Chapter 8 Confidence Intervals

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Chapter 8 Confidence Intervals

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Title: Chapter 8 Confidence Intervals

1
Chapter 8Confidence Intervals

8.3
Confidence Intervals about a Population Proportion

Estimating a Population Proportion
Data are often given in the form of
proportions.

Estimating a Population Proportion
Data are often given in the form of
proportions.
Polls

Estimating a Population Proportion
Data are often given in the form of
proportions.
Polls
Marketing Surveys

Estimating a Population Proportion
Data are often given in the form of
proportions.
Polls
Marketing Surveys
Death Rates for Diseases

Estimating a Population Proportion
Data are often given in the form of
proportions.
Polls
Marketing Surveys
Death Rates for Diseases
Probabilities of events

Estimating a Population Proportion
Data are often given in the form of
proportions.
Polls
Marketing Surveys
Death Rates for Diseases
Probabilities of events
Population proportions are denoted by p.

Estimating a Population Proportion
Population proportions are estimated by sample
proportions.

Estimating a Population Proportion
Population proportions are estimated by sample
proportions.
Take a sample of size n from the population.

Estimating a Population Proportion
Population proportions are estimated by sample
proportions.
Take a sample of size n from the population.
Let X the number of individuals in the sample
with property E.

Estimating a Population Proportion
Population proportions are estimated by sample
proportions.
Take a sample of size n from the population.
Let X the number of individuals in the sample
with property E.
The sample proportion is given by
p X / n

EXAMPLE Computing a Point Estimate
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide

EXAMPLE Computing a Point Estimate
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide
"Will you use the Internet to buy Christmas or
other holiday gifts this year, or not?"

EXAMPLE Computing a Point Estimate
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide
"Will you use the Internet to buy Christmas or
other holiday gifts this year, or not?"
yes 318

EXAMPLE Computing a Point Estimate
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide
"Will you use the Internet to buy Christmas or
other holiday gifts this year, or not?"
yes 318 no 636

EXAMPLE Computing a Point Estimate
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide
"Will you use the Internet to buy Christmas or
other holiday gifts this year, or not?"
yes 318 no 636 no opinion 72

EXAMPLE Computing a Point Estimate
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide
"Will you use the Internet to buy Christmas or
other holiday gifts this year, or not?"
yes 318 no 636 no opinion 72
Let X number of people in sample who will by
gifts over the internet.

EXAMPLE Computing a Point Estimate
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide
"Will you use the Internet to buy Christmas or
other holiday gifts this year, or not?"
yes 318 no 636 no opinion 72
Let X number of people in sample who will by
gifts over the internet.
Estimate p.

EXAMPLE Computing a Point Estimate
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide
"Will you use the Internet to buy Christmas or
other holiday gifts this year, or not?"
yes 318 no 636 no opinion 72
Let X number of people in sample who will by
gifts over the internet.
Estimate p.
p X / n

EXAMPLE Computing a Point Estimate
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide
"Will you use the Internet to buy Christmas or
other holiday gifts this year, or not?"
yes 318 no 636 no opinion 72
Let X number of people in sample who will by
gifts over the internet.
Estimate p.
p X / n 318 / 1026

EXAMPLE Computing a Point Estimate
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide
"Will you use the Internet to buy Christmas or
other holiday gifts this year, or not?"
yes 318 no 636 no opinion 72
Let X number of people in sample who will by
gifts over the internet.
Estimate p.
p X / n 318 / 1026 0.31

EXAMPLE Computing a Point Estimate
Besides the point estimate p, we would also
like some measure of how reliable this estimate
is.

EXAMPLE Computing a Point Estimate
Besides the point estimate p, we would also
like some measure of how reliable this estimate
is.
In other words, we need a confidence interval
for the population proportion.

EXAMPLE Computing a Point Estimate
Besides the point estimate p, we would also
like some measure of how reliable this estimate
is.
In other words, we need a confidence interval
for the population proportion.
To do so, we need to determine the probability
distribution of the sample proportion p X /
n.

EXAMPLE Computing a Point Estimate
Besides the point estimate p, we would also
like some measure of how reliable this estimate
is.
In other words, we need a confidence interval
for the population proportion.
To do so, we need to determine the probability
distribution of the sample proportion p X /
n.
To do this first find the distribution of X.

