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Department of Quantitative Methods & Information Systems
Business Statistics
Chapter 8
Estimating Single
Population Parameters
QMIS 220
Dr. Mohammad Zainal
Chapter Goals
After completing this chapter, you should be
able to:

Distinguish between a point estimate and a confidence
interval estimate

Construct and interpret a confidence interval estimate for a
single population mean using both the z and t distributions

Determine the required sample size to estimate a single
population mean within a specified margin of error

Form and interpret a confidence interval estimate for a
single population proportion
QMIS 220, by Dr. M. Zainal
Chap 8-2
Confidence Intervals
Content of this chapter
 Confidence Intervals for the Population
Mean, μ




when Population Standard Deviation σ is Known
when Population Standard Deviation σ is Unknown
Determining the Required Sample Size
Confidence Intervals for the Population
Proportion, p
QMIS 220, by Dr. M. Zainal
Chap 8-3
Point and Interval Estimates

A point estimate is a single number, used to
estimate an unknown population parameter

a confidence interval provides additional
information about variability
Lower
Confidence
Limit
Point Estimate
Upper
Confidence
Limit
Width of
confidence interval
QMIS 220, by Dr. M. Zainal
Chap 8-4
Point Estimates
We can estimate a
Population Parameter …
with a Sample
Statistic
(a Point Estimate)
Mean
μ
x
Proportion
π
p
QMIS 220, by Dr. M. Zainal
Chap 8-5
Confidence Intervals

How much uncertainty is associated with a
point estimate of a population parameter?

An interval estimate provides more
information about a population characteristic
than does a point estimate

Such interval estimates are called confidence
intervals
QMIS 220, by Dr. M. Zainal
Chap 8-6
Confidence Interval Estimate

An interval gives a range of values:

Takes into consideration variation in sample
statistics from sample to sample

Based on observation from 1 sample

Gives information about closeness to
unknown population parameters

Stated in terms of level of confidence

QMIS 220, by Dr. M. Zainal
Never 100% sure
Chap 8-7
Estimation Process
Random Sample
Population
(mean, μ, is
unknown)
Mean
x = 50
I am 95%
confident that
μ is between
40 & 60.
Sample
QMIS 220, by Dr. M. Zainal
Chap 8-8
General Formula

The general formula for all confidence
intervals is:
Point Estimate  (Critical Value)(Standard Error)
The Margin of Error
QMIS 220, by Dr. M. Zainal
Chap 8-9
Confidence Level

Confidence Level


QMIS 220, by Dr. M. Zainal
Confidence in which the interval
will contain the unknown
population parameter
A percentage (less than 100%)
Chap 8-10
Confidence Level, (1-)
(continued)



Suppose confidence level = 95%
Also written (1 - ) = .95
A relative frequency interpretation:


In the long run, 95% of all the confidence
intervals that can be constructed will contain the
unknown true parameter
A specific interval either will contain or will
not contain the true parameter

QMIS 220, by Dr. M. Zainal
No probability involved in a specific interval
Chap 8-11
Confidence Intervals
Confidence
Intervals
Population
Mean
σ Known
QMIS 220, by Dr. M. Zainal
Population
Proportion
σ Unknown
Chap 8-12
Confidence Interval for μ
(σ Known)

Assumptions
 Population standard deviation σ is known
 Population is normally distributed
 Or, the sample size is large (that is, n  30)

Confidence interval estimate
σ
xz
n
QMIS 220, by Dr. M. Zainal
Chap 8-13
Finding the Critical Value

Consider a 95% confidence
interval:
z  1.96
1   .95
α
 .025
2
α
 .025
2
z units: -z = -1.96
x units:
Lower
Confidence
Limit
xz
QMIS 220, by Dr. M. Zainal
0
σ
n
Point Estimate
x
z = 1.96
Upper
Confidence
Limit
xz
σ
n
Chap 8-14
Common Levels of Confidence

Commonly used confidence levels are
90%, 95%, and 99%
Confidence
Level
80%
90%
95%
98%
99%
99.8%
99.9%
QMIS 220, by Dr. M. Zainal
Confidence
Coefficient,
1 
.80
.90
.95
.98
.99
.998
.999
Critical
value, z
1.28
1.645
1.96
2.33
2.58
3.08
3.27
Chap 8-15
Interval and Level of Confidence
Sampling Distribution of the Mean
/2
Intervals
extend from
σ
xz
n
1 
μx  μ
x
x1
x2
to
σ
xz
n
QMIS 220, by Dr. M. Zainal
/2
100(1-)%
of intervals
constructed
contain μ;
100% do not.
Confidence Intervals
Chap 8-16
Margin of Error

