Download Chapter 7 Hypothesis Testing

ELEMENTARY
STATISTICS
Chapter 7
Hypothesis Testing
MARIO F. TRIOLA
EIGHTH
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
EDITION 1
Chapter 7
Hypothesis Testing
7-1 Overview
7-2 Fundamentals of Hypothesis Testing
7-3 Testing a Claim about a Mean: Large
Samples
7-4 Testing a Claim about a Mean: Small
Samples
7-5 Testing a Claim about a Proportion
7-6 Testing a Claim about a Standard
Deviation
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
7-1
Overview
Definition
Hypothesis
in statistics, is a claim or statement about
a property of a population
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
3
Rare Event Rule for Inferential
Statistics
If, under a given assumption, the
probability of a particular observed event
is exceptionally small, we conclude that
the assumption is probably not correct.
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
4
7-2
Fundamentals of
Hypothesis Testing
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
Figure 7-1
Central Limit Theorem
The Expected Distribution of Sample Means
Assuming that  = 98.6
Likely sample means
µx = 98.6
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
Figure 7-1
Central Limit Theorem
The Expected Distribution of Sample Means
Assuming that  = 98.6
Likely sample means
µx = 98.6
z = - 1.96
or
x = 98.48
z=
1.96
or
x = 98.72
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
Figure 7-1
Central Limit Theorem
The Expected Distribution of Sample Means
Assuming that  = 98.6
Sample data: z = - 6.64
or
x = 98.20
Likely sample means
µx = 98.6
z = - 1.96
or
x = 98.48
z=
1.96
or
x = 98.72
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Components of a
Formal Hypothesis
Test
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
9
Null Hypothesis: H0
 Statement about value
of population parameter
 Must contain condition of equality
 =, , or 
 Test the Null Hypothesis directly
 Reject H0 or fail to reject H0
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Alternative Hypothesis: H1
 Must be true if H0 is false
 , <, >
 ‘opposite’ of Null
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Note about Forming Your Own Claims
(Hypotheses)
If you are conducting a study and want
to use a hypothesis test to support your
claim, the claim must be worded so that
it becomes the alternative hypothesis.
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
Note about Testing the Validity of
Someone Else’s Claim
Someone else’s claim may become the
null hypothesis (because it contains
equality), and it sometimes becomes the
alternative hypothesis (because it does
not contain equality).
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
13
Test Statistic
a value computed from the sample data that is
used in making the decision about the
rejection of the null hypothesis
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
Test Statistic
a value computed from the sample data that is
used in making the decision about the
rejection of the null hypothesis
For large samples, testing claims about
population means
z=
x - µx

