Region
2.33
.33
Zzc  2
c
IfIf zZ<-11.96
.96 or Zz >
 1.96,
1.96,reject
rejectHHo.0 .
o. 0 .
IfIf - 1.96
1.96 Zz 1.96,
1.96,do
donot
notreject
rejectHH
0
zZc c22.33
33
Critical Values
1
2

2 2
22
 1S

22
1 S

n1n nn2
1
2
(70.700
62.187)

70.700 - 62
.187  -(0)
0

2.235.35
256
166.411
.411
256.253
253 166

32
34
32
34
Since
.35 
.96, reject
reject HHo.0 .
Since zZ=22.35
>1
1.96,
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-10
Difference Between Means: Using Excel
z-Test: Two Sample for Means
Adv Mgr
Auditing Mgr
Mean
70.7001
62.187
Known Variance
264.164
166.411
32
34
Observations
Hypothesized Mean Difference
z
P(Z<=z) one-tail
z Critical one-tail
0
2.35
0.0094
1.64
P(Z<=z) two-tail
0.0189
z Critical two-tail
1.960
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-11
Demonstration Problem 10.1 (part 1)
Rejection
Region
Ho: 1   2  0
Ha: 1   2  0
 .001
Non Rejection Region
zZc c3
3..08
0
Critical Value
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-12
Demonstration Problem 10.1 (part 2)
Rejection
Region
Non Rejection Region
c
Men
x2  $5,727
 1  $1,100
 2  $1,700
n2  76
n1  87
 .001
Zzc  33.08.08
Women
x1  $3,352
0
Critical Value
x  x      
z
 
n n
1

If z < - 3.08, reject Ho.
If z   3.08, do not reject Ho.
2
1
2
2
2
1
2
1
2
3352  5727  0  10.42
2
1100  1700
87
76
2
Since z = - 10.42 < - 3.08, reject Ho.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-13
Confidence Interval to Estimate 1 - 2 When
1, 2 are known
x  x   z   
n n
1
2
2
1
2
2
1
2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

 
1

2
 x1  x 2   z

2
1
n
1


2
2
n
2
10-14
Demonstration Problem 10.2
Re gular
Pr emium
n  50
x  21.45
  3.46
n  50
x  24.6
  2.99
1
2
95% Confidence  z = 1.96
2
1
2
1
 x  x   z         x  x   z   
n n
n n
21.45  24.6  1.96 3.46  2.99      21.45  24.6  1.96 3.46
50
50
50
 4.42      1.88
1
2
2
2
2
1
2
1
2
1
2
2
1
1
2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
2
2
2
1
2
1
2
1
2
2
2.992

50
10-15
The t Test for Differences
in Population Means
• Each of the two populations is normally
distributed.
• The two samples are independent.
• The values of the population variances are
unknown.
• The variances of the two populations are equal.
12 = 22
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-16
t Formula to Test the Difference in
Means Assuming 12 = 22
t
( x1  x2 )  ( 1   2 )
s12 (n1  1)  s22 (n2  1)
n1  n2  2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
1
1

n1 n2
10-17
Hernandez Manufacturing Company
(part 1)
Ho: 1   2  0
Ha: 1   2  0

Rejection
Region

.025
2
.05
 .025
2
2
df  n1  n2  2  15  12  2  25

t0.25, 25  2.060
If t < - 2.060 or t > 2.060, reject Ho.
If - 2.060  t  2.060, do not reject Ho.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Rejection
Region

.025
2
Non Rejection Region
t
.025, 25
 2.060
0
t
.025, 25
 2.060
Critical Values
10-18
Hernandez Manufacturing Company
(part 2)
Training Method A
Training Method B
56
51
45
59
57
53
47
52
43
52
56
65
42
53
52
53
55
53
50
42
48
54
64
57
47
44
44
n1  15
x1  47.73
s12  19.495
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
n2  12
x2  56.5
s22  18.273
10-19
Hernandez Manufacturing Company
(part 3)
t