Margin of Error (e): the amount added and
subtracted to the point estimate to form the
confidence interval
Example: Margin of error for estimating μ, σ known:
σ
xz
n
QMIS 220, by Dr. M. Zainal
σ
ez
n
Chap 8-17
Factors Affecting Margin of Error
σ
ez
n

Data variation, σ :
e
as σ

Sample size, n :
e
as n

Level of confidence, 1 -  :
e
if 1 - 
QMIS 220, by Dr. M. Zainal
Chap 8-18
Example

A sample of 11 circuits from a large normal
population has a mean resistance of 2.20
ohms. We know from past testing that the
population standard deviation is 0.35 ohms.

Determine a 95% confidence interval for the
true mean resistance of the population.
QMIS 220, by Dr. M. Zainal
Chap 8-19
Example
(continued)

A sample of 11 circuits from a large normal
population has a mean resistance of 2.20
ohms. We know from past testing that the
population standard deviation is 0.35 ohms.

Solution:
σ
x z
n
 2.20  1.96 (0.35/ 11)
 2.20  .2068
1.9932 .......... ..... 2.4068
QMIS 220, by Dr. M. Zainal
Chap 8-20
Interpretation

We are 95% confident that the true mean
resistance is between 1.9932 and 2.4068 ohms

Although the true mean may or may not be in this
interval, 95% of intervals formed in this manner
will contain the true mean

An incorrect interpretation is that there is 95% probability that this
interval contains the true population mean.
(This interval either does or does not contain the true mean, there is
no probability for a single interval)
QMIS 220, by Dr. M. Zainal
Chap 8-21
Confidence Intervals
Confidence
Intervals
Population
Mean
σ Known
QMIS 220, by Dr. M. Zainal
Population
Proportion
σ Unknown
Chap 8-22
Confidence Interval for μ
(σ Unknown)

If the population standard deviation σ is
unknown, we can substitute the sample
standard deviation, s

This introduces extra uncertainty, since s
is variable from sample to sample

So we use the t distribution instead of the
normal distribution
QMIS 220, by Dr. M. Zainal
Chap 8-23
Confidence Interval for μ
(σ Unknown)
(continued)

Assumptions





Population standard deviation is unknown
Population is (approximately) normally distributed
The sample size is small (that is, n < 30)
Use Student’s t Distribution
Confidence Interval Estimate
s
xt
n
QMIS 220, by Dr. M. Zainal
Chap 8-24
Student’s t Distribution

The t is a family of distributions

The t value depends on degrees of
freedom (d.f.)

Number of observations that are free to vary after
sample mean has been calculated
d.f. = n - 1
QMIS 220, by Dr. M. Zainal
Chap 8-25
Degrees of Freedom (df)
Idea: Number of observations that are free to vary
after sample mean has been calculated
Example: Suppose the mean of 3 numbers is 8.0
Let x1 = 7
Let x2 = 8
What is x3?
If the mean of these three
values is 8.0,
then x3 must be 9
(i.e., x3 is not free to vary)
Here, n = 3, so degrees of freedom = n -1 = 3 – 1 = 2
(2 values can be any numbers, but the third is not free to vary
for a given mean)
QMIS 220, by Dr. M. Zainal
Chap 8-26
Student’s t Distribution
Note: t
z as n increases
Standard Normal
(t with df = )
t (df = 13)
t-distributions are bellshaped and symmetric, but
have ‘fatter’ tails than the
normal
t (df = 5)
0
QMIS 220, by Dr. M. Zainal
t
Chap 8-27
Student’s t Table
Upper Tail Area
df
.25
.10
.05
1 1.000 3.078 6.314
Let: n = 3
df = n - 1 = 2
 = .10
/2 =.05
2 0.817 1.886 2.920
/2 = .05
3 0.765 1.638 2.353
The body of the table
contains t values, not
probabilities
QMIS 220, by Dr. M. Zainal
0
2.920 t
Chap 8-28
Example: Student’s t Table
Find the value of t for n = 10 and .05 area in
the right tail
n = 10
df = n - 1 = 9
/2 =.05
/2 = .05
The required value of t for 9
df and .05 area in the right tail
0
QMIS 220, by Dr. M. Zainal
1.833
Chap 8-29
t
t distribution values
With comparison to the z value
Confidence
t
Level
(10 d.f.)
t
(20 d.f.)
t
(30 d.f.)
z
____
.80
1.372
1.325
1.310
1.28
.90
1.812
1.725
1.697
1.64
.95
2.228
2.086
2.042
1.96
.99
3.169
2.845
2.750
2.58
Note: t
QMIS 220, by Dr. M. Zainal
z as n increases
Chap 8-30
Example: Student’s t Table
A random sample of n = 25 has x = 50 and
s = 8. Form a 95% confidence interval for μ

