Chap 9x

Report
Estimation and
Confidence Intervals
Chapter 9
McGraw-Hill/Irwin
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
Learning Objectives
Chapter 9
1. Define a point estimate.
2. Compute a confidence interval for the population mean
when the population standard deviation is known.
3. Compute a confidence interval for a population mean
when the population standard deviation is unknown.
4. Compute a confidence interval for a population
proportion.
Chapter 10
Conduct a test of hypothesis about a population proportion.
9-2
Point Estimates

A point estimate is
a single value
(point) derived from
a sample and used
to estimate a
population value.
X  
s  
s  
p  
2
2
Disadvantages of point estimates:
1. Point estimates derived from a sample can be quite different from
the corresponding population parameter it represents.
2. Point estimates does not take into consideration of the sample
size and standard deviation.
9-3
Confidence Interval Estimates


A confidence interval estimate is a range
of values constructed from sample data so
that the population parameter is likely to
occur within that range.
The likelihood is controlled by the level of
confidence, denoted 1-α.
C.I. = point estimate ± margin of error
9-4
Confidence Intervals for a Mean – σ Known
X  z / 2

n
x  samplemean
z  z - value for a particularconfidencelevel
σ  thepopulationstandarddeviation
n  thenumber of observations in thesample
The width of the interval is determined by
1. the level of confidence; the higher the level, the wider the CI is.
2. Sample size; the large the size, the narrow the CI is.
3. Population standard deviation; CI is wider for larger σ.
9-5
Example: Confidence Interval for a Mean –
σ Known
The American Management Association wishes to
have information on the mean income of middle
managers in the retail industry. A random sample
of 256 managers reveals a sample mean of
$45,420. The standard deviation of this population
is $2,050. The association would like answers to
the following questions:
1. What is the population mean?
2. What is a reasonable range of values for the
population mean?
3. What do these results mean?
9-6
Example: Confidence Interval for a Mean –
σ Known
The American Management Association wishes to have
information on the mean income of middle managers in the
retail industry. A random sample of 256 managers reveals a
sample mean of $45,420. The standard deviation of this
population is $2,050. The association would like answers to
the following questions:
1. What is the population mean?
In this case, we do not know. We do know the sample mean
is $45,420. Hence, our best estimate of the unknown
population value is the corresponding sample statistic.
The sample mean of $45,420 is a point estimate of the
unknown population mean.
9-7
Example: Confidence Interval for a Mean –
σ Known
The American Management Association wishes to have information
on the mean income of middle managers in the retail industry. A
random sample of 256 managers reveals a sample mean of
$45,420. The standard deviation of this population is $2,050. The
association would like answers to the following questions:
2. What is a reasonable range of values for the population mean?
Suppose the association decides to use the 95 percent level of
confidence:
The confidence limit are $45,169 and $45,671
The ±$251 is referred to as the margin of error
9-8
How to Obtain z value for a Given
Confidence Level
The 95% confidence refers to
the middle 95% of the
observations. Therefore, the
remaining 5% is equally divided
between the two tails and each
tail gets 2.5%. Thus the z value
we are looking for is Z.025.
Option 1: to find the z value in
the table, we look for .025 (the
subscript of Z) inside the table.
Here we can find an exact
match for .025. Thus the
corresponding z value is 1.96.
Option 2: =NORMSINV(1-.025)
9-9
How to Obtain z value for a Given
Confidence Level
Formally the z-value we are looking for is
Zα/2
Example:
Recall that the level of confidence is denoted 1- α. For a 95% level of
confidence,
1- α = 95%
α = 5% =.05
α/2 = .025
Thus
Zα/2 = Z.025
95% C.I. for the population mean:
9-10
Example: Confidence Interval for a Mean Interpretation
The American Management Association wishes to have information
on the mean income of middle managers in the retail industry. A
random sample of 256 managers reveals a sample mean of
$45,420. The standard deviation of this population is $2,050. The
association would like answers to the following questions:
3.
What do these results mean?
The 95% confidence limit are $45,169 and $45,671. This means we
are 95% confidence that the mean income of middle managers in
the retail industry lies in between $45,169 and $45,671.
9-11
Interval Estimates - Interpretation
The meaning of being 95% confidence:
if we construct similar intervals using 100 samples with the same
size from the same population, then 95% of the intervals will include
the true population mean.
9-12
Population Standard Deviation (σ) Unknown
In most sampling situations the population standard deviation (σ) is
not known. Below are some examples where it is unlikely the
population standard deviations would be known.
In such situations, we replace the unknown with sample standard
deviation s, and replace z values with t values. The formula is given
below.
Compute the C.I. using the t - dist. (since is unknow n)
s
X  t / 2,n 1
n
9-13
Example: Confidence Interval for a Mean –
σ Unknown
A tire manufacturer wishes to
investigate the tread life of its
tires. A sample of 10 tires
driven 50,000 miles revealed a
sample mean of 0.32 inch of
tread remaining with a standard
deviation of 0.09 inch.
Construct a 95 percent
confidence interval for the
population mean.
Would it be reasonable for the
manufacturer to conclude that
after 50,000 miles the
population mean amount of
tread remaining is 0.30 inches?
Given in the problem :
n  10
x  0.32
s  0.09
Formula
X  t / 2,n 1
s
n
9-14
Example: Confidence Interval for a
Mean – σ Unknown
Finding t value:
tα/2, n-1= t .025, 9 = 2.262
Option 1: look up in the t table.
Option 2: =tinv(.025*2, 9)
Note: 1 – α = 95%.
9-15
T distribution table
tα/2, n-1= t .025, 9 = 2.262
Confidence Interval Estimates for the Mean
The manager of the Inlet Square Mall, near
Ft. Myers, Florida, wants to estimate the
mean amount spent per shopping visit by
customers. A sample of 20 customers
reveals the following amounts spent.
What is the best estimate of the population
mean? Determine a 95% confidence interval.
Interpret the result. Would it be reasonable to
conclude that he population mean is $50?
What about $60? Data Inlet Square Mall
9-17
Confidence Interval Estimates
for the Mean – By Formula
1. The best estimated of the
population mean is the sample
mean: $49.35.
s
2. The 95% confidence interval is
X  t / 2,n 1
from $45.13 to $53.57.
n
3. This means we are 95%
s
 X  t.05 / 2, 20 1
confidence that the mean
n
amount spent per shopping
9.01
visit lies between $45.13 and
 49.35  t.025 ,19
20
$53.57.
9.01
4. Since $50 is in the interval, it is
 49.35  2.093
reasonable to conclude that the
20
population mean is $50. But
 49.35  4.22
$60 is not in the interval, thus
T heendpoint sof t heconfidenceint ervalare $45.13and $53.57
we conclude that the population
Conclude : It is reasonablet hat t hepopulat ionmean
be $50.to be $60.
meancould
is unlikely
T he value of $60is not in t heconfidenceint erval.Hence, we
Comput et heC.I.
using t he t - dist . (since is unknown)
conclude t hat t hepopulat ionmean is unlikely ot be $60.
9-18
Confidence Interval Estimates for the Mean –
Using Excel
Note: this only
works for
constructing
intervals using t
distribution.
Excel instruction:
see Lind et al.,
p 329, #5 for
more details.
9-19
When to Use the z or t Distribution for
Confidence Interval Computation
X  t / 2,n 1