( x1  x2 )  ( 1   2 )
s12 ( n1  1)  s22 ( n2  1)
n1  n2  2
1
1

n1 n2
47.73  56.50  0
19.49514  18.27311
15  12  2
1
1

15 12
 5.20
If t < - 2.060 or t > 2.060, reject Ho.
If - 2.060  t  2.060, do not reject Ho.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Since t = -5.20 < -2.060, reject Ho.
10-20
MINITAB Output for Hernandez
New-Employee Training Problem
Twosample T for method A vs method B
method A
method B
N
15
12
Mean
47.73
56.60
StDev
4.42
4.27
SE Mean
1.1
1.2
95% C.I. for mu method A - mu method B: (-12.2, -5.3)
T-Test mu method A = mu method B (vs not =): T = -5.20
P=0.0000 DF = 25
Both use Pooled StDev = 4.35
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-21
EXCEL Output for Hernandez
New-Employee Training Problem
t-Test: Two-Sample Assuming Equal Variances
Mean
Variance
Observations
Pooled Variance
Hypothesized Mean Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Variable 1 Variable 2
4 7.73
56.5
19.495
18.27
15
12
18.957
0
25
- 5.20
1.12E-05
1.71
2.23E-05
2.06
10-22
Confidence Interval to Estimate 1 2 when 12 and 22 are unknown and
12 = 22
s (n1  1)  s (n2  1) 1 1
( x1  x2 )  t

n1  n2  2
n1 n2
2
1
2
2
where df  n1  n2  2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-23
Dependent Samples
• Before and after
measurements on
the same
individual
• Studies of twins
• Studies of spouses
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Individual
Before
After
1
32
39
2
11
15
3
21
35
4
17
13
5
30
41
6
38
39
7
14
22
10-24
Formulas for Dependent Samples
d D
t
sd
n
df  n  1
n  number of pairs
d = sample difference in pairs
D = mean population difference
st = standard deviation of sample difference
d 
sd 
d
n
(d  d ) 2
n 1
( d ) 2
d 
n
n 1
2

d = mean sample difference
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-25
P/E Ratios for Nine Randomly Selected
Companies
Company
2001 P/E Ratio
2002 P/E Ratio
1
8.9
12.7
2
38.1
45.4
3
43.0
10.0
4
34.0
27.2
5
34.5
22.8
6
15.2
24.1
7
20.3
32.3
8
19.9
40.1
9
61.9
106.5
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-26
Hypothesis Testing with Dependent
Samples: P/E Ratios for Nine Companies
Ho : D  0
Ha : D  0
Rejection
Region
  .01
df  n  1  9  1  8
t.005,6  3.355
If t < - 3.355 or t > 3.355, reject Ho.
If - 3.355  t  3.355, do not reject Ho.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Rejection
Region

