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Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-1
Business Statistics, 4eby Ken Black
Chapter 9
Statistical Inference: Hypothesis Testing
for Single Populations
Discrete Distributions
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-2
Learning Objectives• Understand the logic of hypothesis testing, and know how
to establish null and alternate hypotheses.• Understand Type I and Type II errors, and know how to
solve for Type II errors.• Know how to implement the HTAB system to test
hypotheses.• Test hypotheses about a single population mean when is
known.• Test hypotheses about a single population mean when is
unknown.• Test hypotheses about a single population proportion.• Test hypotheses about a single population variance.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-3
Types of Hypotheses
• Research Hypothesis– a statement of what the researcher believes will
be the outcome of an experiment or a study.• Statistical Hypotheses
– a more formal structure derived from the research hypothesis.
• Substantive Hypotheses– a statistically significant difference does not
imply or mean a material, substantive difference.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-4
Example Research Hypotheses
• Older workers are more loyal to a company• Companies with more than $1 billion of
assets spend a higher percentage of their annual budget on advertising than do companies with less than $1 billion of assets.
• The price of scrap metal is a good indicator of the industrial production index six months later.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-5
Statistical Hypotheses• Two Parts
– a null hypothesis– an alternative hypothesis
• Null Hypothesis – nothing new is happening
• Alternative Hypothesis – something new is happening
• Notation– null: H0
– alternative: Ha
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-6
Null and Alternative Hypotheses
• The Null and Alternative Hypotheses are mutually exclusive. Only one of them can be true.
• The Null and Alternative Hypotheses are collectively exhaustive. They are stated to include all possibilities. (An abbreviated form of the null hypothesis is often used.)
• The Null Hypothesis is assumed to be true.• The burden of proof falls on the Alternative
Hypothesis.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-7
Null and Alternative Hypotheses: Example
• A manufacturer is filling 40 oz. packages with flour.
• The company wants the package contents to average 40 ounces.
ozH
ozH
a 40:
40:0
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-8
• One-tailed Tests
One-tailed and Two-tailed Tests
40:
40:0
aH
H
18.0:
18.0:0
pH
pH
a
12:
12:0
aH
H
• Two-tailed Test
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-9
HTAB System to Test Hypotheses
Task 1:HYPOTHESIZE
Task 2:TEST
Task 3:TAKE STATISTICAL ACTION
Task 4:DETERMINING THE
BUSINESS IMPLICATIONS
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-10
Steps in Testing Hypotheses
1. Establish hypotheses: state the null and alternative hypotheses.
2. Determine the appropriate statistical test and sampling distribution.
3. Specify the Type I error rate (4. State the decision rule.5. Gather sample data.6. Calculate the value of the test statistic.7. State the statistical conclusion.8. Make a managerial decision.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-11
HTAB Paradigm – Task 1
Task 1: Hypotheses
Step 1. Establish hypotheses: state the null and alternative hypotheses.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-12
HTAB Paradigm – Task 2
Task 2: Test
Step 2. Determine the appropriate statistical test and sampling distribution.
Step 3. Specify the Type I error rate (Step 4. State the decision rule.Step 5. Gather sample data.Step 6. Calculate the value of the test
statistic.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-13
HTAB Paradigm – Task 3
Task 3: Take Statistical Action
Step 7. State the statistical conclusion.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-14
HTAB Paradigm – Task 4
Task 4: Determine the business implications
Step 8. Make a managerial decision.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-15
Rejection and Non Rejection Regions
=40 oz
Non Rejection Region
Rejection Region
Critical Value
Rejection Region
Critical Value
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-16
Type I and Type II Errors
• Type I Error– Rejecting a true null hypothesis – The probability of committing a Type I error is
called , the level of significance.
• Type II Error– Failing to reject a false null hypothesis– The probability of committing a Type II error is
called .
