Download Chapter 7 Hypothesis Testing

Overview
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Definitions
In statistics, a hypothesis is a claim or
statement about a property of a population.
A hypothesis test (or test of significance) is a
standard procedure for testing a claim about a
property of a population.
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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.
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Key Concept
This section presents individual components of a hypothesis
test, and the following sections use those components in
comprehensive procedures.
The role of the following should be understood:
 null hypothesis
 alternative hypothesis
 test statistic
 critical region
 significance level
 critical value
 P-value
 Type I and II error
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Components of a
Formal Hypothesis
Test
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Null Hypothesis:
H0
 The null hypothesis (denoted by H0) is
a statement that the value of a
population parameter (such as
proportion, mean, or standard
deviation) is equal to some claimed
value.
 We test the null hypothesis directly.
 Either reject H0 or fail to reject H0.
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Alternative Hypothesis:
H1
 The alternative hypothesis (denoted
by H1 or Ha or HA) is the statement that
the parameter has a value that
somehow differs from the null
hypothesis.
 The symbolic form of the alternative
hypothesis must use one of these
symbols: , <, >.
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Example: Use the given claims to express the
corresponding null and alternative hypotheses
in symbolic form.
a) The proportion of drivers who admit to running red
lights is greater than 0.5.
b) The mean height of professional basketball players
is at most 7 ft.
c) The standard deviation of IQ scores of actors is
equal to 15.
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a) The proportion of drivers who admit to running
red lights is greater than 0.5.
•
•
•
In Step 1, we express the given claim as p > 0.5.
In Step 2, we see that if p > 0.5 is false, then p 
0.5 must be true.
In Step 3, we see that the expression p > 0.5
does not contain equality, so we let the
alternative hypothesis H1 be p > 0.5, and we let
H0 be p = 0.5.
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b) The mean height of professional basketball
players is at most 7 ft.
• In Step 1, we express “a mean of at most 7 ft”
in symbols as   7.
• In Step 2, we see that if   7 is false, then µ > 7
must be true.
• In Step 3, we see that the expression µ > 7
does not contain equality, so we let the
alternative hypothesis H1 be µ > 7, and we let H0
be µ = 7.
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c) The standard deviation of IQ scores of actors is
equal to 15.
• In Step 1, we express the given claim as  = 15.
• In Step 2, we see that if  = 15 is false, then  
15 must be true.
• In Step 3, we let the alternative hypothesis H1 be
  15, and we let H0 be  = 15.
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Test Statistic
The test statistic is a value used in making
a decision about the null hypothesis, and is
found by converting the sample statistic to
a score with the assumption that the null
hypothesis is true.
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Test Statistic - Formulas

z=p-p

z=
pq
n
x - µx

Test statistic for
proportions
Test statistic
for mean
n
2 =
(n – 1)s2 Test statistic

2
for standard
deviation
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Example: A survey of n = 880 randomly
selected adult drivers showed that 56%
(or p = 0.56) of those respondents
admitted to running red lights. Find the
value of the test statistic for the claim
that the majority of all adult drivers
admit to running red lights.
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Solution: The preceding example showed that
the given claim results in the following null and
alternative hypotheses: H0: p = 0.5 and H1: p >
0.5.
we work under the assumption that the null
hypothesis is true with p = 0.5, we get the
following test statistic:
z = p – p = 0.56 - 0.5 = 3.56


pq
n

(0.5)(0.5)
880
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Interpretation: We know from
previous chapters that a z score of
3.56 is exceptionally large. It appears
that in addition to being “more than
half,” the sample result of 56% is
significantly more than 50%.
See figure following.
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Critical Region, Critical Value,
Test Statistic
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Critical Region
The critical region (or rejection region) is the
set of all values of the test statistic that
cause us to reject the null hypothesis. For
example, see the red-shaded region in the
previous figure.
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Significance Level
The significance level (denoted by ) is the
probability that the test statistic will fall in the
critical region when the null hypothesis is
actually true.
Common choices for  are 0.05, 0.01, and
0.10.
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Two-tailed, Right-tailed,
Left-tailed Tests
The tails in a distribution are the extreme
regions bounded by critical values.
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Two-tailed Test
H0: =
H1:

