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Statistics for
Business and Economics
8th Edition
Chapter 7
Estimation: Single Population
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-1
ESTIMATION: AN INTRODUCTION
Definition
The assignment of value(s) to a population parameter
based on a value of the corresponding sample statistic is
called estimation.
Definition
The value(s) assigned to a population parameter based
on the value of a sample statistic is called an estimate.
The sample statistic used to estimate a population
parameter is called an estimator.
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Properties of Unbiased Point
Estimators
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-3
POINT AND INTERVAL ESTIMATES
 A Point Estimate:
The value of a sample statistic that is used to
estimate a population parameter is called a point
estimate.
 An Interval Estimate:
In interval estimation, an interval is constructed
around the point estimate, and it is stated that this
interval is likely to contain the corresponding
population parameter.
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
A Point Estimate
 Usually, whenever we use point estimation, we
calculate the margin of error associated with that
point estimation.
 The margin of error is calculated as follows:
Margin of error  1.96 x
or
 1.96sx
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Interval estimation
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Point and Interval Estimates
 A point estimate is a single number,
 a confidence interval provides additional
information about variability
Lower
Confidence
Limit
Point Estimate
Upper
Confidence
Limit
Width of
confidence interval
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-7
Point Estimates
We can estimate a
Population Parameter …
with a Sample
Statistic
(a Point Estimate)
Mean
μ
x
Proportion
P
p̂
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-8
Confidence Intervals
 Confidence Interval Estimator for a
population parameter is a rule for determining
(based on sample information) a range or an
interval that is likely to include the parameter.
 The corresponding estimate is called a
confidence interval estimate.
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-9
Confidence Interval and
Confidence Level
 If P(a <  < b) = 1 -  then the interval from a
to b is called a 100(1 - )% confidence
interval of .
 The quantity (1 - ) is called the confidence
level of the interval ( between 0 and 1)
 In repeated samples of the population, the true value
of the parameter  would be contained in 100(1 )% of intervals calculated this way.
 The confidence interval calculated in this manner is
written as a <  < b with 100(1 - )% confidence
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-10
Confidence Level, (1-)
(continued)
 Suppose confidence level = 95%
 Also written (1 - ) = 0.95
 A relative frequency interpretation:
 From repeated samples, 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
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-11
General Formula
 The general formula for all confidence
intervals is:
Point Estimate ± (Reliability Factor)(Standard Error)
 The value of the reliability factor
depends on the desired level of
confidence
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-12
Confidence Intervals
Confidence
Intervals
Population
Mean
σ2 Known
Population
Proportion
σ2 Unknown
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-13
ESTIMATION OF A POPULATION
MEAN: Population Variance KNOWN
Three Possible Cases
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
ESTIMATION OF A POPULATION MEAN:
Population Variance NOT KNOWN
Three Possible Cases
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
ESTIMATION OF A POPULATION
MEAN: Population Variance KNOWN
Confidence Interval for μ
(σ2 Known)
 Assumptions
 Population variance σ2 is known
 Population is normally distributed
 If population is not normal, use large sample
 Confidence interval estimate:
x  z α/2
σ
σ
 μ  x  z α/2
n
n
(where z/2 is the normal distribution value for a probability of /2 in
each tail)
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-16
Margin of Error

Keep in mind that in any time sampling occurs, one expects the possibility of a difference
between the particular value of an estimator and the parameter's true value. The true value of an
unknown parameter  might be somewhat greater or somewhat less than the value determined by
even the best point estimator  .

So, The confidence
form-
interval estimate for a parameter takes on the general
Point Estimate ± (Reliability Factor)(Standard Error)
x  z α/2
σ
σ
 μ  x  z α/2
n
n
x  ME

Can also be written as

where ME is called the margin of error (error factor)

The interval width, w, is equal to twice the margin of error.
 w = 2(ME)
The maximum distance between an estimator and the true value of a parameter is called the
margin of error.