.005
2

.005
2
Non Rejection Region
t
.01,11
 3.355
0
t
.01,11
 3.355
Critical Value
10-27
Hypothesis Testing with Dependent
Samples: P/E Ratios for Nine Companies
Company
2001 P/E
Ratio
2002 P/E
Ratio
d
1
8.9
12.7
-3.8
2
38.1
45.4
-7.3
3
43.0
10.0
33.0
4
34.0
27.2
6.8
5
34.5
22.8
11.7
6
15.2
24.1
-8.9
7
20.3
32.3
-12.0
8
19.9
40.1
-20.2
9
61.9
106.5
-44.6
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-28
Hypothesis Testing with Dependent
Samples: P/E Ratios for Nine Companies
d  5.033
sd  21.599
 5.033  0
t
 0.70
21.599
9
Since -3.355  t = -0.70  3.355, do not reject Ho
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-29
Hypothesis Testing with Dependent
Samples: P/E Ratios for Nine Companies
t-Test: Paired Two Sample for Means
2001 P/E
Ratio
2002 P/E
Ratio
Mean
30.64
35.68
Variance
268.1
837.5
9
9
Observations
Pearson Correlation
0.674
Hypothesized Mean Difference
0
df
8
t Stat
-0.7
P(T<=t) one-tail
0.252
t Critical one-tail
1.86
P(T<=t) two-tail
0.504
t Critical two-tail
2.306
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-30
Hypothesis Testing with Dependent
Samples: Demonstration Problem 10.5
Individual
Before
After
1
32
39
-7
2
11
15
-4
3
21
35
-14
4
17
13
4
5
30
41
-11
6
38
39
-1
7
14
22
-8
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
d
10-31
Hypothesis Testing with Dependent
Samples: Demonstration Problem 10.5
Ho: D  0
Ha: D  0
  .05
df  n  1  7  1  6
t.05,6  1.943
If t  - 1.943, reject Ho.
If t  -1.943, do not reject Ho.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Rejection
Region
  .05
Non Rejection Region
t
.05, 6
 1.943
0
Critical Value
10-32
Hypothesis Testing with Dependent
Samples: Demonstration Problem 10.5
d  5.857
sd  6.0945
t
 5.857  0
 2.54
6.0945
7
Since t = - 2.54  t c  -1.943, reject H 0 .
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-33
Confidence Intervals for Mean Difference
for Related Samples
s
d t
d
s
 D  d t
n
df  n  1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
d
n
10-34
Difference in Number of New-House Sales
d  3.39
sd  3.27
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Realtor
May 2001
May 2002
d
1
8
11
-3
2
19
30
-11
3
5
6
-1
4
9
13
-4
5
3
5
-2
6
0
4
-4
7
13
15
-2
8
11
17
-6
9
9
12
-3
10
5
12
-7
11
8
6
2
12
2
5
-3
13
11
10
1
14
14
22
-8
15
7
8
-1
16
12
15
-3
17
6
12
-6
18
10
10
0
10-35
Confidence Interval for Mean Difference
in Number of New-House Sales
df  n  1  18  1  17
t .005,17  2.898
d t
s
d
n
 D  d t
s
d
n
3.27
3.27
 D  3.39  2.898
18
18
 3.39  2.23  D  3.39  2.23
 3.39  2.898
 5.62  D  1.16
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-36
Sampling Distribution of Differences
in Sample Proportions
For large samples
n  pˆ  5,
2. n  q
ˆ  5,
3. n  p
ˆ  5, and
4. n  q
ˆ  5 where qˆ = 1 - pˆ
1.
1
1
1
1
2
2
2
2
the difference in sample proportion s is normally distribute d with
 pˆ

pp

p
ˆ
1
2
p q
n
1
σ pˆ

pˆ 2
1
1
1
1
and
2

p q
n
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
2
2
2
10-37
Z Formula for the Difference
in Two Population Proportions

pˆ  pˆ   p  p 
Z
1
2
1
p q
n
1
1
1

2
p q
n
2
2
2
pˆ  proportion from sample 1
pˆ  proportion from sample 2
n  size of sample 1
n  size of sample 2
p  proportion from population 1
p  proportion from population 2
q 1- p
q 1- p
1
2
1
2
1
2
1
2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
1
2
10-38
Z Formula to Test the Difference
in Population Proportions

pˆ  pˆ   p  p 
Z
1
P
2
1
2
 1

1
 p  q   
 n1 n2 
x1  x2
n n
pˆ  n pˆ
n

n n
1
1
2
1
2
1
2
2
q  1 p
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-39
Testing the Difference in Population
Proportions (Demonstration Problem 10.6)
pp
H :pp
Ho :
1
2
a
1
2
0
0
Rejection
Region
Rejection
Region

.005
2

.01

 .005
2
2
z.005  2.575
If z < - 2.575 or z > 2.575, reject Ho.
If - 2.575  z  2.575, do not reject Ho.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

.005
2
Non Rejection Region
Zzc  2.575
c
0
Zzc  2.575
c
Critical Values
10-40
Testing the Difference in Population
Proportions (Demonstration Problem 10.6)
n  95
x  39
 .41
ˆp  39
95
n  100
x  24
24
 .24
ˆp  100
2
1
1
2
1