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-17
Decision Table for Hypothesis Testing
(
( )
Null True Null False
Fail toreject null
CorrectDecision
Type II error)
Reject null Type I error
Correct Decision
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-18
One-tailed Tests
40:
40:0
aH
H
40:
40:0
aH
H
=40 oz
Rejection Region
Non Rejection Region
Critical Value
=40 oz
Rejection Region
Non Rejection Region
Critical Value
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-19
Two-tailed Tests
40:
40:
a
o
H
H
=12 oz
Rejection Region
Non Rejection Region
Critical Values
Rejection Region
40:
40:0
aH
H
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-20
CPA Net Income Example: Two-tailed Test (Part 1)
914,74$:
914,74$:
a
0
H
HRejection Region
Non Rejection Region
=0
Zc 196.
Rejection Region
Zc 196.
2
025.2
025.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-21
CPA Net Income Example: Two-tailed Test (Part 2)
.Hreject not do ,96.1 If
.Hreject ,96.1 If
0
0
c
c
zz
zz
75.2
112
530,14914,74695,78
n
xz
.Hreject 1.96, = z 2.75 = z 0c
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-22
CPA Net Income Example:Critical Value Method (Part 1)
605,77112
530,1496.1914,74
n
zx
Upper
cc
914,74$:
914,74$:0
aH
H
223,72112
530,1496.1914,74
n
zx
Lower
cc
Rejection Region
Non Rejection Region
=0 Zc 196.
Rejection Region
Zc 196.
2
025.2
025.
72,223 77,605
96.1cz 0z 96.1cz
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-23
CPA Net Income Example:Critical Value Method (Part 2)
.Hreject not do ,605,7777,223 If
.Hreject ,605,77or 223,77 If
0
0
x
x x
.Hreject ,605,77695,78 Since o cxx
Rejection Region
Non Rejection Region
=0 Zc 196.
Rejection Region
Zc 196.
2
025.2
025.
72,223 77,605
96.1cz 0z 96.1cz
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-24
Demonstration Problem 9.1: z Test (Part 1)
30.4:
30.4:0
aH
H
Rejection Region
Non Rejection Region
0
=.05
Zc 1645.645.1cz
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-25
Demonstration Problem 9.1: z Test (Part 2)
Rejection Region
Non Rejection Region
0
=.05
Zc 1645.
.reject not do ,6451 If
.reject ,6451 If
0
0
H.z
H.z
42.1
32
574.030.4156.4
n
xxz
.reject not do
,645142.1
0H
.z
645.1cz
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-26
Demonstration Problem 9.1: Critical Value (Part 1)
30.4:
30.4:0
aH
HRejection Region
Non Rejection Region
0
=.05
Zc 1645.
cx 4133. 4.30
133.432
574.0)645.1(30.4
n
zxc
645.1cz
133.4cx
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-27
Demonstration Problem 9.1: Critical Value (Part 2)
Rejection Region
Non Rejection Region
0
=.05
Zc 1645.
cx 4133. 4.30
.reject not do ,133.4 If
.reject ,133.4 If
0
0
Hx
Hx
.reject not do ,133.4156.4 0
Hx
645.1cz
133.4cx
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-28
Rejection Region
Non Rejection Region
0
=.05
Demonstration Problem 9.1: Using the p-Value
30.4:
30.4:0
aH
H
.reject not do , value- If
.reject , < value- If
0
0
Hp
Hp
.reject not do
.05, = > .0778 = value- Since
0H
p
0778.)42.1(
42.1
32
574.030.4156.4
zpn
xz
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-29
Demonstration Problem 9.1: MINITAB
Test of mu = 4.300 vs mu < 4.300The assumed sigma = 0.574
Variable N MEAN STDEV SE MEAN Z P VALUERatings 32 4.156 0.574 0.101 -1.42 0.078
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-30
Demonstration Problem 9.1: Excel (Part 1)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-31
Demonstration Problem 9.1: Excel (Part 2)
H0: = 4.3
Ha: < 4.3
3 4 5 5 4 5 5 4
4 4 4 4 4 4 4 5
4 4 4 3 4 4 4 3
5 4 4 5 4 4 4 5
n = =COUNT(A4:H7)
= 0.05
Mean = =AVERAGE(A4:H7)
S = =STDEV(A4:H7)
Std Error = =B12/SQRT(B9)
Z = =(B11-B1)/B13
p-Value =NORMSDIST(B14)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-32
Two-tailed Test: Unknown, = .05 (Part 1)
Weights in Pounds of a Sample of 20 Plates
22.622.2 23.2 27.4 24.527.026.6 28.1 26.9 24.926.225.3 23.1 24.2 26.125.830.4 28.6 23.5 23.6
20 = and 2.1933,= ,51.25 nsx
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-33
Two-tailed Test: Unknown, = .05 (part 2)
Critical Values
Non Rejection Region
Rejection Regions
ct 2 093. ct 2 093.
