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Chapter Eight
Hypothesis Testing
McGraw-Hill/Irwin Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.
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Hypothesis Testing
8.1 Null and Alternative Hypotheses and Errors in Testing
8.2 Large Sample Tests about a Mean:Testing a One-Sided Alternative Hypothesis
8.3 Large Sample Tests about a Mean:Testing a Two-Sided Alternative Hypothesis
8.4 Small Sample Tests about a Population Mean
8.5 Hypothesis Tests about a Population Proportion
*8.6 Type II Error Probabilities and Sample Size Determination
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8.1 Null and Alternative Hypotheses
The null hypothesis, denoted H0, is a statement of the basic proposition being tested. The statement generally represents the status quo and is not rejected unless there is convincing sample evidence that it is false.
The alternative or research hypothesis, denoted Ha, is an alternative (to the null hypothesis) statement that will be accepted only if there is convincing sample evidence that it is true.
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Types of Hypothesis
One-Sided, Less Than
H0: 19.5 Ha: < 19.5 (Accounts Receivable)
One-Sided, Greater Than
H0: 50 Ha: > 50 (Trash Bag)
Two-Sided, Not Equal To
H0: = 4.5 Ha: 4.5 (Camshaft)
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Type I and Type II Errors
State of Nature
Conclusion H0 True H0 False
Reject H0 Type I Error
Correct Decision
Do not Reject H0 Correct Decision
Type II Error
Type I Error Rejecting H0 when it is trueType II Error Failing to reject H0 when it is false
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8.2 Large Sample Tests about a Mean:Testing a One-Sided Alternative Hypothesis
n/
-x=z 0
Test Statistic
If the sampled population is normal or if n is large, we can reject H0: = 0 at the level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds.
Alternative Reject H0 if:
If unknown and n is large, estimate by s.
:0
0:
a
a
HH
zz
zz
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Example: One-Sided, Greater Than
Testing H0: 50 versus Trash Bag
Ha: > 50 for = 0.05 and = 0.01
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Example: The p-Value for “Greater Than”
Testing H0: 50 vs Ha: > 50 using rejection points and p-value. Trash Bag
The p-value or the observed level of significance is the probability of observing a value of the test statistic greater than or equal to z when H0 is true. It measures the weight of the evidence against the null hypothesis and is also the smallest value of for which we can reject H0.
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Example: One-Sided, Less Than
Testing H0: 19.5 versus Accts Rec
Ha: < 19.5 for = 0.01
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Large Sample Tests about Mean: p-Values
If the sampled population is normal or if n is large, we can reject H0: = 0 at the level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than .
n/
-x=z 0
Test Statistic
If unknown and n is large, estimate by s.
Alternative Reject H0 if: p-Value
0
0
0
:::
a
a
a
HHH
2/2/
2/
or
isthat,
zzzz
zz
zz
zz
zofrightcurve normalstdunderarea Twice
zofleftcurvenormalstdunderArea
zofrightcurvenormalstdunderArea
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Five Steps of Hypothesis Testing
1) Determine null and alternative hypotheses2) Specify level of significance (probability of Type
I error) 3) Select the test statistic that will be used. Collect
the sample data and compute the value of the test statistic.
4) Use the value of the test statistic to make a decision using a rejection point or a p-value.
5) Interpret statistical result in (real-world) managerial terms
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Example: Two-Sided, Not Equal to
Testing H0: = 4.5 versus Camshaft
Ha: 4.5 for = 0.05
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8.5 Small Sample Tests about a Population Mean
If the sampled population is normal, we can reject H0: = 0 at the level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than .
ns /
-x=t 0
Test Statistic
t, t/2 and p-values are based on n – 1 degrees of freedom.
Alternative Reject H0 if:
0
0
0
:::
a
a
a
HHH
2/2/
2/
or
isthat,
tttt
tt
tt
tt
p-Value
tofrightondistributit underarea Twice
tofleftondistributit underArea
tofrightondistributit underArea
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Example: Small Sample Test about a Mean
Testing H0: 18.8 vs Ha: < 18.8 using rejection points and p-value.Credit Card Interest Rates
97.415/538.1
18.8-16.827=
/
-x=t 0
ns
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8.5 Hypothesis Tests about a Population Proportion
npp )1(
p-p̂=z
00
0
Test Statistic
If the sample size n is large, we can reject H0: p = p0 at the level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than .
Alternative Reject H0 if:
0
0
0
:::
ppHppHppH
a
a
a
2/2/
2/
or
isthat,
zzzz
zz
zz
zz
p-Value
zofrightcurve normalstdunderarea Twice
zofleftcurvenormalstdunderArea
zofrightcurvenormalstdunderArea
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Example: Hypothesis Tests about a Proportion
Testing H0: p 0.70 versus Ha: p > 0.70
using rejection points and p-value.Using Phantol, proportion of patients with reduced severity and duration of viral infections.
65.2
300)70.01(70.0
0.70-0.77=
)1(
p-p̂=z
00
0
npp
09.365.2z,33.265.2z,645.165.2z 001.01.05. zzz
004.0)4960.05.0()65.2P(z value-p
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*8.6 Type II Error Probabilities
Testing H0: vs Ha: < 3
(Amount of Coffee in 3-Pound Can)
, Probability of Type II Error, Given = 2.995, = 0.05.
995299591262
35
0147064513
050
050
.μ|.xP
..xP
nzμxP
zn/
- xPβ=
.
.
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How Type II Error Varies Against Alternatives
Testing H0: vs Ha: < 3
(Amount of Coffee in 3-Pound Can)
, Probability of Type II Error ( = 0.05)
Given = 2.995, Given = 2.990, Given = 2.985,
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Summary: Selecting an Appropriate Test Statistic
for a Test about a Population Mean
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Hypothesis Testing
8.1 Null and Alternative Hypotheses and Errors in Testing
8.2 Large Sample Tests about a Mean:Testing a One-Sided Alternative Hypothesis
8.3 Large Sample Tests about a Mean:Testing a Two-Sided Alternative Hypothesis
8.4 Small Sample Tests about a Population Mean
8.5 Hypothesis Tests about a Population Proportion
*8.6 Type II Error Probabilities and Sample Size Determination
Summary:
