Business Statistics
Chapter 8
Estimating Single
Population Parameters
Dr. Mohammad Zainal
Department of Economics
ECON 509
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
ECON 509, 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
ECON 509, 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
Point Estimate
Lower
Confidence
Limit
Upper
Confidence
Limit
Width of confidence interval
ECON 509, By Dr. M. Zainal Chap 8-4
Point Estimates
We can estimate a
Population Parameter …
with a Sample
Statistic
(a Point Estimate)
Mean
Proportion p π
x μ
ECON 509, 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
ECON 509, 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
Never 100% sure
ECON 509, By Dr. M. Zainal Chap 8-7
Estimation Process
(mean, μ, is
unknown)
Population
Random Sample
Mean
x = 50
Sample
I am 95%
confident that
μ is between
40 & 60.
ECON 509, By Dr. M. Zainal Chap 8-8
General Formula
The general formula for all confidence
intervals is:
Point Estimate (Critical Value)(Standard Error)
ECON 509, By Dr. M. Zainal Chap 8-9
The Margin of Error
Confidence Level
Confidence Level
Confidence in which the interval
will contain the unknown
population parameter
A percentage (less than 100%)
ECON 509, By Dr. M. Zainal Chap 8-10
Confidence Level, (1-)
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
No probability involved in a specific interval
(continued)
ECON 509, By Dr. M. Zainal Chap 8-11
Confidence Intervals
Population
Mean
σ Unknown
Confidence
Intervals
Population
Proportion
σ Known
ECON 509, By Dr. M. Zainal 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
n
σzx
ECON 509, By Dr. M. Zainal Chap 8-13
Finding the Critical Value
Consider a 95% confidence
interval:
n
σzx
-z = -1.96 z = 1.96
.951
.0252
α .025
2
α
Point Estimate Lower Confidence Limit
Upper Confidence Limit
z units:
x units: Point Estimate
0
1.96z
n
σzx x
ECON 509, By Dr. M. Zainal Chap 8-14
Common Levels of Confidence
Commonly used confidence levels are
90%, 95%, and 99%
Confidence
Level
Confidence
Coefficient,
Critical
value, z
1.28
1.645
1.96
2.33
2.58
3.08
3.27
.80
.90
.95
.98
.99
.998
.999
80%
90%
95%
98%
99%
99.8%
99.9%
1
ECON 509, By Dr. M. Zainal Chap 8-15
μμx
Interval and Level of Confidence
Confidence Intervals
Intervals extend from
to
100(1-)%
of intervals
constructed
contain μ;
100% do not.
Sampling Distribution of the Mean
n
σzx
n
σzx
x
x1
x2
/2 /21
ECON 509, By Dr. M. Zainal Chap 8-16
Margin of Error
Margin of Error (e): the amount added and
subtracted to the point estimate to form the
confidence interval
n
σzx
n
σze
Example: Margin of error for estimating μ, σ known:
ECON 509, By Dr. M. Zainal Chap 8-17
Factors Affecting Margin of Error
Data variation, σ : e as σ
Sample size, n : e as n
Level of confidence, 1 - : e if 1 -
n
σze
ECON 509, 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.
ECON 509, By Dr. M. Zainal Chap 8-19
2.4068 ...............1.9932
.2068 2.20
)11(0.35/ 1.96 2.20
n
σz x
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.
Solution:
(continued)
ECON 509, 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)
ECON 509, By Dr. M. Zainal Chap 8-21
Confidence Intervals
Population
Mean
σ Unknown
Confidence
Intervals
Population
Proportion
σ Known
ECON 509, By Dr. M. Zainal Chap 8-22
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
Confidence Interval for μ (σ Unknown)
ECON 509, By Dr. M. Zainal Chap 8-23
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
Confidence Interval for μ (σ Unknown)
n
stx
(continued)
ECON 509, 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
ECON 509, By Dr. M. Zainal Chap 8-25
If the mean of these three
values is 8.0,
then x3 must be 9
(i.e., x3 is not free to vary)
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?