ME  z α/2
σ
n
Chap 7-17
Reducing the Margin of Error
ME  z α/2
σ
n
The margin of error can be reduced if-
 The sample size is increased (n↑)
 the population standard deviation can be reduced (σ↓)
 The confidence level is decreased, (1 – ) ↓
 A 95% confidence interval estimate for a population mean is
determined to be 62.8 to 73.4. If the confidence level is reduced to
90%, the confidence interval for becomes narrower
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-18
Finding the Reliability Factor, z/2
 Consider a 95% confidence interval:
1   .95
α
 .025
2
Z units:
α
 .025
2
z = -1.96
X units:
Lower
Confidence
Limit
0
Point Estimate
z = 1.96
Upper
Confidence
Limit
 Find z.025 = 1.96 from the standard normal distribution table
(In 1.96 we find value.9750 from table-1)(So,1-.9750=.025)(.9750-.025=.95 or,95%)
1
α
 1  0.025  0.9750
2
In the value 0.9750 from table-1 we find 1.96, Hence, z=1.96
Chap 7-19
Common Levels of Confidence
 Commonly used confidence levels are 90%,
95%, and 99%
Confidence
Level
80%
90%
95%
98%
99%
99.8%
99.9%
Confidence
Coefficient,
Z/2 value
.80
.90
.95
.98
.99
.998
.999
1.28
1.645
1.96
2.33
2.58
3.08
3.27
1 
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-20
Intervals and Level of Confidence
Sampling Distribution of the Mean
/2
Intervals
extend from
σ
xz
n
1 
/2
x
μx  μ
x1
x2
to
σ
xz
n
Confidence Intervals
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
100(1-)%
of intervals
constructed
contain μ;
100()% do
not.
Chap 7-21
Figure 8.4 Confidence intervals.
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Example-1: Refined Sugar
(Confidence Interval)
 A process produces bags of refined sugar. The
weights of the content of these bags are
normally distributed with standard deviation 1.2
ounces. The contents of a random sample of 25
bags has a mean weight of 19.8 ounces.
 Find the upper and lower confidence limits of a
99% confidence interval for the true mean
weight for all bags of sugar produced by the
process.
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-23
Example 2 (practice)
 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 90% confidence interval for the true
mean resistance of the population.
 Here, Confidence level is 90% or .90, so The area in each tail
of the normal distribution curve is α/2=(1-.90)/2=.05
α
 1  0.05  0.950
2
 In the value 0.950 from table-1 we find 1.65 Hence, z = 1.65
1
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-24
Example-2 (practice)
(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 .35 ohms.
 Solution:
σ
xz
n
 2.20  1.65 (.35/ 11)
 2.20  .1741
2.0259  μ  2.3741
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-25
Example 2(practice)Interpretation
 We are 90% confident that the true mean
resistance is between 2.0259 and 2.3741
ohms
 Although the true mean may or may not be
in this interval, 90% of intervals formed in
this manner will contain the true mean
 Also See Ex-7.3 from book
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-26
Figure 8.4 Confidence intervals.
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
ESTIMATION OF A POPULATION MEAN:
Population Variance NOT KNOWN
Three Possible Cases
Prem Mann, Introductory Statistics, 7/E
Copyright © 2010 John Wiley & Sons. All right reserved
Confidence Intervals
Confidence
Intervals
Population
Mean
σ2 Known
Population
Proportion
σ2 Unknown
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-29
Student’s t Distribution
 Consider a random sample of n observations
 with mean x and standard deviation s
 from a normally distributed population with mean μ
 Then the variable
x μ
t
s/ n
follows the Student’s t distribution with (n - 1) degrees
of freedom
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-30
Confidence Interval for μ
(σ2 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

The Student’s t distribution is the ratio of the standard normal distribution to the square root of
the chi-square distribution divided by its degrees of freedom.
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-31
Confidence Interval for μ
(σ Unknown)
(continued)
 Assumptions
 Population standard deviation is unknown
 Population is normally distributed
 If population is not normal, use large sample
 Use Student’s t Distribution
 Confidence Interval Estimate:
x  t n-1,α/2
S
S
 μ  x  t n-1,α/2
n
n
where tn-1,α/2 is the critical value of the t distribution with n-1 d.f.
and an area of α/2 in each tail: P(t  t
)  α/2
n1
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
n1,α/2
Chap 7-32
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
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-33
Student’s t Table-8 (appendix)
Upper Tail Area
df
.10
.05
.025
1 3.078 6.314 12.706
Let: n = 3
df = n - 1 = 2
 = .10
/2 =.05
2 1.886 2.920 4.303
/2 = .05
3 1.638 2.353 3.182
The body of the table
contains t values, not
probabilities
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
0
2.920 t
Chap 7-34
Example 7.4 (book)
 Gasoline prices rose drastically during the early
years of this century. Suppose that a recent
study was conducted using truck drivers with
equivalent years of experience to test run 24
trucks of a particular model over the same
highway. The sample mean and standard
deviation is 18.68 and 1.69526 respectively.
 Estimate the population mean fuel consumption
for this truck model with 90% confidence
interval.
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-35
Example-3 (practice)
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 n1,α/2  t 24,.025  2.0639
The confidence interval is
S
S
x  t n-1,α/2
 μ  x  t n-1,α/2
n
n
8
8
50  (2.0639)
 μ  50  (2.0639)
25
25
46.698  μ  53.302
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-36
Confidence Intervals
Confidence
Intervals
Population
Mean
σ Known
Population
Proportion
σ Unknown
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-37
Confidence Intervals for the
Population Proportion, p
 An interval estimate for the population
proportion ( P ) can be calculated by
adding an allowance for uncertainty to
the sample proportion ( p̂ )
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-38
Confidence Intervals for the
Population Proportion, p
(continued)
 Recall that the distribution of the sample
proportion is approximately normal if the
sample size is large, with standard deviation
P(1 P)
σP 
n
 We will estimate this with sample data:
pˆ (1  pˆ )
n
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-39
Confidence Interval Endpoints
 Upper and lower confidence limits for the
population proportion are calculated with the
formula
pˆ  z α/2
ˆ (1 pˆ )
pˆ (1 pˆ )
p
 P  pˆ  z α/2
n
n
 where
 z/2 is the standard normal value for the level of confidence desired
 p̂ is the sample proportion
 n is the sample size
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-40
Example-4
 A random sample of 100 people
shows that 25 are left-handed.
 Form a 95% confidence interval for
the true proportion of left-handers
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-41
Example-4
(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.
pˆ  z α/2
ˆ (1  pˆ )
pˆ (1  pˆ )
p
 P  pˆ  z α/2
n
n
25
.25(.75)
25
.25(.75)
 1.96
 P 
 1.96
100
100
100
100
0.1651  P  0.3349
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-42
Ex-4 Interpretation
 We are 95% confident that the true
percentage of left-handers in the population
is between
16.51% and 33.49%.
 Although the interval from 0.1651 to 0.3349
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.
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.
Chap 7-43