Estimating a Population Proportion
What type of random variable is X ?

Estimating a Population Proportion
What type of random variable is X ?
X is the number of occurrences of an event E in a
sample of n.

Estimating a Population Proportion
What type of random variable is X ?
X is the number of occurrences of an event E in a
sample of n.
The proportion of the population with property E
is p.

Estimating a Population Proportion
What type of random variable is X ?
X is the number of occurrences of an event E in a
sample of n.
The proportion of the population with property E
is p.
X Bin(n, p)

Estimating a Population Proportion
What type of random variable is X ?
X is the number of occurrences of an event E in a
sample of n.
The proportion of the population with property E
is p.
X Bin(n, p)
This is true as long as the sample size n is
much smaller than the population size N.

Estimating a Population Proportion
What type of random variable is X ?
X is the number of occurrences of an event E in a
sample of n.
The proportion of the population with property E
is p.
X Bin(n, p)
This is true as long as the sample size n is
much smaller than the population size N.
(rule of thumb n

Estimating a Population Proportion
X Bin(n, p)

Estimating a Population Proportion
X Bin(n, p)
E(X) np

Estimating a Population Proportion
X Bin(n, p)
E(X) np SD(X) sqrtnp(1-p)

Estimating a Population Proportion
X Bin(n, p)
E(X) np SD(X) sqrtnp(1-p)
Also recall that if np(1-p) 10, then X is
approximately normal

Estimating a Population Proportion
X Bin(n, p)
E(X) np SD(X) sqrtnp(1-p)
Also recall that if np(1-p) 10, then X is
approximately normal
X N(np, sqrtnp(1-p))

Estimating a Population Proportion
X Bin(n, p)
E(X) np SD(X) sqrtnp(1-p)
Also recall that if np(1-p) 10, then X is
approximately normal
X N(np, sqrtnp(1-p))
Since p X / n

Estimating a Population Proportion
X Bin(n, p)
E(X) np SD(X) sqrtnp(1-p)
Also recall that if np(1-p) 10, then X is
approximately normal
X N(np, sqrtnp(1-p))
Since p X / n
E(p) p

Estimating a Population Proportion
X Bin(n, p)
E(X) np SD(X) sqrtnp(1-p)
Also recall that if np(1-p) 10, then X is
approximately normal
X N(np, sqrtnp(1-p))
Since p X / n,
E(p) p SD(p) sqrtp(1-p)/n

Estimating a Population Proportion
X Bin(n, p)
E(X) np SD(X) sqrtnp(1-p)
Also recall that if np(1-p) 10, then X is
approximately normal
X N(np, sqrtnp(1-p))
Since p X / n,
E(p) p SD(p) sqrtp(1-p)/n
p N(p, sqrtp(1-p)/n)

41
(No Transcript)
42

Estimating a Population Proportion
Since p N(p, sqrtp(1-p)/n), we can use the
confidence interval derived in Section 8.1

Estimating a Population Proportion
Since p N(p, sqrtp(1-p)/n), we can use the
confidence interval derived in Section 8.1
(p - z?/2 sqrtp(1-p)/n, p z?/2
sqrtp(1-p)/n)

Estimating a Population Proportion
Since p N(p, sqrtp(1-p)/n), we can use the
confidence interval derived in Section 8.1
(p - z?/2 sqrtp(1-p)/n, p z?/2
sqrtp(1-p)/n)
This is the (1 - ?)100 CI for the population
proportion p.

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"
p X / n 318 / 1026 0.31

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"
p X / n 318 / 1026 0.31
Construct a 95 CI for the population
proportion

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"
p X / n 318 / 1026 0.31
Construct a 95 CI for the population
proportion
(n)(p)(1-p) (1026)(0.31)(1-0.31) 219 10

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"
p X / n 318 / 1026 0.31
Construct a 95 CI for the population
proportion
(n)(p)(1-p) (1026)(0.31)(1-0.31) 219 10
(p - z?/2 sqrtp(1-p)/n, p z?/2
sqrtp(1-p)/n)

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"
p X / n 318 / 1026 0.31
Construct a 95 CI for the population
proportion
(n)(p)(1-p) (1026)(0.31)(1-0.31) 219 10
(0.31 - z?/2 sqrtp(1-p)/n, 0.31 z?/2
sqrtp(1-p)/n)

