Chapter 10 Hypothesis Testing

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Chapter 10Hypothesis Testing

• 10.1
• The Language of Hypothesis Testing

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Objectives

• Determine the null and alternative hypotheses
  from a claim
• Understand Type I and Type II errors
• Understand the probability of making Type I and
  Type II errors
• State conclusions to hypothesis tests

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Steps in Hypothesis Testing 1. A claim is made.
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Steps in Hypothesis Testing 1. A claim is
made. 2. Evidence (sample data) is collected in
order to test the claim.
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Steps in Hypothesis Testing 1. A claim is
made. 2. Evidence (sample data) is collected in
order to test the claim. 3. The data is analyzed
in order to support or refute the claim.
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Hypothesis Testing
A hypothesis is a statement or claim regarding a
characteristic of one or more populations. In
this chapter, we look at hypotheses regarding a
single population.
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Examples of Claims Regarding a Characteristic of
a Single Population

• In 1997, 43 of Americans 18 years or older
  participated in some form of charity work. A
  researcher believes that this percentage
  different today.

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Examples of Claims Regarding a Characteristic of
a Single Population

• In 1997, 43 of Americans 18 years or older
  participated in some form of charity work. A
  researcher believes that this percentage
  different today.
• In June, 2001 the mean length of a phone call
  on a cellular telephone was 2.62 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 1997, 43 of Americans 18 years or older
  participated in some form of charity work. A
  researcher believes that this percentage
  different today.
• In June, 2001 the mean length of a phone call
  on a cellular telephone was 2.62 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 claims using sample data
because it is usually impossible or impractical
to gain access to the entire population. If
population data is available, then inferential
statistics is not necessary.
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Example of Claims Regarding a Characteristic of a
Single Population

• Consider the researcher who believes that the
  mean length of a cell phone call has increased
  from its June, 2001 mean of 2.62 minutes.
• To test this claim, the researcher might obtain a
  simple random sample of 36 cell phone calls.
• Suppose he determines the mean length of the
  phone calls is 2.70 minutes.
• Is this enough evidence to conclude the length of
  a phone call has increased?

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Example of Claims Regarding a Characteristic of a
Single Population
Assuming the length of the phone call is 2.62
minutes and the standard deviation of the phone
call is known to be 0.78 minutes in June, 2001 .

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What if our sample resulted in a sample mean of
2.95 minutes?
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Hypothesis Testing
Hypothesis testing is a procedure, based on
sample evidence and probability, used to test
claims regarding a characteristic of one or more
populations. Hypothesis testing is based on two
types of hypothesis.
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Hypothesis Testing
The null hypothesis, denoted Ho (read
H-naught), is a statement to be tested. The
null hypothesis is assumed true until evidence
indicates otherwise. In this chapter, it will be
a statement regarding the value of a population
parameter. The alternative hypothesis, denoted,
H1 (read H-one), is a claim to be tested. We
are trying to find evidence for the alternative
hypothesis. In this chapter, it will be a claim
regarding the value of a population parameter.
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In this chapter, there are three ways to set up
the null and alternative hypothesis. 1. Equal
versus not equal hypothesis (two-tailed
test) Ho 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 hypothesis. 1. Equal
versus not equal hypothesis (two-tailed
test) Ho parameter some value H1 parameter
? some value 2. Equal versus less than
(left-tailed test) Ho parameter some
value H1 parameter lt some value
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In this chapter, there are three ways to set up
the null and alternative hypothesis. 1. Equal
versus not equal hypothesis (two-tailed
test) Ho parameter some value H1 parameter
? some value 2. Equal versus less than
(left-tailed test) Ho parameter some
value H1 parameter lt some value 3. Equal
versus greater than (right-tailed test) Ho
parameter some value H1 parameter gt some value
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EXAMPLE Forming Hypotheses For each of the
following claims, determine the null and
alternative hypothesis.

• In 1997, 43 of Americans 18 years or older
  participated in some form of charity work. A
  researcher believes that this percentage
  different today.

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EXAMPLE Forming Hypotheses For each of the
following claims, determine the null and
alternative hypothesis.

• In 1997, 43 of Americans 18 years or older
  participated in some form of charity work. A
  researcher believes that this percentage
  different today.
• Ho p 0.43
• H1 p ? 0.43
• Two tailed test

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EXAMPLE Forming Hypotheses For each of the
following claims, determine the null and
alternative hypothesis.

• In June, 2001 the mean length of a phone call on
  a cellular telephone was 2.62 minutes. A
  researcher believes that the mean length of a
  call has increased since then.

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EXAMPLE Forming Hypotheses For each of the
following claims, determine the null and
alternative hypothesis.

• In June, 2001 the mean length of a phone call on
  a cellular telephone was 2.62 minutes. A
  researcher believes that the mean length of a
  call has increased since then.
• Ho µ 2.62
• H1 µ gt 2.62
• Right tailed test

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EXAMPLE Forming Hypotheses For each of the
following claims, determine the null and
alternative hypothesis.

