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Transcript of 1 Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001....
1Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH
EDITIONEDITION
ELEMENTARY STATISTICSChapter 7 Hypothesis Testing
2Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Chapter 7Hypothesis 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
3Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7-1 Overview
Definition
Hypothesis
in statistics, is a claim or statement about a property of a population
4Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.
5Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7-2
Fundamentals of
Hypothesis Testing
6Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
µx = 98.6
Figure 7-1 Central Limit TheoremThe Expected Distribution of Sample Means
Assuming that = 98.6
Likely sample means
7Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
z = - 1.96
x = 98.48
or
z = 1.96
x = 98.72
or
µx = 98.6
Figure 7-1 Central Limit TheoremThe Expected Distribution of Sample Means
Assuming that = 98.6
Likely sample means
8Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Figure 7-1 Central Limit TheoremThe Expected Distribution of Sample Means Assuming that
= 98.6
z = - 1.96
x = 98.48
or
z = 1.96
x = 98.72
or
Sample data: z = - 6.64
x = 98.20 or
µx = 98.6
Likely sample means
9Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Components of aFormal Hypothesis
Test
10Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
11Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Alternative Hypothesis: H1
Must be true if H0 is false
, <, >
‘opposite’ of Null
12Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.
13Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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).
14Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Test Statistic
a value computed from the sample data that is used in making the decision about the
rejection of the null hypothesis
15Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
16Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Critical RegionSet of all values of the test statistic that
would cause a rejection of the null hypothesis
17Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Critical RegionSet of all values of the test statistic that
would cause a rejection of thenull hypothesis
CriticalRegion
18Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Critical RegionSet of all values of the test statistic that
would cause a rejection of the null hypothesis
CriticalRegion
19Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Critical RegionSet of all values of the test statistic that
would cause a rejection of the null hypothesis
CriticalRegions
20Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
21Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Critical ValueValue 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
22Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Critical Value
Critical Value( z score )
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
23Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Critical Value
Critical Value( z score )
Fail to reject H0Reject H0
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
24Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Two-tailed,Right-tailed,Left-tailed Tests
The tails in a distribution are the extreme regions bounded
by critical values.
25Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Two-tailed TestH0: µ = 100
H1: µ 100
26Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Two-tailed TestH0: µ = 100
H1: µ 100 is divided equally between
the two tails of the critical region
27Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Two-tailed TestH0: µ = 100
H1: µ 100
Means less than or greater than
is divided equally between the two tails of the critical
region
28Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Two-tailed TestH0: µ = 100
H1: µ 100
Means less than or greater than
100
Values that differ significantly from 100
is divided equally between the two tails of the critical
region
Fail to reject H0Reject H0 Reject H0
29Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Right-tailed TestH0: µ 100
H1: µ > 100
30Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Right-tailed TestH0: µ 100
H1: µ > 100
Points Right
31Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Right-tailed TestH0: µ 100
H1: µ > 100
Values that differ significantly
from 100100
Points Right
Fail to reject H0 Reject H0
32Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Left-tailed Test
H0: µ 100
H1: µ < 100
33Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Left-tailed Test
H0: µ 100
H1: µ < 100Points Left
34Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Left-tailed Test
H0: µ 100
H1: µ < 100
100
Values that differ significantly
from 100
Points Left
Fail to reject H0Reject H0
35Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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
36Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
FIGURE 7-4 Wording of Final Conclusion
Does theoriginal claim contain
the condition ofequality
Doyou rejectH0?.
Yes
(Original claim
contains equality
and becomes H0)
No(Original claimdoes not containequality and
becomes H1)
Yes
(Reject H0)
“There is sufficientevidence to warrantrejection of the claimthat. . . (original claim).”
“There is not sufficientevidence to warrantrejection of the claimthat. . . (original claim).”
“The sample datasupports the claim that . . . (original claim).”
“There is not sufficientevidence to support the claimthat. . . (original claim).”
Doyou reject
H0?
Yes
(Reject H0)
No(Fail to
reject H0)
No(Fail to
reject H0)
(This is theonly case inwhich theoriginal claimis rejected).
(This is theonly case inwhich theoriginal claimis supported).
Start
37Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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)
38Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Type I ErrorThe 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
39Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Type II Errorthe 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
40Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Table 7-2 Type I and Type II Errors
True State of Nature
We decide to
reject the
null hypothesis
We fail to
reject the
null hypothesis
The null
hypothesis is
true
The null
hypothesis is
false
Type I error
(rejecting a true
null hypothesis)
Type II error
(rejecting a false
null hypothesis)
Correct
decision
Correct
decision
Decision
41Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.
42Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.