 is divided equally between
the two tails of the critical
region
Means less than or greater than
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Right-tailed Test
H0: =
H1: >
Points Right
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Left-tailed Test
H0: =
H1: <
Points Left
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P-Value
The P-value (or p-value or probability value)
is the probability of getting a value of the test
statistic that is at least as extreme as the one
representing the sample data, assuming that
the null hypothesis is true. The null
hypothesis is rejected if the P-value is very
small, such as 0.05 or less.
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Decision Criterion
Traditional method:
Reject H0 if the test statistic falls
within the critical region.
Fail to reject H0 if the test statistic
does not fall within the critical
region.
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Decision Criterion - cont
P-value method:
Reject H0 if the P-value   (where 
is the significance level, such as
0.05).
Fail to reject H0 if the P-value > .
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Decision Criterion - cont
Confidence Intervals:
Because a confidence interval
estimate of a population parameter
contains the likely values of that
parameter, reject a claim that the
population parameter has a value
that is not included in the
confidence interval.
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Procedure for Finding P-Values
Figure 8.6
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Procedure for Finding P-Values
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Example: Finding P-values. First determine
whether the given conditions result in a righttailed test, a left-tailed test, or a two-tailed test,
then find the P-values and state a conclusion
about the null hypothesis.
a) A significance level of  = 0.05 is used in testing the
claim that p > 0.25, and the sample data result in a test
statistic of z = 1.18.
b) A significance level of  = 0.05 is used in testing the
claim that p  0.25, and the sample data result in a test
statistic of z = 2.34.
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Example: Finding P-values. First determine
whether the given conditions result in a righttailed test, a left-tailed test, or a two-tailed test,
then find the P-values and state a conclusion
about the null hypothesis.
a) With a claim of p > 0.25, the test is right-tailed.
Because the test is right-tailed, Figure 8-6 shows that
the P-value is the area to the right of the test statistic
z = 1.18. We refer to Table A-2 and find that the area
to the right of z = 1.18 is 0.1190. The P-value of 0.1190
is greater than the significance level  = 0.05, so we
fail to reject the null hypothesis. The P-value of
0.1190 is relatively large, indicating that the sample
results could easily occur by chance.
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Example: Finding P-values. First determine
whether the given conditions result in a righttailed test, a left-tailed test, or a two-tailed test,
then find the P-values and state a conclusion
about the null hypothesis.
b) With a claim of p  0.25, the test is two-tailed.
Because the test is two-tailed, and because the test
statistic of z = 2.34 is to the right of the center,
Figure 8-6 shows that the P-value is twice the area to
the right of z = 2.34. We refer to Table A-2 and find that
the area to the right of z = 2.34 is 0.0096, so P-value =
2 x 0.0096 = 0.0192. The P-value of 0.0192 is less than
or equal to the significance level, so we reject the null
hypothesis. The small P-value o 0.0192 shows that the
sample results are not likely to occur by chance.
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Wording of Final Conclusion
Figure 8-7
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Accept Versus Fail to Reject
 Some texts use “accept the null
hypothesis”
 We are not proving the null hypothesis.
 The sample evidence is not strong
enough to warrant rejection
(such as not enough evidence to
convict a suspect).
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Type I Error
 A Type I error is the mistake of
rejecting the null hypothesis when it
is true.
 The symbol  (alpha) is used to
represent the probability of a type I
error.
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Type II Error
 A Type II error is the mistake of failing
to reject the null hypothesis when it is
false.
 The symbol  (beta) is used to
represent the probability of a type II
error.
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Suppose that we are interested in the burning rate of a solid propellant.
Now burning rate is a random variable that can be described
by a probability distribution. we are interested in deciding whether or not
the mean burning rate is 50 centimeters per second.
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How to Compute Type I and II
Errors
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When n = 10:
Increasing n to 16:
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Considering a mean value closer
to the hypothetical mean (50.5)
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Summary
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Definition
The 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 population parameter that is an
alternative to the value assumed true in the
null hypothesis.
That is, the power of the hypothesis test is the
probability of supporting an alternative
hypothesis that is true.
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Comprehensive
Hypothesis Test –
P-Value Method
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Comprehensive
Hypothesis Test –
Traditional Method
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Comprehensive
Hypothesis Test - cont
A confidence interval estimate of a population
parameter contains the likely values of that
parameter. We should therefore reject a claim
that the population parameter has a value that
is not included in the confidence interval.
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Testing a Claim About a
Proportion
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Test Statistic for Testing
a Claim About a Proportion

