Statistics for
Business and Economics
6th Edition
Chapter 8
Estimation: Single Population
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-1
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
Form and interpret a confidence interval estimate
for a single population proportion
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-2
Confidence Intervals
Content of this chapter
Confidence Intervals for the Population
Mean, μ
when Population Variance σ2 is Known
when Population Variance σ2 is Unknown
Confidence Intervals for the Population
Proportion, p̂ (large samples)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-3
Definitions
An estimator of a population parameter is
a random variable that depends on sample
information . . .
whose value provides an approximation to this
unknown parameter
A specific value of that random variable is
called an estimate
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-4
Point and Interval Estimates
A point estimate is a single number,
a confidence interval provides additional
information about variability
Lower Upper
Confidence Confidence
Point Estimate
Limit Limit
Width of
confidence interval
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-5
Point Estimates
We can estimate a with a Sample
Population Parameter … Statistic
(a Point Estimate)
Mean μ x
Proportion P p̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-6
Unbiasedness
A point estimator θ̂ is said to be an
unbiased estimator of the parameter if the
expected value, or mean, of the sampling
distribution of θ̂ is ,
E(θˆ ) θ
Examples:
The sample mean is an unbiased estimator of μ
The sample variance is an unbiased estimator of σ2
The sample proportion is an unbiased estimator of P
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-7
Unbiasedness
(continued)
θ̂1 is an unbiased estimator, θ̂ 2 is biased:
θ̂1 θ̂ 2
θ θ̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-8
Bias
Let θ̂ be an estimator of
The bias in θ̂ is defined as the difference
between its mean and
Bias(θˆ ) E(θˆ ) θ
The bias of an unbiased estimator is 0
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-9
Consistency
Let θ̂ be an estimator of
θ̂ is a consistent estimator of if the
difference between the expected value of θ̂
and decreases as the sample size increases
Consistency is desired when unbiased
estimators cannot be obtained
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-10
Most Efficient Estimator
Suppose there are several unbiased estimators of
The most efficient estimator or the minimum variance
unbiased estimator of is the unbiased estimator with the
smallest variance
Let θ̂1 and θ̂ 2 be two unbiased estimators of , based on
the same number of sample observations. Then,
ˆ ) Var( θˆ )
θ̂1 is said to be more efficient than θ̂ 2 if Var( θ1 2
The relative efficiency of θ̂1 with respect to θ̂ 2 is the ratio
of their variances:
Var( θˆ 2 )
Relative Efficiency
Var( θˆ )
1
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-11
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-12
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
Can never be 100% confident
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-13
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, 6e © 2007 Pearson Education, Inc. Chap 8-14
Estimation Process
Random Sample I am 95%
confident that
Population μ is between
Mean 40 & 60.
(mean, μ, is X = 50
unknown)
Sample
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-15
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, 6e © 2007 Pearson Education, Inc. Chap 8-16
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, 6e © 2007 Pearson Education, Inc. Chap 8-17
Confidence Intervals
Confidence
Intervals
Population Population
Mean Proportion
σ2 Known σ2 Unknown
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-18
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:
σ σ
x z α/2 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, 6e © 2007 Pearson Education, Inc. Chap 8-19
Margin of Error
The confidence interval,
σ σ
x z α/2 μ x z α/2
n n
Can also be written as x ME
where ME is called the margin of error
σ
ME z α/2
n
The interval width, w, is equal to twice the margin of
error
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-20
Reducing the Margin of Error
σ
ME z α/2
n
The margin of error can be reduced if
the population standard deviation can be reduced (σ↓)
The sample size is increased (n↑)
The confidence level is decreased, (1 – ) ↓
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-21
Finding the Reliability Factor, z/2
Consider a 95% confidence interval:
1 .95
α α
.025 .025
2 2
Z units: z = -1.96 0 z = 1.96
Lower Upper
X units: Confidence Point Estimate Confidence
Limit Limit
Find z.025 = 1.96 from the standard normal distribution table
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-22
Common Levels of Confidence
Commonly used confidence levels are 90%,
95%, and 99%
Confidence
Confidence
Coefficient, Z/2 value
Level
1
80% .80 1.28
90% .90 1.645
95% .95 1.96
98% .98 2.33
99% .99 2.58
99.8% .998 3.08
99.9% .999 3.27
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-23
Intervals and Level of Confidence
Sampling Distribution of the Mean
/2 1 /2
x
Intervals μx μ
extend from x1
σ x2 100(1-)%
xz of intervals
n
constructed
to
σ contain μ;
xz
n 100()% do
Confidence Intervals not.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-24
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.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-25
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 .35 ohms.