• 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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EXAMPLE Forming Hypotheses For each of the
following claims, determine the null and
alternative hypothesis.

• 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.
• Ho s 0.23
• H1 s lt 0.23
• Left tailed test

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Four Outcomes from Hypothesis Testing 1. We could
reject Ho when in fact H1 is true. This would be
a correct decision.
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Four Outcomes from Hypothesis Testing 1. We could
reject Ho when in fact H1 is true. This would be
a correct decision. 2. We could not reject Ho
when in fact Ho is true. This would be a correct
decision.
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Four Outcomes from Hypothesis Testing 1. We could
reject Ho when in fact H1 is true. This would be
a correct decision. 2. We could not reject Ho
when in fact Ho is true. This would be a correct
decision. 3. We could reject Ho when in fact Ho
is true. This would be an incorrect decision.
This type of error is called a Type I error.
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Four Outcomes from Hypothesis Testing 1. We could
reject Ho when in fact H1 is true. This would be
a correct decision. 2. We could not reject Ho
when in fact Ho is true. This would be a correct
decision. 3. We could reject Ho when in fact Ho
is true. This would be an incorrect decision.
This type of error is called a Type I error. 4.
We could not reject Ho when in fact H1 is true.
This would be an incorrect decision. This type
of error is called a Type II error.
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• EXAMPLE Type I and Type II Errors
• For the following claim explain what it would
  mean to make a Type I error. What would it mean
  to make a Type II error?
• In 1997, 43 of Americans 18 years or older
  participated in some form of charity work. A
  researcher believes that this percentage
  different today.

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• EXAMPLE Type I and Type II Errors
• For the following claim explain what it would
  mean to make a Type I error. What would it mean
  to make a Type II error?
• In 1997, 43 of Americans 18 years or older
  participated in some form of charity work. A
  researcher believes that this percentage
  different today.
• Type I error to reject that 43 of the
  population participated in charity work when in
  fact this is true.

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• EXAMPLE Type I and Type II Errors
• For the following claim explain what it would
  mean to make a Type I error. What would it mean
  to make a Type II error?
• In 1997, 43 of Americans 18 years or older
  participated in some form of charity work. A
  researcher believes that this percentage
  different today.
• Type I error to reject that 43 of the
  population participated in charity work when in
  fact this is true.
• Type II error to not reject that 43
  participated in charity work when in fact this is
  false.

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• EXAMPLE Type I and Type II Errors
• For the following claim explain what it would
  mean to make a Type I error. What would it mean
  to make a Type II error?
• In June, 2001 the mean length of a phone call on
  a cellular telephone was 2.62 minutes. A
  researcher believes that the mean length of a
  call has increased since then.

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• EXAMPLE Type I and Type II Errors
• For the following claim explain what it would
  mean to make a Type I error. What would it mean
  to make a Type II error?
• In June, 2001 the mean length of a phone call on
  a cellular telephone was 2.62 minutes. A
  researcher believes that the mean length of a
  call has increased since then.
• Type I error to reject that the mean length of a
  cell phone call is 2.62 minutes when this is true.

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• EXAMPLE Type I and Type II Errors
• For the following claim explain what it would
  mean to make a Type I error. What would it mean
  to make a Type II error?
• In June, 2001 the mean length of a phone call on
  a cellular telephone was 2.62 minutes. A
  researcher believes that the mean length of a
  call has increased since then.
• Type I error to reject that the mean length of a
  cell phone call is 2.62 minutes when this is
  true.
• Type II error to not reject that the mean length
  of a cell phone call is 2.62 minutes when this is
  false.

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CAUTION!
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EXAMPLE Wording the Conclusion In June, 2001
the mean length of a phone call on a cellular
telephone was 2.62 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.
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• EXAMPLE Wording the Conclusion
• In June, 2001 the mean length of a phone call on
  a cellular telephone was 2.62 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.
• We conclude that there is sufficient evidence to
  support the claim that the mean length of a cell
  phone call is more than 2.62 minutes.

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EXAMPLE Wording the Conclusion In June, 2001
the mean length of a phone call on a cellular
telephone was 2.62 minutes. A researcher
believes that the mean length of a call has
increased since then. (b) Suppose the sample
evidence indicates that the null hypothesis
should not be rejected. State the wording of the
conclusion.
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• EXAMPLE Wording the Conclusion
• In June, 2001 the mean length of a phone call on
  a cellular telephone was 2.62 minutes. A
  researcher believes that the mean length of a
  call has increased since then.
• (b) Suppose the sample evidence indicates that
  the null hypothesis should not be rejected.
  State the wording of the conclusion.
• We conclude that there is not sufficient evidence
  to support the claim that the mean length of a
  cell phone call is more than 2.62 minutes.