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"
p X / n 318 / 1026 0.31
Construct a 95 CI for the population
proportion
(n)(p)(1-p) (1026)(0.31)(1-0.31) 219 10
(0.31 1.96 sqrtp(1-p)/n, 0.31 1.96
sqrtp(1-p)/n)

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"
p X / n 318 / 1026 0.31
Construct a 95 CI for the population
proportion
(n)(p)(1-p) (1026)(0.31)(1-0.31) 219 10
(0.31-1.96 sqrt0.31(1- 0.31)/n, 0.311.96
sqrt0.31(1-0.31)/n)

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"
p X / n 318 / 1026 0.31
Construct a 95 CI for the population
proportion
(n)(p)(1-p) (1026)(0.31)(1-0.31) 219 10
(0.31-1.96 sqrt0.31(1-0.31)/1026,0.311.96sqrt0
.31(1-0.31)/1026)

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"
p X / n 318 / 1026 0.31
Construct a 95 CI for the population
proportion
(n)(p)(1-p) (1026)(0.31)(1-0.31) 219 10
(0.31 - 1.96(0.014), 0.31 1.96(0.014))

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"
p X / n 318 / 1026 0.31
Construct a 95 CI for the population
proportion
(n)(p)(1-p) (1026)(0.31)(1-0.31) 219 10
(0.31 - 0.028, 0.31 0.028)

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"
p X / n 318 / 1026 0.31
Construct a 95 CI for the population
proportion
(n)(p)(1-p) (1026)(0.31)(1-0.31) 219 10
(0.28, 0.34)

EXAMPLE Constructing a Confidence Interval
for a Population Proportion
ABC News Poll. Nov. 19-23, 2003. n1,026 adults
nationwide "Will you use the Internet to buy
Christmas or other holiday gifts this year, or
not?"
p X / n 318 / 1026 0.31
Construct a 95 CI for the population
proportion
(n)(p)(1-p) (1026)(0.31)(1-0.31) 219 10
(0.28, 0.34)
A 95 CI for the population proportion is
(0.28,0.34)

58
Chapter 9Hypothesis Testing

9.1
The Language of Hypothesis Testing

59
Steps in Hypothesis Testing 1. A claim is made.
60
Steps in Hypothesis Testing 1. A claim is made.
2. Evidence (sample data) is collected in order
to test the claim.
61
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 ord
er to support or refute the claim.
62

Examples
A drug company claims its cancer drug prolongs
the lifespan of lung cancer patients.

Examples
A drug company claims its cancer drug prolongs
the lifespan of lung cancer patients.
An engineer claims a new material improves auto
safety in crashes.

Examples
A drug company claims its cancer drug prolongs
the lifespan of lung cancer patients.
An engineer claims a new material improves auto
safety in crashes.
A social scientist claims a new teaching
technique improves reading scores.

Examples
A drug company claims its cancer drug prolongs
the lifespan of lung cancer patients.
An engineer claims a new material improves auto
safety in crashes.
A social scientist claims a new teaching
technique improves reading scores.
A new diet is claimed to reduce weight better
than other diets.

66
A hypothesis is a statement or claim regarding a
characteristic of one or more populations.

67
A hypothesis is a statement or claim regarding a
characteristic of one or more populations.
This chapter looks at hypotheses regarding some
parameter of a single population.
68
Examples of Claims Regarding a Characteristic of
a Single Population
69
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.

70
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 is
different today.

71
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 is
different today.
In June, 2001 the mean length of a phone call
on a cellular telephone was 2.62 minutes.

72
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 is
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.

73
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 is
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.

74
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 is
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.

75
We test these types of claims using sample data
because it is usually impossible or impractical
to gain access to the entire population.

76
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 inferentia
l statistics is not necessary.
77

A researcher believes the mean length of a cell
phone call has increased from mean of 2.62
minutes.

A researcher believes the mean length of a cell
phone call has increased from mean of 2.62
minutes.
He obtains a random sample of 36 cell phone
calls.

A researcher believes the mean length of a cell
phone call has increased from mean of 2.62
minutes.
He obtains a random sample of 36 cell phone
calls.
The mean length of calls is 2.70 minutes.