σ
Solution: x z
n
2.20 1.96 (.35/ 11 )
2.20 .2068
1.9932 μ 2.4068
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-26
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-27
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-28
The Wall Street Journal reported that automobile crashes cost the United States
$162 billion annually (The Wall Street Journal, March 5, 2008). The average cost
per person for crashes in the Tampa, Florida, area was reported to be $1599.
Suppose this average cost was based on a sample of 50 persons who had been
involved in car crashes and that the population standard deviation is σ=$600. What
is the margin of error for a 95% confidence interval? What would you recommend if
the study required a margin of error of $150 or less?
A simple random sample of 60 items resulted in a sample mean of 80. The
population standard deviation is σ=15.
a. Compute the 95% confidence interval for the population mean.
b. Assume that the same sample mean was obtained from a sample of 120 items.
Provide a 95% confidence interval for the population mean.
c. What is the effect of a larger sample size on the interval estimate?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-29
Confidence Intervals
Confidence
Intervals
Population Population
Mean Proportion
σ2 Known σ2 Unknown
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-30
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, 6e © 2007 Pearson Education, Inc. Chap 8-31
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-32
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:
S S
x t n-1,α/2 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 n1 t n1,α/2 ) α/2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-33
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, 6e © 2007 Pearson Education, Inc. Chap 8-34
Student’s t Distribution
Note: t Z as n increases
Standard
Normal
(t with df = ∞)
t (df = 13)
t-distributions are bell-
shaped and symmetric, but
have ‘fatter’ tails than the t (df = 5)
normal
0 t
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-35
Student’s t Table
Upper Tail Area
Let: n = 3
df .10 .05 .025 df = n - 1 = 2
= .10
1 3.078 6.314 12.706 /2 =.05
2 1.886 2.920 4.303
3 1.638 2.353 3.182 /2 = .05
The body of the table
contains t values, not 0 2.920 t
probabilities
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-36
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 1.282
.90 1.812 1.725 1.697 1.645
.95 2.228 2.086 2.042 1.960
.99 3.169 2.845 2.750 2.576
Note: t Z as n increases
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-37
Example
A random sample of n = 25 has x = 50 and
s = 8. Form a 95% confidence interval for μ
t n1,α/2 t 24,.025 2.0639
d.f. = n – 1 = 24, so
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, 6e © 2007 Pearson Education, Inc. Chap 8-38
Confidence Intervals
Confidence
Intervals
Population Population
Mean Proportion
σ Known σ Unknown
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-39
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, 6e © 2007 Pearson Education, Inc. Chap 8-40
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, 6e © 2007 Pearson Education, Inc. Chap 8-41
Confidence Interval Endpoints
Upper and lower confidence limits for the
population proportion are calculated with the
formula
pˆ (1 pˆ ) ˆ (1 pˆ )
p
pˆ z α/2 P pˆ z α/2
n n
where
p̂
z/2 is the standard normal value for the level of confidence desired
is the sample proportion
n is the sample size
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-42
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-43
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.
pˆ (1 pˆ ) ˆ (1 pˆ )
p
pˆ z α/2 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, 6e © 2007 Pearson Education, Inc. Chap 8-44
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, 6e © 2007 Pearson Education, Inc. Chap 8-45
Chapter Summary
Introduced the concept of confidence
intervals
Discussed point estimates
Developed confidence interval estimates
Created confidence interval estimates for the
mean (σ2 known)
Introduced the Student’s t distribution
Determined confidence interval estimates for
the mean (σ2 unknown)
Created confidence interval estimates for the
proportion
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 8-46
