chapter 10 ken black
DESCRIPTION
Chapter 10 Ken BlackTRANSCRIPT
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-1
Business Statistics, 4e by Ken Black
Chapter 10
Statistical Inferences about Two Populations
Discrete Distributions
-
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 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. 10-3
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. 10-4
Sampling Distribution of the
Difference Between Two Sample
Means
nx
x
1
1
Population 1
Population 2
nx
x
2
2
1 2X X
1X
2X
1 2X X
1x
1x
1x 2x
2x
2x
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-5
Sampling Distribution of the
Difference between Two Sample
Means
1 2X X1 2X X
1 2
1
2
1
2
2
2X X n n
1 2 1 2X X
2121
xx2
2
2
1
2
1
21 nnxx
21 xx 21 xx
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-6
Z Formula for the Difference
in Two Sample Means
nn
xxz
2
2
2
1
2
1
2121
When 12 and2
2 are known and Independent Samples
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-7
Hypothesis Testing for Differences Between
Means: The Wage Example (part 1)
1 2X X
Rejection
Region
Non Rejection Region
Critical Values
Rejection
Region
1 2X X
025.2
025.
2
H
H
o
a
:
:
1 2
1 2
0
0
21 xx
21 xx
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-8
Hypothesis Testing for Differences Between
Means: The Wage Example (part 2)
.Hz
.Hzz
o
o
reject not do 1.96, 1.96- If
reject 1.96, > or 1.96- < If
Rejection
Region
Non Rejection Region
Critical Values
Rejection
Region
96.1Z c0
96.1Z c
025.2
025.2
96.1cz 96.1cz
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-9
Hypothesis Testing for Differences
Between Means: The Wage Example
(part 3) Advertising Managers
74.256 57.791 71.115
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
75.932 74.194 86.741
80.742 65.360 57.351
39.672 73.904
45.652 54.270
93.083 59.045
63.384 68.508
164.264
253.16
700.70
32
2
1
1
1
1
x
n
411.166
900.12
187.62
34
2
2
2
2
2
x
n
Auditing Managers
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
99.198 67.160 71.804
61.254 37.386 72.401
73.065 59.505 56.470
48.036 72.790 67.814
60.053 71.351 71.492
66.359 58.653
61.261 63.508
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-10
Hypothesis Testing for Differences between
Means: The Wage Example (part 4)
35.2
34
411.166
32
253.256
0187.62700.70
2
2
2
1
2
1
2121
nS
nS
XXZ
.Hreject not do 1.96, Z 1.96- If
.Hreject 1.96, > or Z 1.96- < ZIf
o
o
.Hreject 1.96, > 2.35 = ZSince o
Rejection
Region
Non Rejection Region
Critical Values
Rejection
Region
cZ 2 33.
025.2
0 cZ 2 33.
025.2
.reject not do ,96.196.1 If
.reject ,96.1or 96.1 If
0
0
Hz
Hzz
35.2
34
411.166
32
253.256
(0)-62.187)-(70.700
()(
2
2
2
1
2
1
)2121
nn
xxz
.reject ,96.135.2 Since 0Hz
33.2cz 33.2cz
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-11
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
Observations 32 34
Hypothesized Mean Difference 0
z 2.35
P(Z
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-12
Demonstration Problem 10.1 (part 1)
H
H
o
a
:
:
1 2
1 2
0
0
Non Rejection Region
Critical Value
Rejection
Region
.001
cZ 3 08. 0 08.3cz
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-13
Demonstration Problem 10.1 (part 2)
Non Rejection Region
Critical Value
Rejection
Region
.001
cZ 3 08. 0
.H
.H
o
o
reject not do ,08.3 z If
reject 3.08,-
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-14
Confidence Interval to Estimate 1 - 2 When 1, 2 are known
nn
zxxnn
zxx2
2
2
1
2
1
21212
2
2
1
2
1
21
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-15
Demonstration Problem 10.2
88.142.4
50
99.2 2
50
46.396.16.2445.21
505096.16.2445.21
21
2
21
22
2
2
2
1
2
1
21212
2
2
1
2
1
21
99.246.3
nn
xxnn
xx zz
46.3
45.21
50
Re
1
1
1
x
n
gular
99.2
6.24
50
Pr
2
2
2
x
n
emium
1.96 = Confidence %95 z
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-16
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. 1
2 = 22
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-17
t Formula to Test the Difference in
Means Assuming 12 = 2
2
2121
2
2
21
2
1
2121
11
2
)1()1(
)()(
nnnn
nsns
xxt
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-18
Hernandez Manufacturing Company
(part 1)
H
H
o
a
:
:
1 2
1 2
0
0
If t < -2.060 or t > 2.060, reject H .
If - 2.060 t 2.060, do not reject H .
o
o
060.2
25212152
025.2
05.
2
25,25.0
21
t
nndf
Rejection
Region
Non Rejection Region
Critical Values
Rejection
Region
2025.
0 . , .025 25 2 060t
2025.
. ,.
025 252 060t
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-19
Hernandez Manufacturing Company
(part 2)
Training Method A
56 51 45
47 52 43
42 53 52
50 42 48
47 44 44
Training Method B
59
52
53
54
57
56
55
64
53
65
53
57
495.19
73.47
15
2
1
1
1
s
x
n
273.18
5.56
12
2
2
2
2
s
x
n
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-20
Hernandez Manufacturing Company
(part 3)
.Ht oreject -2.060, or 2.060- < If
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-21
MINITAB Output for Hernandez
New-Employee Training Problem
Twosample T for method A vs method B
N Mean StDev SE Mean
method A 15 47.73 4.42 1.1
method B 12 56.60 4.27 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. 10-22
EXCEL Output for Hernandez
New-Employee Training Problem
t-Test: Two-Sample Assuming Equal Variances
Variable 1 Variable 2
Mean 4 7.73 56.5
Variance 19.495 18.27
Observations 15 12
Pooled Variance 18.957
Hypothesized Mean Difference 0
df 25
t Stat - 5.20
P(T
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-23
Confidence Interval to Estimate 1 - 2 when 1
2 and 22 are unknown and
12 = 2
2
2 where
11
2
)1()1()(
21
2121
2
2
21
2
121
nndf
nnnn
nsnstxx
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-24
Dependent Samples
Before and after measurements on the same individual
Studies of twins
Studies of spouses
Individual
1
2
3
4
5
6
7
Before
32
11
21
17
30
38
14
After
39
15
35
13
41
39
22
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-25
Formulas for Dependent Samples
difference samplemean =
difference sample ofdeviation standard =
difference populationmean =
pairsin difference sample =
pairs ofnumber
1
d
s
D
d
n
ndf
n
s
Ddt
t
d
1
)(
1
)(
22
2
n
n
dd
n
dds
n
dd
d
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-26
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. 10-27
Hypothesis Testing with Dependent
Samples: P/E Ratios for Nine Companies
0:
0:
DH
DH
a
o
.Hreject not do 3.355, 3.355- If
.Hreject 3.355, > or 3.355- < If
o
o
t
tt
355.3
8191
01.
6,005.
t
ndf
Rejection
Region
Non Rejection Region
Critical Value
0 355.311,01. t
2005.
355.311,01.
t
Rejection
Region
2005.
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-28
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. 10-29
Hypothesis Testing with Dependent
Samples: P/E Ratios for Nine Companies
70.0
9
599.21
0033.5
599.21
033.5
t
s
d
d
oHreject not do ,355370.03553 . t = -.-Since
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-30
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
Observations 9 9
Pearson Correlation 0.674
Hypothesized Mean Difference 0
df 8
t Stat -0.7
P(T
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-31
Hypothesis Testing with Dependent
Samples: Demonstration Problem 10.5
Individual
1
2
3
4
5
6
7
Before
32
11
21
17
30
38
14
After
39
15
35
13
41
39
22
d
-7
-4
-14
4
-11
-1
-8
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-32
Hypothesis Testing with Dependent
Samples: Demonstration Problem 10.5
H D
H D
o
a
:
:
0
0
.reject not do -1.943, If
.reject 1.943,- If
o
o
Ht
Ht
943.1
6171
05.
6,05.
t
ndf
Rejection
Region
Non Rejection Region
Critical Value
0 943.1
6,05.t
05.
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-33
Hypothesis Testing with Dependent
Samples: Demonstration Problem 10.5
54.2
7
0945.6
0857.5
0945.6
857.5
t
s
d
d
.reject 1.943,- 2.54- = 0HttSince c
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-34
Confidence Intervals for Mean Difference
for Related Samples
1
ndf
ntdD
ntd ss dd
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-35
Difference in Number of New-House Sales 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
27.3
39.3
ds
d
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-36
Confidence Interval for Mean Difference
in Number of New-House Sales
16.162.5
23.239.323.239.3
18
27.3898.239.3
18
27.3898.239.3
898.2
171181
17,005.