ˆ  pˆ   p  p 
p
z
1
2

P
x x
n n
1
2
1
2
2
1
2
 1
1 

 p  q   
 n1 n2 
.24  .41  0
.323.677 
1
1 


100
95


 .17
.067
 2.54

24  39
100  95
 .323

Since - 2.575  z = - 2.54  2.575, do not reject Ho.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-41
Confidence Interval to Estimate p1 - p2
 pˆ  pˆ  z
1
2
pˆ qˆ  pˆ qˆ
n
n
1
1
1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
2
2
2

pp
1
2

 pˆ  pˆ  z
1
2
pˆ qˆ  pˆ qˆ
n
n
1
1
1
2
2
2
10-42
Example Problem:
When do men shop
for groceries?
n  400
n  480
x  48
x  187
48

 .39
pˆ 400  .12 pˆ  187
480
qˆ  1  pˆ  .88 qˆ  1  pˆ  .61
1
2
1
2
1
2
1
1
2
 pˆ  pˆ  Z
pˆ qˆ
n
1
2
For a 98% level of confidence , z = 2.33.
2
1
1
1

pˆ qˆ
n
2
2

pp
1
2

 pˆ  pˆ  Z
1
2
2
pˆ qˆ
n
1
1
1

pˆ qˆ
n
2
2
2
.12  .39  2.33 .12.88  .39.61  p1  p2  .12  .39  2.33 .12.88  .39.61
400
480
400
480
 .27  .064 
pp
 .334  p  p
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
1
2
1
2
 .27  .064
 .206
10-43
F Test for Two Population Variances
s12
F 2
s2
dfnumerator  1  n1  1
dfdeno min ator   2  n2  1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-44
F Distribution with 1 = 10 and 2 = 8
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.00
1.00
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
2.00
3.00
4.00
5.00
6.00
10-45
A Portion of the F Distribution Table
for  = 0.025
F
.025,9 ,11
Numerator Degrees of Freedom
1
2
3
4
Denominator
5
Degrees of Freedom 6
7
8
9
10
11
12
1
647.79
38.51
17.44
12.22
10.01
8.81
8.07
7.57
7.21
6.94
6.72
6.55
2
799.48
39.00
16.04
10.65
8.43
7.26
6.54
6.06
5.71
5.46
5.26
5.10
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
3
864.15
39.17
15.44
9.98
7.76
6.60
5.89
5.42
5.08
4.83
4.63
4.47
4
899.60
39.25
15.10
9.60
7.39
6.23
5.52
5.05
4.72
4.47
4.28
4.12
5
921.83
39.30
14.88
9.36
7.15
5.99
5.29
4.82
4.48
4.24
4.04
3.89
6
937.11
39.33
14.73
9.20
6.98
5.82
5.12
4.65
4.32
4.07
3.88
3.73
7
948.20
39.36
14.62
9.07
6.85
5.70
4.99
4.53
4.20
3.95
3.76
3.61
8
956.64
39.37
14.54
8.98
6.76
5.60
4.90
4.43
4.10
3.85
3.66
3.51
9
963.28
39.39
14.47
8.90
6.68
5.52
4.82
4.36
4.03
3.78
3.59
3.44
10-46
Sheet Metal Example: Hypothesis Test for
Equality of Two Population Variances (Part 1)
Ho :  12   22
Ha :  12   22
  0.05
n1  10
n 2  12
s12
F 2
s2
dfnumerator  1  n1  1
dfdeno min ator   2  n2  1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
F.025,9,11  3.59
F.05,9,11 =
1
F.05,9,11
1
3.59
 0.28

If F < 0.28 or F > 3.59, reject Ho.
If 0.28  F  3.59, do reject Ho.
10-47
Sheet metal Manufacturer (Part 2)
Rejection Regions
If F < 0.28 or F > 3.59, reject Ho.
If 0.28  F  3.59, do reject Ho.
Non Rejection
Region
F
.975,11,9
 0.28
F
.025,9 ,11
 359
.
Critical Values
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-48
Sheet Metal Example (Part 3)
Machine 1
n1  10
s12  0.1138
Machine 2
22.3
21.8
22.2
22.0
22.2
22.0
21.8
21.9
21.6
22.1
22.0
22.1
22.3
22.4
21.8
21.7
21.9
21.6
22.5
21.9
21.9
22.1
Fs
s
2
1
2
2
0.1138

 5.63
0.0202
n2  12
s22  0.0202
Since F = 5.63 > Fc = 3.59, reject Ho.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
10-49