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)
ECON 509, By Dr. M. Zainal Chap 8-26
Student’s t Distribution
t 0
t (df = 5)
t (df = 13) t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal
Standard Normal (t with df = )
Note: t z as n increases
ECON 509, By Dr. M. Zainal Chap 8-27
Student’s t Table
Upper Tail Area
df
.25 .10 .05
1 1.000 3.078 6.314
2 0.817 1.886
3 0.765 1.638 2.353
t 0 2.920
The body of the table
contains t values, not
probabilities
Let: n = 3
df = n - 1 = 2
= .10
/2 =.05
/2 = .05
ECON 509, By Dr. M. Zainal Chap 8-28
2.920
Example: Student’s t Table
t 0 1.833
n = 10
df = n - 1 = 9
/2 =.05
/2 = .05
ECON 509, By Dr. M. Zainal Chap 8-29
Find the value of t for n = 10 and .05 area in the right tail
The required value of t for 9
df and .05 area in the right tail
t distribution values
With comparison to the z value
Confidence t t t z
Level (10 d.f.) (20 d.f.) (30 d.f.) ____
.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 z as n increases
ECON 509, By Dr. M. Zainal 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
The confidence interval is
2.0639tt .025,241n,/2
25
8(2.0639)50
n
stx
46.698 …………….. 53.302
ECON 509, 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
n
stx
n
szx
Correct formula, σ unknown
Approximation for very large n
ECON 509, By Dr. M. Zainal 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
222
e
σz
e
σzn
ECON 509, By Dr. M. Zainal Chap 8-33
Required Sample Size Example
If = 45, what sample size is needed to be
90% confident of being correct within ± 5?
(Always round up)
219.195
(45)1.645
e
σzn
2
22
2
22
So the required sample size is n = 220
ECON 509, By Dr. M. Zainal Chap 8-34
If σ is unknown
If unknown, σ can be estimated when
using the required sample size formula
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)
ECON 509, By Dr. M. Zainal Chap 8-35
Confidence Intervals
Population
Mean
σ Unknown
Confidence
Intervals
Population
Proportion
σ Known
ECON 509, By Dr. M. Zainal 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 )
ECON 509, By Dr. M. Zainal Chap 8-37
Confidence Intervals for the Population Proportion, π
Recall that the distribution of the sample
proportion is approximately normal if the
sample size is large, with standard deviation
We will estimate this with sample data:
(continued)
n
p)p(1sp
n
π)π(1σπ
ECON 509, By Dr. M. Zainal Chap 8-38
Confidence interval endpoints
Upper and lower confidence limits for the population proportion are calculated with the formula
where
z is the standard normal value for the level of confidence desired
p is the sample proportion
n is the sample size
n
p)p(1zp
ECON 509, 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
ECON 509, By Dr. M. Zainal Chap 8-40
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.
1.
2.
3.
0.3349 . . . . . 0.1651
(.0433) 1.96 .25
(continued)
ECON 509, By Dr. M. Zainal Chap 8-41
.0433 00.25(.75)/1p)/np(1Sp
P = 25/100 = 0.25
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.
ECON 509, 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
ECON 509, By Dr. M. Zainal Chap 8-43
Finding the Required Sample Size for proportion problems
n
π)π(1ze
Solve for n:
Define the margin of error:
2
2
e
π)(1πzn
π can be estimated with a pilot sample, if necessary (or conservatively use π = .50)
ECON 509, 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)
ECON 509, By Dr. M. Zainal Chap 8-45
What sample size...?
Solution:
For 95% confidence, use Z = 1.96
E = .03
p = .12, so use this to estimate π
So use n = 451
450.74(.03)
.12)(.12)(1(1.96)
e
π)(1πzn
2
2
2
2
(continued)
ECON 509, By Dr. M. Zainal Chap 8-46
Flowchart
ECON 509, By Dr. M. Zainal Chap 8-47
Problems
ECON 509, By Dr. M. Zainal Chap 8-48
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.
Problems
ECON 509, By Dr. M. Zainal Chap 8-49
Problems
ECON 509, By Dr. M. Zainal Chap 8-50
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.
Problems
ECON 509, By Dr. M. Zainal Chap 8-51
Problems
ECON 509, By Dr. M. Zainal Chap 8-52
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?
Problems
ECON 509, By Dr. M. Zainal Chap 8-53
Problems
ECON 509, By Dr. M. Zainal Chap 8-54
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?
Problems
ECON 509, By Dr. M. Zainal Chap 8-55
Problems
ECON 509, By Dr. M. Zainal Chap 8-56
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.
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 ECON 509, 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.
ECON 509, By Dr. M. Zainal Chap 8-58