Transcript Slides for Chapter 9

9- 1
Chapter
Nine
McGraw-Hill/Irwin
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
9- 2
Chapter Nine
Estimation and Confidence Intervals
GOALS
When you have completed this chapter, you will be able to:
ONE
Define a what is meant by a point estimate.
TWO
Define the term level of level of confidence.
THREE
Construct a confidence interval for the population mean when
the population standard deviation is known.
FOUR
Construct a confidence interval for the population mean when
the population standard deviation is unknown.
Goals
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Chapter Nine
continued
Estimation and Confidence Intervals
GOALS
When you have completed this chapter, you will be able to:
FIVE
Construct a confidence interval for the population proportion.
SIX
Determine the sample size for attribute and variable sampling.
Goals
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A point estimate is
a single value
(statistic) used to
e s t i m a t e a
population value
(parameter).
An Interval Estimate
states the range within
which a population
parameter probably
l
i
e
s
.
A confidence interval is
a range of values
within which the
population parameter is
e x p e ct ed t o o c c u r.
The two confidence
intervals that are used
extensively are the
9 5 % and th e 99 %.
Point and Interval Estimates
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Factors that
determine the
width of a
confidence
interval
The sample size, n
The desired level of confidence
The variability in the population,
usually estimated by s
Point and Interval Estimates
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For a 95% confidence
interval about 95% of
the similarly constructed
intervals will contain the
parameter being
estimated.
95% of the sample means
for a specified sample
size will lie within 1.96
standard deviations of
the hypothesized
population mean.
For the 99% confidence
interval, 99% of the sample
means for a specified sample
size will lie within 2.58 standard
deviations of the hypothesized
population mean.
Interval Estimates
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Standard Error of the Sample Mean
Standard
deviation of
the sampling
distribution of
the sample
means
x


n
 x symbol for the standard error
of the sample mean
 the standard deviation of the population
n is the size of the sample
Standard Error of the Sample
Means
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If s is not known and n
>30, the standard
deviation of the sample,
designated s, is used to
approximate the
population standard
deviation.
X z
s
n
The standard error
sx 
s
n
If the population standard
deviation is known or the
sample is greater than 30
we use the z distribution.
Standard Error of the Sample
Means
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If the population
standard deviation
is unknown, the
underlying
s
population is
X t
approximately
n
normal, and the
sample size is less
than 30 we use the
t distribution.
The value of t for a given confidence level depends
upon its degrees of freedom.
Point and Interval Estimates
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Characteristics of the t
distribution
It is a continuous
distribution.
There is a family of t
distributions.
Assumption: the
population is normal
or nearly normal
It is bell-shaped and
symmetrical.
The t distribution is more
spread out and flatter at the
center than is the standard
normal distribution,
differences that diminish as n
increases.
Point and Interval Estimates
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Confidence interval for the mean
X  z
s
n
95% CI for the population mean
X  1.96
s
n
99% CI for the population mean
s
X  2 .58
n
Constructing General Confidence
Intervals for µ
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The Dean of the Business
School wants to estimate the
mean number of hours worked
per week by students. A sample
of 49 students showed a mean
of 24 hours with a standard
deviation of 4 hours. What is
the population mean?
The value of the population mean is not known. Our
best estimate of this value is the sample mean of 24.0
hours. This value is called a point estimate.
Example 3
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95 percent confidence interval
for the population mean
X  1.96
s
 24 .00  1.96
n
4
49
 24 .00  1.12
The confidence
limits range from
22.88 to 25.12.
About 95 percent of the similarly
constructed intervals include the
population parameter.
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The confidence interval for a
population proportion
p(1  p)
pz
n
Confidence Interval for a
Population Proportion
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A sample of 500
executives who own
their own home
revealed 175 planned to
sell their homes and
retire to Arizona.
Develop a 98%
confidence interval for
the proportion of
executives that plan to
sell and move to
Arizona.
(.35 )(. 65 )
.35  2.33
 .35  .0497
500
Example 4
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Finite population
Adjust the standard
errors of the sample
means and the proportion

x 
n
N n
N 1
fixed upper bound
Finite-Population
Correction Factor
N, total number of objects
n, sample size
Finite-Population
Correction Factor
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Standard error of the sample proportions
p 
p (1  p )
n
N n
N 1
Ignore finite-population
correction factor if n/N < .05.
Finite-Population Correction
Factor
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95% confidence interval for the mean number of
hours worked per week by the students if there
are only 500 students on campus
n/N = 49/500 = .098 > .05
Use finite population correction factor
500  49
24  1.96(
)(
)  24.00  1.0648
500  1
49
4
EXAMPLE 4 revisited
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3 factors that determine the size of a sample
The degree of confidence selected
The maximum allowable error
The variation in the population
Selecting a Sample Size
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Calculating the sample size
 zs
n

 E 
2
where
E is the allowable error
z the z- value corresponding to the selected level
of confidence
s the sample deviation of the pilot survey
Selecting a Sample Size
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A consumer group
would like to estimate
the mean monthly
electricity charge for a
single family house in
July within $5 using a
99 percent level of
confidence. Based on
similar studies the
standard deviation is
estimated to be $20.00.
How large a sample is
required?
2
 (2.58)( 20) 
n
  107
5


Example 6
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The formula for
determining the
sample size in the case
of a proportion is
 Z
n  p(1  p) 
 E
2
where
p is the estimated proportion, based on past
experience or a pilot survey
z is the z value associated with the degree of
confidence selected
E is the maximum allowable error the
researcher will tolerate
Sample Size for Proportions
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The American Kennel Club
wanted to estimate the
proportion of children that
have a dog as a pet. If the
club wanted the estimate to
be within 3% of the
population proportion, how many children would they
need to contact? Assume a 95% level of confidence and
that the club estimated that 30% of the children have a dog
as a pet.
2
 1.96 
n  (.30 )(. 70 )
  897
 .03 
Example 7
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What happens when
the population has less
members than the
sample size
calculated requires?
Step One: Calculate the sample
size as before.
no
Step Two: Calculate
the new sample size.
n=
where no is the sample size
calculated in step one.
no
1+ N
Optional method, not covered in text:
Sample Size for Small Populations
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An auditor wishes to survey
employees in an organization to
determine compliance with
federal regulations. The auditor
estimates that 80% of the
employees would say that the
organization is in compliance.
The organization has 200
employees. The auditor wishes
to be 95% confident in the
results, with a margin of error no
greater than 3%. How many
employees should the auditor
survey?
Example 8 Optional
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Step One
Calculate the sample size as before.
 Z
n  p(1  p) 
 E
2
= (.80)(.20) 1.96
.03
2
= 683
Step Two
Calculate the new sample size.
no
n=
1 + no
N
=
683
1 + 683
200
= 155
Example 8 continued