D
D
D
ntdD
ntd
t
ndf
ss dd
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-37
Sampling Distribution of Differences
in Sample Proportions
n
qp
n
qp
pp
qn
pn
qn
pn
pp
pp
pq
2
22
1
11
21
22
22
11
11
and
withddistributenormally is sproportion samplein difference the
- 1 = where5 4.
and ,5 3.
,5 2.
,5 1.
samples largeFor
21
21
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-38
Z Formula for the Difference
in Two Population Proportions
pq
pq
p
p
n
n
p
p
n
qp
n
qp
ppppZ
22
11
2
1
2
1
2
1
2
22
1
11
2121
- 1
- 1
2 population from proportion
1 population from proportion
2 sample of size
1 sample of size
2 sample from proportion
1 sample from proportion
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-39
Z Formula to Test the Difference
in Population Proportions
pq
P
qp
Z
nn
pnpn
nnxx
nn
pppp
1
11
21
2211
21
21
21
2121
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-40
Testing the Difference in Population
Proportions (Demonstration Problem 10.6)
0:
0:
21
21
pp
pp
a
o
H
H
.reject not do 2.575, 2.575- If
.reject 2.575, > or 2.575- < If
o
o
Hz
Hzz
575.2
005.2
01.
2
005.
z
Rejection
Region
Non Rejection Region
Critical Values
Rejection
Region
2005.
0 cZ 2 575.
2005.
cZ 2 575.czcz
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-41
Testing the Difference in Population
Proportions (Demonstration Problem 10.6)
24.100
24
24
100
1
1
1
p
x
n
41.95
39
39
95
2
2
2
p
x
n
323.
95100
3924
21
21
nnxxP
54.2
067.
17.
95
1
100
1677.323.
041.24.
11
21
2121
nn
pppp
qp
z
.Horeject not do 2.575, 2.54- = z 2.575- Since
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-42
Confidence Interval to Estimate p1 - p2
n
qp
n
qppppp
n
qp
n
qppp zz
2
22
1
11
21212
22
1
11
21
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-43
Example Problem:
When do men shop
for groceries?
88.1
12.400
48
48
400
11
1
1
1
pq
p
x
n
61.1
39.480
187
187
480
22
2
2
2
pq
p
x
n
206.334.
064.27.064.27.
480
61.39.
400
88.12.33.239.12.
480
61.39.
400
88.12.33.239.12.
21
21
21
2
22
1
11
21212
22
1
11
21
pp
pp
pp
n
qp
n
qppppp
n
qp
n
qppp ZZ
2.33. = z ,confidence of level 98% aFor
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-44
F Test for Two Population Variances
1
1
22min
11
2
2
2
1
ndf
ndf
s
sF
atordeno
numerator
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-45
F Distribution with 1 = 10 and 2 = 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.00 1.00 2.00 3.00 4.00 5.00 6.00
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-46
A Portion of the F Distribution Table
for = 0.025
Numerator Degrees of Freedom
Denominator
Degrees of Freedom
. , ,025 9 11F
1 2 3 4 5 6 7 8 9
1 647.79 799.48 864.15 899.60 921.83 937.11 948.20 956.64 963.28
2 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39
3 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47
4 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90
5 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68
6 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52
7 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82
8 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36
9 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03
10 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78
11 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59
12 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-47
Sheet Metal Example: Hypothesis Test for
Equality of Two Population Variances (Part 1)
22
21
22
21
:
:
a
o
H
H 59.3 11,9,025. F
.HFIf
.HFFIf
o
o
reject do ,59.3 0.28
reject ,3.59 > or 0.28<
28.0
59.3
1
1 =
11,9,05.
11,9,05.
FF
1
1
22min
11
2
2
2
1
ndf
ndf
s
sF
atordeno
numerator
12
10
05.0
2
1
n
n
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-48
Sheet metal Manufacturer (Part 2)
Rejection Regions
Critical Values
. , ,.
025 9 11359F
Non Rejection
Region
. , ,.
975 11 90 28F
.reject do ,59.3 0.28
.reject ,3.59 > or 0.28<
o
o
HFIf
HFFIf
-
Business Statistics, 4e, by Ken Black. 2003 John Wiley & Sons. 10-49
Sheet Metal Example (Part 3)
Machine 1
22.3 21.8 22.2
21.8 21.9 21.6
22.3 22.4
21.6 22.5
Machine 2
22.0
22.1
21.8
21.9
22.2
22.0
21.7
21.9
22.0
22.1
21.9
22.1
1138.0
10
2
1
1
s
n
0202.0
12
2
2
2
s
n63.5
0202.0
1138.02
2
2
1
ssF
.HFF oc reject 3.59, = > 5.63 = Since