n
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
15
Critical Region
Set of all values of the test statistic that
would cause a rejection of the
null hypothesis
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
16
Critical Region
Set of all values of the test statistic that
would cause a rejection of the
null hypothesis
Critical
Region
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
17
Critical Region
Set of all values of the test statistic that
would cause a rejection of the
null hypothesis
Critical
Region
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
18
Critical Region
Set of all values of the test statistic that
would cause a rejection of the
null hypothesis
Critical
Regions
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
19
Significance Level
 denoted by 
 the probability that the
test statistic will fall in the
critical region when the null
hypothesis is actually true.
 common choices are 0.05,
0.01, and 0.10
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
20
Critical Value
Value or values that separate the critical region
(where we reject the null hypothesis) from the
values of the test statistics that do not lead
to a rejection of the null hypothesis
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
21
Critical Value
Value or values that separate the critical region
(where we reject the null hypothesis) from the
values of the test statistics that do not lead
to a rejection of the null hypothesis
Critical Value
( z score )
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
22
Critical Value
Value or values that separate the critical region
(where we reject the null hypothesis) from the
values of the test statistics that do not lead
to a rejection of the null hypothesis
Reject H0
Fail to reject H0
Critical Value
( z score )
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
23
Two-tailed,Right-tailed,
Left-tailed Tests
The tails in a distribution are the
extreme regions bounded
by critical values.
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Two-tailed Test
H0: µ = 100
H1: µ  100
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
25
Two-tailed Test
H0: µ = 100
H1: µ  100
 is divided equally between
the two tails of the critical
region
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
26
Two-tailed Test
H0: µ = 100
H1: µ  100
 is divided equally between
the two tails of the critical
region
Means less than or greater than
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
27
Two-tailed Test
H0: µ = 100
 is divided equally between
the two tails of the critical
region
H1: µ  100
Means less than or greater than
Reject H0
Fail to reject H0
Reject H0
100
Values that differ significantly from 100
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
28
Right-tailed Test
H0: µ  100
H1: µ > 100
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
29
Right-tailed Test
H0: µ  100
H1: µ > 100
Points Right
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
30
Right-tailed Test
H0: µ  100
H1: µ > 100
Points Right
Fail to reject H0
100
Reject H0
Values that
differ significantly
from 100
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
31
Left-tailed Test
H0: µ  100
H1: µ < 100
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
32
Left-tailed Test
H0: µ  100
H1: µ < 100
Points Left
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
33
Left-tailed Test
H0: µ  100
H1: µ < 100
Points Left
Reject H0
Values that
differ significantly
from 100
Fail to reject H0
100
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
34
Conclusions
in Hypothesis Testing
always test the null hypothesis
1. Reject the H0
2. Fail to reject the H0
need to formulate correct wording of final
conclusion
See Figure 7-4
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
35
Wording of Final Conclusion
FIGURE 7-4
Start
Does the
original claim contain
the condition of
equality
Yes
(Original claim
contains equality
and becomes H0)
No
Do
you reject
H0?.
“There is sufficient
evidence to warrant
(Reject H0) rejection of the claim
that. . . (original claim).”
Yes
No
(Fail to
reject H0)
(Original claim
does not contain
equality and
becomes H1)
Do
you reject
H0?
Yes
(Reject H0)
“There is not sufficient
evidence to warrant
rejection of the claim
that. . . (original claim).”
“The sample data
supports the claim that
. . . (original claim).”
No
(Fail to
reject H0)
(This is the
only case in
which the
original claim
is rejected).
(This is the
only case in
which the
original claim
is supported).
“There is not sufficient
evidence to support
the claim
that. . . (original claim).”
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
36
Accept versus Fail to Reject
some texts use “accept the null
hypothesis
we are not proving the null hypothesis
sample evidence is not strong enough
to warrant rejection (such as not
enough evidence to convict a suspect)
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
37
Type I Error
The mistake of rejecting the null hypothesis
when it is true.
 (alpha) is used to represent the probability
of a type I error
Example: Rejecting a claim that the mean
body temperature is 98.6 degrees when the
mean really does equal 98.6
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
38
Type II Error
the mistake of failing to reject the null
hypothesis when it is false.
ß (beta) is used to represent the probability of
a type II error
Example: Failing to reject the claim that the
mean body temperature is 98.6 degrees when
the mean is really different from 98.6
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
39
Table 7-2
Type I and Type II Errors
True State of Nature
We decide to
reject the
null hypothesis
The null
hypothesis is
true
The null
hypothesis is
false
Type I error
(rejecting a true
null hypothesis)

Correct
decision
Correct
decision
Type II error
(rejecting a false
null hypothesis)

Decision
We fail to
reject the
null hypothesis
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
40
Controlling Type I and Type II Errors
For any fixed , an increase in the sample
size n will cause a decrease in 
For any fixed sample size n , a decrease in 
will cause an increase in . Conversely, an
increase in  will cause a decrease in  .
To decrease both  and , increase the
sample size.
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
41
Definition
Power of a Hypothesis Test
is the probability (1 - ) of rejecting a
false null hypothesis, which is
computed by using a particular
significance level  and a particular
value of the mean that is an alternative
to the value assumed true in the null
hypothesis.
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
42