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Chapter
10
Hypothesis
Tests Regarding
a Parameter
Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc.
Section
10.1
The Language of
Hypothesis
Testing
Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc.
Objectives
1. Determine the null and alternative
hypotheses
2. Explain Type I and Type II errors
3. State conclusions to hypothesis tests
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Objective 1
• Determine the Null and Alternative
Hypotheses
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A hypothesis is a statement regarding a
characteristic of one or more populations.
Ex: A newspaper headline makes the claim
that most workers get their jobs through
networking.
In this chapter, we look at hypotheses
regarding a single population parameter.
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Examples of Claims Regarding a
Characteristic of a Single Population
• In 2008, 62% of American adults regularly volunteered their
time for charity work. A researcher believes that this percentage
is different today.
Source: ReadersDigest.com poll created on 2008/05/02
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Examples of Claims Regarding a
Characteristic of a Single Population
• In 2008, 62% of American adults regularly volunteered their time
for charity work. A researcher believes that this percentage is
different today.
• According to a study published in March, 2006 the mean length
of a phone call on a cellular telephone was 3.25 minutes. A
researcher believes that the mean length of a call has increased
since then.
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Examples of Claims Regarding a
Characteristic of a Single Population
• In 2008, 62% of American adults regularly volunteered their time
for charity work. A researcher believes that this percentage is
different today.
• According to a study published in March, 2006 the mean length
of a phone call on a cellular telephone was 3.25 minutes. A
researcher believes that the mean length of a call has increased
since then.
• Using an old manufacturing process, the standard deviation of the
amount of wine put in a bottle was 0.23 ounces. With new
equipment, the quality control manager believes the standard
deviation has decreased.
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CAUTION!
We test these types of statements using sample
data because it is usually impossible or
impractical to gain access to the entire
population. If population data are available,
there is no need for inferential statistics.
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Hypothesis testing is a procedure, based
on sample evidence and probability, used to
test statements regarding a characteristic of
one or more populations.
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Steps in Hypothesis Testing
1. Make a statement regarding the nature of the
population.
2. Collect evidence (sample data) to test the
statement.
3. Analyze the data to assess the plausibility of the
statement.
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The null hypothesis, denoted H0, is a
statement to be tested. The null hypothesis
is a statement of no change, no effect or no
difference and is assumed true until
evidence indicates otherwise.
Its states that the value of a population
parameter(mean,proportion…) is equal to
some claimed value
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The alternative hypothesis, denoted H1, is
a statement that we are trying to find
evidence to support.
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In this chapter, there are three ways to set up the
null and alternative hypotheses:
1.Equal versus not equal hypothesis (two-tailed test)
H0: parameter = some value
H1: parameter ≠ some value
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In this chapter, there are three ways to set up the
null and alternative hypotheses:
1. Equal versus not equal hypothesis (two-tailed test)
H0: parameter = some value
H1: parameter ≠ some value
2. Equal versus less than (left-tailed test)
H0: parameter = some value
H1: parameter < some value
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In this chapter, there are three ways to set up the
null and alternative hypotheses:
1. Equal versus not equal hypothesis (two-tailed test)
H0: parameter = some value
H1: parameter ≠ some value
2. Equal versus less than (left-tailed test)
H0: parameter = some value
H1: parameter < some value
3. Equal versus greater than (right-tailed test)
H0: parameter = some value
H1: parameter > some value
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“In Other Words”
The null hypothesis is a statement of status
quo or no difference and always contains a
statement of equality. The null hypothesis is
assumed to be true until we have evidence to
the contrary. We seek evidence that supports
the statement in the alternative hypothesis.
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Parallel Example 2: Forming Hypotheses
For each of the following claims, determine the null and alternative
hypotheses. State whether the test is two-tailed, left-tailed or righttailed.
a)
In 2008, 62% of American adults regularly volunteered their
time for charity work. A researcher believes that this
percentage is different today.
b)
According to a study published in March, 2006 the mean length
of a phone call on a cellular telephone was 3.25 minutes. A
researcher believes that the mean length of a call has increased
since then.
c)
Using an old manufacturing process, the standard deviation of
the amount of wine put in a bottle was 0.23 ounces. With new
equipment, the quality control manager believes the standard
deviation has decreased.
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Solution
a)
In 2008, 62% of American adults regularly volunteered
their time for charity work. A researcher believes that this
percentage is different today.
The hypothesis deals with a population proportion, p. If
the percentage participating in charity work is no different
than in 2008, it will be 0.62 so the null hypothesis is H0:
p=0.62.
Since the researcher believes that the percentage is
different today, the alternative hypothesis is a two-tailed
hypothesis: H1: p≠0.62.
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Solution
b)
According to a study published in March, 2006 the mean
length of a phone call on a cellular telephone was 3.25
minutes. A researcher believes that the mean length of a
call has increased since then.
The hypothesis deals with a population mean, μ. If the
mean call length on a cellular phone is no different than in
2006, it will be 3.25 minutes so the null hypothesis is
H0: μ = 3.25.
Since the researcher believes that the mean call length has
increased, the alternative hypothesis is:
H1: μ > 3.25, a right-tailed test.