A researcher believes the mean length of a cell
phone call has increased from mean of 2.62
minutes.
He obtains a random sample of 36 cell phone
calls.
The mean length of calls is 2.70 minutes.
Is this enough evidence to conclude the length
of a phone call has increased?

A researcher believes the mean length of a cell
phone call has increased from mean of 2.62
minutes.
He obtains a random sample of 36 cell phone
calls.
The mean length of calls is 2.70 minutes.
Is this enough evidence to conclude the length
of a phone call has increased?
How likely/unlikely would it be to get a sample
mean of 2.70 minutes if the population mean
were still actually 2.62 minutes?

Assume we know that the population s.d. ?
0.78.

Assume we know that the population s.d. ?
0.78.
The population mean used to be ? 2.62.

Assume we know that the population s.d. ?
0.78.
The population mean used to be ? 2.62.
The sample mean X 2.70 (n36)

Assume we know that the population s.d. ?
0.78.
The population mean used to be ? 2.62.
The sample mean X 2.70 (n36)
We know that X N(? , ?/sqrt(n))

Assume we know that the population s.d. ?
0.78.
The population mean used to be ? 2.62.
The sample mean X 2.70 (n36)
We know that X N(? , 0.78/sqrt(36))

Assume we know that the population s.d. ?
0.78.
The population mean used to be ? 2.62.
The sample mean X 2.70 (n36)
We know that X N(? , 0.13)

Assume we know that the population s.d. ?
0.78.
The population mean used to be ? 2.62.
The sample mean X 2.70 (n36)
We know that X N(? , 0.13)
What is ??

Assume we know that the population s.d. ?
0.78.
The population mean used to be ? 2.62.
The sample mean X 2.70 (n36)
We know that X N(? , 0.13)
What is ?? We dont know what ? is now.

Assume we know that the population s.d. ?
0.78.
The population mean used to be ? 2.62.
The sample mean X 2.70 (n36)
We know that X N(? , 0.13)
What is ?? We dont know what ? is now.
The researcher is claiming that the old status
quo has changed (? is now a different value).

Assume we know that the population s.d. ?
0.78.
The population mean used to be ? 2.62.
The sample mean X 2.70 (n36)
We know that X N(? , 0.13)
What is ?? We dont know what ? is now.
The researcher is claiming that the old status
quo has changed (? is now a different value).
Because of this, we place the burden of proof
on the researcher.

Assume we know that the population s.d. ?
0.78.
The population mean used to be ? 2.62.
The sample mean X 2.70 (n36)
We know that X N(? , 0.13)
What is ?? We dont know what ? is now.
The researcher is claiming that the old status
quo has changed (? is now a different value).
Because of this, we place the burden of proof
on the researcher. He must provide overwhelming
evidence that his hypothesis is true before it is
accepted.

So we assume ? is still equal to 2.62 and ask
whether the sample collected strongly contradicts
this assumption.

So we assume ? is still equal to 2.62 and ask
whether the sample collected strongly contradicts
this assumption.
So assume X N(2.62, 0.13)

So we assume ? is still equal to 2.62 and ask
whether the sample collected strongly contradicts
this assumption.
So assume X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.70 if this is true?

So we assume ? is still equal to 2.62 and ask
whether the sample collected strongly contradicts
this assumption.
So assume X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.70 if this is true?
Z (X 2.62) / 0.13

So we assume ? is still equal to 2.62 and ask
whether the sample collected strongly contradicts
this assumption.
So assume X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.70 if this is true?
Z (X 2.62) / 0.13
z ( 2.70 2.62) / 0.13

So we assume ? is still equal to 2.62 and ask
whether the sample collected strongly contradicts
this assumption.
So assume X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.70 if this is true?
Z (X 2.62) / 0.13
z 0.62

So we assume ? is still equal to 2.62 and ask
whether the sample collected strongly contradicts
this assumption.
So assume X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.70 if this is true?
Z (X 2.62) / 0.13
z 0.62
How unlikely would it be to get a value of z
this large or larger?

100

So we assume ? is still equal to 2.62 and ask
whether the sample collected strongly contradicts
this assumption.
So assume X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.70 if this is true?
Z (X 2.62) / 0.13
z 0.62
How unlikely would it be to get a value of z
this large or larger? I.e., what is P(Z
0.62) ?