n
X  z / 2

n
9-20
A Confidence Interval for a Population
Proportion (π)

The examples below illustrate the situations that involve
proportion.
1.
2.
3.

The career services director at Southern Technical Institute reports that 80
percent of its graduates enter the job market in a position related to their
field of study.
A company representative claims that 45 percent of Burger King sales are
made at the drive-through window.
A survey of homes in the Chicago area indicated that 85 percent of the
new construction had central air conditioning.
Inference on population proportion is useful for nominal
(categorical) data
Proportion: The fraction, ratio, or percent indicating the part of
the sample or the population having a particular trait of interest.
9-21
Confidence Interval for a Population
Proportion - Formula
Where X is the number of “success,” that is, the number of observations
that fall into the category we are interested in.
9-22
Confidence Interval for a Population ProportionExample
The union representing the
Bottle Blowers of America
(BBA) is considering a proposal
to merge with the Teamsters
Union. According to BBA union
bylaws, at least three-fourths of
the union membership must
approve any merger. A random
sample of 2,000 current BBA
members reveals 1,600 plan to
vote for the merger proposal.
What is the estimate of the
population proportion?
Develop a 95 percent
confidence interval for the
population proportion. Basing
your decision on this sample
information, can you conclude
that the necessary proportion of
BBA members favor the
merger? Why?
First,computethesampleproportion:
x 1,600
p 
 0.80
n 2000
Computethe95% C.I.
C.I.  p  z / 2
p (1  p )
n
 0.80  1.96
.80(1  .80)
 .80  .018
2,000
 (0.782, 0.818)
Conclude : T hemerger proposalwill likely pass
because theintervalestimateincludes valuesgreater
than75 percentof theunion membership.
9-23

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