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Solution
c)
Using an old manufacturing process, the standard
deviation of the amount of wine put in a bottle was 0.23
ounces. With new equipment, the quality control manager
believes the standard deviation has decreased.
The hypothesis deals with a population standard deviation,
σ. If the standard deviation with the new equipment has
not changed, it will be 0.23 ounces so the null hypothesis
is H0: σ = 0.23.
Since the quality control manager believes that the
standard deviation has decreased, the alternative
hypothesis is: H1: σ < 0.23, a left-tailed test.
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Objective 2
• Explain Type I and Type II Errors
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Four Outcomes from Hypothesis Testing
1. Reject the null hypothesis when the alternative
hypothesis is true. This decision would be correct.
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Four Outcomes from Hypothesis Testing
1. Reject the null hypothesis when the alternative
hypothesis is true. This decision would be correct.
2. Do not reject the null hypothesis when the null
hypothesis is true. This decision would be correct.
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Four Outcomes from Hypothesis Testing
1. Reject the null hypothesis when the alternative
hypothesis is true. This decision would be correct.
2. Do not reject the null hypothesis when the null
hypothesis is true. This decision would be correct.
3. Reject the null hypothesis when the null hypothesis is
true. This decision would be incorrect. This type of
error is called a Type I error.
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Four Outcomes from Hypothesis Testing
1. Reject the null hypothesis when the alternative
hypothesis is true. This decision would be correct.
2. Do not reject the null hypothesis when the null
hypothesis is true. This decision would be correct.
3. Reject the null hypothesis when the null hypothesis is
true. This decision would be incorrect. This type of
error is called a Type I error.
4. Do not reject the null hypothesis when the alternative
hypothesis is true. This decision would be incorrect.
This type of error is called a Type II error.
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Parallel Example 3: Type I and Type II Errors
For each of the following claims, explain what it would
mean to make a Type I error. What would it mean to make
a Type II error?
a) In 2008, 62% of American adults regularly
volunteered their time for charity work. A researcher
believes that this percentage is different today.
b) According to a study published in March, 2006 the
mean length of a phone call on a cellular telephone
was 3.25 minutes. A researcher believes that the mean
length of a call has increased since then.
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Solution
a)
In 2008, 62% of American adults regularly volunteered
their time for charity work. A researcher believes that this
percentage is different today.
H0: p=0.62
H1: p≠0.62
A Type I error is made if the researcher concludes that
p ≠ 0.62 when the true proportion of Americans 18 years
or older who participated in some form of charity work is
currently 62%.
A Type II error is made if the sample evidence leads the
researcher to believe that the current percentage of
Americans 18 years or older who participated in some
form of charity work is still 62% when, in fact, this
percentage differs from 62%.
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Solution
b)
According to a study published in March, 2006 the mean
length of a phone call on a cellular telephone was 3.25
minutes. A researcher believes that the mean length of a
call has increased since then.
A Type I error occurs if the sample evidence leads the
researcher to conclude that μ > 3.25 when, in fact, the
actual mean call length on a cellular phone is still 3.25
minutes.
A Type II error occurs if the researcher fails to reject the
hypothesis that the mean length of a phone call on a
cellular phone is 3.25 minutes when, in fact, it is longer
than 3.25 minutes.
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α = P(Type I Error)
= P(rejecting H0 when H0 is true)
β = P(Type II Error)
= P(not rejecting H0 when H1 is true)
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The probability of making a Type I error, α,
is chosen by the researcher before the
sample data is collected.
The level of significance, α, is the
probability of making a Type I error.
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“In Other Words”
As the probability of a Type I error
increases, the probability of a Type II
error decreases, and vice-versa.
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Objective 3
• State Conclusions to Hypothesis Tests
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CAUTION!
We never “accept” the null hypothesis,
because, without having access to the entire
population, we don’t know the exact value of
the parameter stated in the null. Rather, we
say that we do not reject the null hypothesis.
This is just like the court system. We never
declare a defendant “innocent”, but rather say
the defendant is “not guilty”.
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Parallel Example 4: Stating the Conclusion
According to a study published in March, 2006 the mean
length of a phone call on a cellular telephone was 3.25
minutes. A researcher believes that the mean length of a
call has increased since then.
a) Suppose the sample evidence indicates that the null
hypothesis should be rejected. State the wording of
the conclusion.
b) Suppose the sample evidence indicates that the null
hypothesis should not be rejected. State the wording
of the conclusion.
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Solution
a) Suppose the sample evidence indicates that the null
hypothesis should be rejected. State the wording of
the conclusion.
The statement in the alternative hypothesis is that the
mean call length is greater than 3.25 minutes. Since the
null hypothesis (μ = 3.25) is rejected, there is sufficient
evidence to conclude that the mean length of a phone call
on a cell phone is greater than 3.25 minutes.
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Solution
b) Suppose the sample evidence indicates that the null
hypothesis should not be rejected. State the wording
of the conclusion.
Since the null hypothesis (μ = 3.25) is not rejected, there is
insufficient evidence to conclude that the mean length of a
phone call on a cell phone is greater than 3.25 minutes. In
other words, the sample evidence is consistent with the
mean call length equaling 3.25 minutes.
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