101
What is P(Z 0.62) ?
Z
102
What is P(Z 0.62) ?
0.62
Z
103
What is P(Z 0.62) ?
0.27
0.62
Z
104
What is P(Z 0.62) 0.27
0.27
0.62
Z
105

So we assume ? is still equal to 2.62 and ask
whether the sample collected strongly contradicts
this assumption.
So assume X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.70 if this is true?
Z (X 2.62) / 0.13
z 0.62
How unlikely would it be to get a value of z
this large or larger? P(Z 0.62)
0.27

106

So we assume ? is still equal to 2.62 and ask
whether the sample collected strongly contradicts
this assumption.
So assume X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.70 if this is true?
Z (X 2.62) / 0.13
z 0.62
How unlikely would it be to get a value of z
this large or larger? P(Z 0.62)
0.27
The event is not all that unusual.

107

So we assume ? is still equal to 2.62 and ask
whether the sample collected strongly contradicts
this assumption.
So assume X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.70 if this is true?
Z (X 2.62) / 0.13
z 0.62
How unlikely would it be to get a value of z
this large or larger? P(Z 0.62)
0.27
The event is not all that unusual. There is
not enough evidence to claim that the status quo
has changed.

108

What if instead the sample mean X 2.90?

109

What if instead the sample mean X 2.90?
X N(2.62, 0.13)

110

What if instead the sample mean X 2.90?
X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.90 if this is true?

111

What if instead the sample mean X 2.90?
X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.90 if this is true?
z (2.90 2.62) / 0.13

112

What if instead the sample mean X 2.90?
X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.90 if this is true?
z (2.90 2.62) / 0.13 2.15

113

What if instead the sample mean X 2.90?
X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.90 if this is true?
z (2.90 2.62) / 0.13 2.15
How unlikely would it be to get a value of z
this large or larger?

114

What if instead the sample mean X 2.90?
X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.90 if this is true?
z (2.90 2.62) / 0.13 2.15
How unlikely would it be to get a value of z
this large or larger? P(Z 2.15) ?

115
What is P(Z 2.15) ?
Z
116
What is P(Z 2.15) ?
2.15
Z
117
What is P(Z 2.15) ?
0.016
2.15
Z
118
What is P(Z 2.15) 0.016
0.016
2.15
Z
119

What if instead the sample mean X 2.90?
X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.90 if this is true?
z (2.90 2.62) / 0.13 2.15
How unlikely would it be to get a value of z
this large or larger? P(Z 2.15) 0.016

120

What if instead the sample mean X 2.90?
X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.90 if this is true?
z (2.90 2.62) / 0.13 2.15
How unlikely would it be to get a value of z
this large or larger? P(Z 2.15)
0.016
The event is unusual.

121

What if instead the sample mean X 2.90?
X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.90 if this is true?
z (2.90 2.62) / 0.13 2.15
How unlikely would it be to get a value of z
this large or larger? P(Z 2.15)
0.016
The event is unusual. There is enough evidence
to claim that the status quo has changed.

122

What if instead the sample mean X 2.90?
X N(2.62, 0.13)
How unlikely would it be to get a sample mean
of 2.90 if this is true?
z (2.90 2.62) / 0.13 2.15
How unlikely would it be to get a value of z
this large or larger? P(Z 2.15)
0.016
The event is unusual. There is enough evidence
to claim that the status quo has changed.
Note that the researcher determines at what
level a result is considered to be unusual.

123
Hypothesis testing is a procedure used to test
claims regarding a characteristic of a
population.
124
Hypothesis testing is a procedure used to test
claims regarding a characteristic of a
population.
It consists of
125
Hypothesis testing is a procedure used to test
claims regarding a characteristic of a
population. It consists of (1) sample evidenc
e.
126
Hypothesis testing is a procedure used to test
claims regarding a characteristic of a
population. It consists of (1) sample evidenc
e. (2) A rejection or acceptance.
127
Hypothesis testing is a procedure used to test
claims regarding a characteristic of a
population. It consists of (1) sample evidenc
e. (2) A rejection or acceptance. (3) A probab
ility statement.
128

The null hypothesis, denoted Ho is a statement
to be tested.

129

The null hypothesis, denoted Ho is a statement
to be tested.
It is the status quo situation.