z=
p–p
pq
n
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P-Value Method
Use the same method as described in Section 8-2
and in Figure 8-8.
Use the standard normal distribution (Table A-2).
Traditional Method
Use the same method as described in Section
8-2 and in Figure 8-9.
Confidence Interval Method
Use the same method as described in Section
8-2 and in Table 8-2.
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Example: An article distributed by the
Associated Press included these results from
a nationwide survey: Of 880 randomly
selected drivers, 56% admitted that they run
red lights. The claim is that the majority of all
Americans run red lights. That is, p > 0.5.

The sample data are n = 880, and p = 0.56.
np = (880)(0.5) = 440  5
nq = (880)(0.5) = 440  5
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Example: An article distributed by the Associated
Press included these results from a nationwide
survey: Of 880 randomly selected drivers, 56%
admitted that they run red lights. The claim is that
the majority of all Americans run red lights. That is,

p > 0.5. The sample data are n = 880, and p = 0.56.
We will use the P-value Method.
H0: p = 0.5
H1: p > 0.5
 = 0.05
z=

p–p
= 0.56 – 0.5
pq
(0.5)(0.5)
n
880
= 3.56
Referring to Table A-2, we see that for values of z = 3.50
and higher, we use 0.9999 for the cumulative area to the left
of the test statistic. The P-value is 1 – 0.9999 = 0.0001.
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Example: An article distributed by the Associated
Press included these results from a nationwide
survey: Of 880 randomly selected drivers, 56%
admitted that they run red lights. The claim is that
the majority of all Americans run red lights. That is,

p > 0.5. The sample data are n = 880, and p = 0.56.
We will use the P-value Method.
H0: p = 0.5
H1: p > 0.5
 = 0.05
z=

p–p
= 0.56 – 0.5
pq
(0.5)(0.5)
n
880
= 3.56
Since the P-value of 0.0001 is less than the significance
level of  = 0.05, we reject the null hypothesis.
There is sufficient evidence to support the claim.
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Example: An article distributed by the Associated
Press included these results from a nationwide
survey: Of 880 randomly selected drivers, 56%
admitted that they run red lights. The claim is that
the majority of all Americans run red lights. That is,

p > 0.5. The sample data are n = 880, and p = 0.56.
We will use the P-value Method.
H0: p = 0.5
H1: p > 0.5
 = 0.05
z = 3.56
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Example: An article distributed by the Associated
Press included these results from a nationwide
survey: Of 880 randomly selected drivers, 56%
admitted that they run red lights. The claim is that
the majority of all Americans run red lights. That is,

p > 0.5. The sample data are n = 880, and p = 0.56.
We will use the Traditional Method.
H0: p = 0.5
H1: p > 0.5
 = 0.05
z=

p–p
= 0.56 – 0.5
pq
(0.5)(0.5)
n
880
= 3.56
This is a right-tailed test, so the critical region is an area of
0.05. We find that z = 1.645 is the critical value of the
critical region. We reject the null hypothesis.
There is sufficient evidence to support the claim.
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Example: An article distributed by the Associated
Press included these results from a nationwide
survey: Of 880 randomly selected drivers, 56%
admitted that they run red lights. The claim is that
the majority of all Americans run red lights. That is,

p > 0.5. The sample data are n = 880, and p = 0.56.
We will use the confidence interval method.
For a one-tailed hypothesis test with significance level ,
we will construct a confidence interval with a confidence
level of 1 – 2. We construct a 90% confidence interval.
We obtain 0.533 < p < 0.588. We are 90% confident that
the true value of p is contained within the limits of 0.533
and 0.588. Thus we support the claim that p > 0.5.
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CAUTION
When testing claims about a population proportion,
the traditional method and the P-value method are
equivalent and will yield the same result since they
use the same standard deviation based on the claimed
proportion p. However, the confidence interval uses
an estimated standard deviation based upon the sample

proportion p. Consequently, it is possible that the
traditional and P-value methods may yield a different
conclusion than the confidence interval method.
A good strategy is to use a confidence interval to
estimate a population proportion, but use the P-value
or traditional method for testing a hypothesis.
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