2
025.2
025.
H
H
o
a
:
:
25
25
df n 1 19
25:
25:0
aH
H
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-34
Two-tailed Test: Unknown, = .05 (part 3)
tX
S
n
2551 25 0
2 1933
20
104. .
. .
Since t do not reject H .o 104 2 093. . ,Critical Values
Non Rejection Region
Rejection Regions
ct 2 093. ct 2 093.
2
025.2
025.
If t reject H .
If t do not reject H .
o
o
2 093
2 093
. ,
. , .reject not do 2.093, If
.reject 2.093, If
0
0
Ht
Ht
04.1
20
1933.20.2551.25
n
sx
t
.reject not do ,093.204.1 Since0
Ht
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-35
MINITAB Computer Printout for the Machine Plate Example
Test of mu = 25.000 vs mu not = 25.000
Variable N MEAN STDEV SE MEAN T P VALUEPlatewt 20 25.510 2.193 0.490 1.04 0.31
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-36
Machine Plate Example: Excel(Part 1)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-37
Machine Plate Example: Excel(Part 2)
A B C D E
1 H0: = 25
2 Ha: 253
4 22.6 22.2 23.2 27.4 24.5
5 27 26.6 28.1 26.9 24.9
6 26.2 25.3 23.1 24.2 26.1
7 25.8 30.4 28.6 23.5 23.68
9 n = =COUNT(A4:E7)
10 = 0.05
11 Mean = =AVERAGE(A4:E7)
12 S = =STDEV(A4:E7)
13 Std Error = =B12/SQRT(B9)
14 t = =(B11-B1)/B13
15 p-Value =TDIST(B14,B9-1,2)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-38
Demonstration Problem 9.2 (Part 1)
Size in Acres of 23 Farms
445 489 474 505 553 477 545463 466 557 502 449 438 500466 477 557 433 545 511 590561 560
23 = and 46.94,= ,78.498 nsx
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-39
Demonstration Problem 9.2 (Part 2)
471:
471:0
aH
H
df n 1 22
Critical Value
Non Rejection Region
Rejection Region
ct 1717.
.05
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-40
Demonstration Problem 9.2 (Part 3)
.reject not do ,717.1 If
.reject ,717.1 If
0
0
Ht
Ht
84.2
23
94.4647178.498
n
sx
t
.reject ,717.184.2 Since0
Ht Critical Value
Non Rejection Region
Rejection Region
ct 1717.
.05
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-41
z Test of Population Proportion
pq
p
pn
qp
ppz
-1
proportion population
proportion sampleˆ :where
ˆ
5
and ,5
qn
pn
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-42
Testing Hypotheses about a Proportion: Manufacturer Example
(Part 1)
08.:
08.:
a
0
pH
pH
cZ 1645.
Critical Values
Non Rejection Region
Rejection Regions
cZ 1645.
2
05. 2
05.
cz cz
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-43
Testing Hypotheses about a Proportion: Manufacturer Example
(Part 2)
.
. .
(. )(. ).
p
Zp P
P Qn
33
200165
165 08
08 92200
4 43
If Z reject H .
If Z do not reject H .
o
o
1645
1645
. ,
. ,
Since Z reject H .o 4 43 1645. . ,
cZ 1645.
