Download Chapter 7 Hypothesis Testing

STATISTICS
Chapter 8
Hypothesis Testing
C.M. Pascual
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
1
Chapter 8
Hypothesis Testing
8-1 Overview
8-2 Fundamentals of Hypothesis Testing
8-3 Testing a Claim about a Mean: Large
Samples
8-4 Testing a Claim about a Mean: Small
Samples
8-5 Testing a Claim about a Proportion
8-6 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
8-1
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
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8-2
Fundamentals of
Hypothesis Testing
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Figure 8-1
Central Limit Theorem
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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Figure 8-1
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
7
Figure 8-1
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
8
Figure 8-1
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
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Components of a
Formal Hypothesis
Test
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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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
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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
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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
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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
15
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
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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
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
Region
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
19
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
20
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
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
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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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
23
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
24
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
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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
27
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
28
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
29
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
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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
31
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
32
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
33
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
34
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
35
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 8-4
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
36
Wording of Final Conclusion
FIGURE 8-4
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
37
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
38
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
39
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
40
Table 8-2
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
41
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
42
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
43
Steps in Hypothesis Testing
1. State the null and alternative hypothesis;
2. Select the level of significance;
3. Determine the critical value and the rejection
region/s;
4. State the decision rule;
5. Compute the test statistics; and
6. Make a decision, whether to reject or not to
reject the null hypothesis.
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
44
Example 1
• A manufacturer claims that the average
lifetime of his lightbulbs is 3 years or 36
months. The stabdard deviation is 8 months.
Fifty (50) bulbs are selected, and the average
lifetime is found to be 32 months. Should the
manufacturer’s statement be rejected at =
001?
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
45
Example 1
• Solution:
• Step 1. State the hypothesis:
– Ho: µ = 36 months
– Ha : µ  36 months
• Step 2. Level of significance =001
• Step 3. Determine critical values and rejection
region
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
46
Example 1
• Solution:
• Step 3. Determine critical values and rejection
region
– Z = +/- 2.575 (from Appendix B of z values)
• Step 4. State the decision rule
– Reject the null hypothesis if Zc > 2.575 or Zc = 2.575
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
47
Example 1
• Solution:
• Step 5. Compute the test statistic.
zc =
x - µx

n
Zc = (32-36)/ (8/(50)0.5 = - 3.54
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
48
Example 1
• Solution:
• Step 6. Make a decision.
Zc = - 3.54 is less than Z = -2.575
And it falls in the rejection region in the left tail.
Therefore, reject Ho and conclude that the average
lifetime of lightbulbs is not equal to 36 months.
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
49
Example 1
• Solution:
• Step 6. Make a decision.
Zc = - 3.54 is less than Z = -2.575
And it falls in the rejection region in the left tail.
Therefore, reject Ho and conclude that the average
lifetime of lightbulbs is not equal to 36 months.
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
50
Example 2
• A test on car braking reaction times for men
between 18 to 36 years old have produced a
mean and standard deviation of 0.610 second
and 0.123 second, respectively. When 40 male
drivers of this age group were randomly
selected and tested for their breaking reaction
times, a mean of 0.587 second came out. At
the  = 0.10, test the claim of the driving
instructor that his graduates had faster reaction
times.
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
51
Example 2
• Solution:
• Step 1. State the hypothesis:
– Ho: µ = 0.610 second
– Ha: µ < 0.610 second
• Step 2. Level of significance =010
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
52
Example 2
• Solution:
• Step 3. Determine critical values and rejection
region
Z = - 1/.28 (from Appendix B of z values)
• Step 4. State the decision rule
– Reject the null hypothesis if Zc < - 1.28
.
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
53
Example 2
• Solution:
• Step 5. Compute the test statistic.
zc =
x - µx

n
Zc = (0.587-0.610)/ (0.123/(40)0.5 = - 1.18
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
54
Example 2
• Solution:
• Step 6.
– Since the test statistics falls within the non-critical
region, do not reject Ho. There is enough evidence
to support the instructor’s claim.; accept Ho.
.
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
55
Test on Small Sample Mean
• The t-test is a statistical test for the mean of a
population and is used when the population is
normally distributed, σ is unknown, and n <
30. The formula for the t-test with degrees of
freedom are d.f. = n – 1 is
t=
x-µ
s
n
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
56
Example 3
• In order to increase customer service, a muffler
repair shop claim its mechanics can replace a
muffler in 12 minutes. A time management
specialist selected 6 repair jobs and found their
mean time to be 11.6 minutes. The standard
deviation of the sample was 2.1 minutes. A 
= 0.025, is there enough evidence to conclude
that the mean time in changing a muffler is
less than 12 minutes?
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
57
Example 3
Solution
1. State the hypothesis:
1. Ho: µ = 12
2. Ha: µ < 12
2. Step 2. Level of significance =0025
3. Step 3. Since and d.f. = 6 – 1 = 5, then at =
0025,Appendix C at t-value = - 2.571
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
58
Example 1
Solution
4. Step 4. Reject Ho if tc < - 2.571
5. Compute for the test statistic
t=
x-µ
s
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
59
Example 3
t = (11.6 – 12)/(2.1(6)0.5 = - 0.47
• Step 6. Since the critical value fall within the
non-critical region, do not reject Ho. Accept
Ho.
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
60
Sit Work (Submit after class)
1. A diet clinic states that there is an average
loss of 24 pounds for those who stay on the
program for 20 weeks. The standard deviation
is 5 pounds. The clinic tries a new diet,
reducing salt intake to see whether that
strategy will produce a greater weight loss. A
group of 40 volunteers loses an average of
16.3 pounds each over 20 weeks. Should the
clinic change the new diet? Use  = 0.05
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
61
Assignment (Submit next meeting)
2. A recent survey stated that household received
an average 37 telephone calls per month. To
test the claim, a researcher surveyed 29
households and found that the average
number of calls was 34.9. The standard
deviation of the sample was 6. At = 0.05,
can the claim be substantiated?
Chapter 7. Section 7-1 and 7-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
62