d.f. = n – 1 = 24, so t /2 , n1  t.025,24  2.0639
The confidence interval is
s
8
xt
 50  (2.0639)
n
25
46.698 …………….. 53.302
QMIS 220, by Dr. M. Zainal
Chap 8-31
Approximation for Large Samples



Since t approaches z as the sample size increases,
an approximation is sometimes used when n is very
large (n  30)
The text t-table provides t values up to 500 degrees
of freedom
Computer software will provide the correct t-value for
any degrees of freedom
Correct formula,
σ unknown
s
xt
n
QMIS 220, by Dr. M. Zainal
Approximation
for very large n
s
xz
n
Chap 8-32
Determining Sample Size

The required sample size can be found to
reach a desired margin of error (e) and
level of confidence (1 - )

Required sample size, σ known:
2
zσ
z σ
  2
n  
e
 e 
QMIS 220, by Dr. M. Zainal
2
2
Chap 8-33
Required Sample Size Example
If  = 45, what sample size is needed to be
90% confident of being correct within ± 5?
zσ
1.645 (45)
n 2 
 219.19
2
e
5
2
2
2
2
So the required sample size is n = 220
(Always round up)
QMIS 220, by Dr. M. Zainal
Chap 8-34
If σ is unknown

If unknown, σ can be estimated when
using the required sample size formula



QMIS 220, by Dr. M. Zainal
Use a value for σ that is expected to be at least
as large as the true σ
Select a pilot sample and estimate σ with the
sample standard deviation, s
Use the range R to estimate the standard deviation
using σ = R/6 (or R/4 for a more conservative
estimate, producing a larger sample size)
Chap 8-35
Confidence Intervals
Confidence
Intervals
Population
Mean
σ Known
QMIS 220, by Dr. M. Zainal
Population
Proportion
σ Unknown
Chap 8-36
Confidence Intervals for the
Population Proportion, π

An interval estimate for the population
proportion ( π ) can be calculated by
adding an allowance for uncertainty to
the sample proportion ( p )
QMIS 220, by Dr. M. Zainal
Chap 8-37
Confidence Intervals for the
Population Proportion, π
(continued)

Recall that the distribution of the sample
proportion is approximately normal if the
sample size is large, with standard deviation
π(1 π)
σπ 
n

We will estimate this with sample data:
p(1 p)
sp 
n
QMIS 220, by Dr. M. Zainal
Chap 8-38
Confidence interval endpoints

Upper and lower confidence limits for the
population proportion are calculated with the
formula
p(1 p)
pz
n

where



z is the standard normal value for the level of confidence desired
p is the sample proportion
n is the sample size
QMIS 220, by Dr. M. Zainal
Chap 8-39
Example

A random sample of 100 people
shows that 25 are left-handed.

Form a 95% confidence interval for
the true proportion of left-handers
QMIS 220, by Dr. M. Zainal
Chap 8-40
Example
(continued)

A random sample of 100 people shows
that 25 are left-handed. Form a 95%
confidence interval for the true proportion
of left-handers.
1. P = 25/100 = 0.25
2. Sp 
3.
p(1  p)/n  .25(.75)/1 00  .0433
.25  1.96 (.0433)
0.1651 . . . . . 0.3349
QMIS 220, by Dr. M. Zainal
Chap 8-41
Interpretation

We are 95% confident that the true
percentage of left-handers in the population
is between
16.51% and 33.49%.