Critical Values
Non Rejection Region
Rejection Regions
cZ 1645.
2
05. 2
05.
cz cz
.0
0
reject not do 1.645, If
.reject ,645.1 If
Hz
Hz
43.4
200)92)(.08(.
08.165.ˆ
165.200
33ˆ
nqp
ppz
p
.reject ,645.143.4 Since0
Hz
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-44
Demonstration Problem 9.3 (Part 1)
H P
H P
o
a
: .
: .
17
17
Critical Value
Non Rejection Region
Rejection Region
cZ 1645.
.0517.:
17.:0
pH
pH
a
cz
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-45
Demonstration Problem 9.3 (Part 2)
.
. .
(. )(. ).
p
Zp P
P Qn
115
550209
209 17
17 83550
2 44
If reject H .
If do not reject H .
o
o
Z
Z
1645
1645
. ,
. ,
Since Z = 2.44 reject H .o1645. ,Critical Value
Non Rejection Region
Rejection Region
cZ 1645.
.05
cz
.reject not do ,645.1 If
.reject ,645.1 If
0
0
Hz
Hz
44.2
550)83)(.17(.
17.209.ˆ
209.550
115ˆ
nqp
ppz
p
.reject ,645.144.2 Since 0Hz
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-46
Hypothesis Test for 2:
Demonstration Problem 9.4 (Part 1)
0
df = 15
.05
.05
.95
7.26094 24.9958
H
H
o
a
:
:
2
2
25
25
25:
25:2
2
0
0
H
H
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-47
Hypothesis Test for 2:
Demonstration Problem 9.4 (Part 2)
0
df = 15
.05
.05
.95
7.26094 24.9958
If or reject H .
If 7.26094 do not reject H .
2 2o
2o
7 26094 24 9958
24 9958
. . ,
. ,
22
2
1 15 281
251686
n S .
.
Since
do not reject H .
2
o
1686 24 995805 15
2. . ,
. ,
86.1625
)1.28)(15()1(2
22
sn
.0
0
reject not do ,9958.2427.26094 If
.reject ,9958.242or 26094.72 If
H
H
.reject not do
,9958.2486.16 Since
0
2
15,05.
2
H
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-48
Solving for Type II Errors: The Beverage Example
H
H
o
a
:
:
12
12
Rejection Region
Non Rejection Region=0
=.05
Zc 1645.
c cX Z n
12 1645010
6011979
( . ).
.
If X reject H .
If X do not reject H .
o
o
11979
11979
. ,
. ,
cz
12:
12:0
aH
H
11.979 60
10.0)645.1(12
n
zx cc
.reject not do ,979.11 If
.reject ,979.11 If
0
0
Hx
Hx
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-49
Type II Error for Beverage Example with =11.99 oz
=.05
Reject Ho Do Not Reject Ho
Ho is True
Ho is False
95%
=.8023
CorrectDecision
Type IError
Type IIError
CorrectDecision 19.77%
X
Z0
Z1
0z
1z
x
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-50
Type II Error for Demonstration Problem 9.5, with =11.96 oz
=.05
Ho is True
Ho is False
95%
Reject Ho Do Not Reject Ho
=.0708
CorrectDecision
Type IError
Type IIError
CorrectDecision 92.92%
X
Z0
Z1
0z
1z
x
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-51
Values and Power Values for the Soft-Drink Example
Power
11.999 .94 .06
11.995 .89 .11
11.990 .80 .20
11.980 .53 .47
11.970 .24 .76
11.960 .07 .93
11.950 .01 .99
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-52
Operating Characteristic Curve for the Soft-Drink Example
0
0.10.2
0.3
0.4
0.50.6
0.7
0.80.9
1
11.95 11.96 11.97 11.98 11.99 12
Pro
bab
ilit
y
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9-53
Power Curve for the Soft-Drink Example
0
0.10.2
0.3
0.4
0.50.6
0.7
0.80.9
1
11.95 11.96 11.97 11.98 11.99 12
Pro
bab
ilit
y
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