130

The null hypothesis, denoted Ho is a statement
to be tested.
It is the status quo situation.
The null hypothesis is assumed true until
evidence overwhelmingly indicates otherwise.

131

The null hypothesis, denoted Ho is a statement
to be tested.
It is the status quo situation.
The null hypothesis is assumed true until
evidence overwhelmingly indicates otherwise.
It is usually a statement regarding the value
of a population parameter.

132

The alternative hypothesis H1 is a claim to be
tested.

133

The alternative hypothesis H1 is a claim to be
tested.
It is a statement that the status quo has
changed.

134

The alternative hypothesis H1 is a claim to be
tested.
It is a statement that the status quo has
changed.
The researcher want to provide overwhelming
evidence for the alternative hypothesis.

135

The alternative hypothesis H1 is a claim to be
tested.
It is a statement that the status quo has
changed.
The researcher want to provide overwhelming
evidence for the alternative hypothesis.
It is a claim regarding the value of a
population parameter.

136
There are three ways to set up the Ho and H1
137
There are three ways to set up the Ho and H1
1. Equal versus not equal hypotheses
138

There are three ways to set up the Ho and H1
Equal versus not equal hypotheses
Ho parameter some value

139

There are three ways to set up the Ho and H1
Equal versus not equal hypotheses
Ho parameter some value
H1 parameter ? some value

140

There are three ways to set up the Ho and H1
Equal versus not equal hypotheses
Ho parameter some value
H1 parameter ? some value
Example A researcher claims that the percentage
of Americans involved in charity work is
different than it was 6 years ago (43).

141

There are three ways to set up the Ho and H1
Equal versus not equal hypotheses
Ho parameter some value
H1 parameter ? some value
Example A researcher claims that the percentage
of Americans involved in charity work is
different than it was 6 years ago (43).
Ho pop. prop. 43

142
There are three ways to set up the Ho and H1
1. Equal versus not equal hypotheses
Ho parameter some value H1 parameter ? so
me value Example A researcher claims that t
he percentage of Americans involved in charity
work is different than it was 6 years ago (43).
Ho pop. prop. 43 H1 pop. prop. ? 43

143
There are three ways to set up the Ho and H1

2. Equal versus less than
144
There are three ways to set up the Ho and H1
2. Equal versus less than Ho parameter s
ome value

145
There are three ways to set up the Ho and H1
2. Equal versus less than Ho parameter s
ome value
H1 parameter
146
There are three ways to set up the Ho and H1
2. Equal versus less than Ho parameter s
ome value H1 parameter e A researcher claims that the mean number of
defects on a manufactured item (0.12) is smaller
when using a new machine tool.
147
There are three ways to set up the Ho and H1
2. Equal versus less than Ho parameter s
ome value H1 parameter e A researcher claims that the mean number of
defects on a manufactured item (0.12) is smaller
when using a new machine tool. Ho ? 0.12
148
There are three ways to set up the Ho and H1
2. Equal versus less than Ho parameter s
ome value H1 parameter e A researcher claims that the mean number of
defects on a manufactured item (0.12) is smaller
when using a new machine tool.
Ho ? 0.12 H1 ?
149
There are three ways to set up the Ho and H1

3. Equal versus greater than
150
There are three ways to set up the Ho and H1
3. Equal versus greater than Ho parameter
some value

151
There are three ways to set up the Ho and H1
3. Equal versus greater than Ho parameter
some value
H1 parameter some value
152
There are three ways to set up the Ho and H1
3. Equal versus greater than Ho parameter
some value H1 parameter some value Exa
mple A researcher claims that a new drug
increases the mean lifespan of lung cancer
patients (? 23 months).
153
There are three ways to set up the Ho and H1
3. Equal versus greater than Ho parameter
some value H1 parameter some value Exa
mple A researcher claims that a new drug
increases the mean lifespan of lung cancer
patients (? 23 months). Ho ? 23
154
There are three ways to set up the Ho and H1
3. Equal versus greater than Ho parameter
some value H1 parameter some value Exa
mple A researcher claims that a new drug
increases the mean lifespan of lung cancer
patients (? 23 months). Ho ? 23 H1 ?
23
155
Four Outcomes from Hypothesis Testing
1. We could reject Ho when in fact H1 is true.
This would be a correct decision.
156
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.
157
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.
158
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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