Although this range may or may not contain
the true proportion, 95% of intervals formed
from samples of size 100 in this manner will
contain the true proportion.
QMIS 220, by Dr. M. Zainal
Chap 8-42
Changing the sample size

Increases in the sample size reduce
the width of the confidence interval.
Example:

If the sample size in the above example is
doubled to 200, and if 50 are left-handed in the
sample, then the interval is still centered at .25,
but the width shrinks to
.19 …… .31
QMIS 220, by Dr. M. Zainal
Chap 8-43
Finding the Required Sample Size
for proportion problems
Define the
margin of error:
π(1 π)
ez
n
z π (1  π)
2
Solve for n:
n
e
2
π can be estimated with a pilot sample, if
necessary (or conservatively use π = .50)
QMIS 220, by Dr. M. Zainal
Chap 8-44
What sample size...?

How large a sample would be necessary
to estimate the true proportion defective in
a large population within 3%, with 95%
confidence?
(Assume a pilot sample yields p = .12)
QMIS 220, by Dr. M. Zainal
Chap 8-45
What sample size...?
(continued)
Solution:
For 95% confidence, use Z = 1.96
E = .03
p = .12, so use this to estimate π
n
z 2 π (1 π)
e2
(1.96) 2 (.12)(1  .12)

 450.74
2
(.03)
So use n = 451
QMIS 220, by Dr. M. Zainal
Chap 8-46
Flowchart
QMIS 220, by Dr. M. Zainal
Chap 8-47
Problems
According to CardWeb.com, the mean bank credit card debt for
households was $7868 in 2004. Assume that this mean was
based on a random sample of 900 households and that the
standard deviation of such debts for all households in 2004
was $2070. Make a 99% confidence interval for the 2004 mean
bank credit card debt for all households.
QMIS 220, by Dr. M. Zainal
Chap 8-48
Problems
QMIS 220, by Dr. M. Zainal
Chap 8-49
Problems
According to a 2002 survey, 20% of Americans needed legal
advice during the past year to resolve such thorny issues as
family trusts and landlord disputes. Suppose a recent sample
of 1000 adult Americans showed that 20% of them needed
legal advice during the past year to resolve such family-related
issues.
(a) What is the point estimate of the population proportion?
What is the margin of error for this estimate?
(b) Construct a 99% confidence interval for all adults
Americans who needed legal advice during the past year.
QMIS 220, by Dr. M. Zainal
Chap 8-50
Problems
QMIS 220, by Dr. M. Zainal
Chap 8-51
Problems
An alumni association wants to estimate the mean debt of this
year's college graduates. It is known that the population
standard deviation of the debts of this year's college graduates
is $11,800. How large a sample should be selected so that the
estimate with a 99% confidence level is within $800 of the
population mean?
QMIS 220, by Dr. M. Zainal
Chap 8-52
Problems
QMIS 220, by Dr. M. Zainal
Chap 8-53
Problems
An electronics company has just installed a new machine that
makes a part that is used in clocks. The company wants to
estimate the proportion of these parts produced by this
machine that are defective. The company manager wants this
estimate to be within .02 of the population proportion for a 95%
confidence level. What is the most conservative estimate of the
sample size that will limit the maximum error to within .02 of the
population proportion?
QMIS 220, by Dr. M. Zainal
Chap 8-54
Problems
QMIS 220, by Dr. M. Zainal
Chap 8-55
Problems
A doctor wanted to estimate the mean cholesterol level for all
adult men living in Dasmah. He took a sample of 25 adult men
from Dasmah and found that the mean cholesterol level for this
sample is 186 with a standard deviation of 12. Assume that the
cholesterol level for all adult men in Dasmah are
(approximately) normally distributed. Construct a 95%
confidence interval for the population mean µ.
QMIS 220, by Dr. M. Zainal
Chap 8-56
Chapter Summary







Illustrated estimation process
Discussed point estimates
Introduced interval estimates
Discussed confidence interval estimation for
the mean (σ known)
Addressed determining sample size
Discussed confidence interval estimation for
the mean (σ unknown)
Discussed confidence interval estimation for
the proportion
QMIS 220, by Dr. M. Zainal
Chap 8-57
Copyright
The materials of this presentation were mostly
taken from the PowerPoint files accompanied
Business Statistics: A Decision-Making Approach,
7e © 2008 Prentice-Hall, Inc.
QMIS 220, by Dr. M. Zainal
Chap 8-58