Download Estimation

Statistics for
Business and Economics
7th Edition
Chapter 8
Estimation: Additional Topics
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-1
Chapter Goals
After completing this chapter, you should be able to:

Form confidence intervals for the difference between
two means from dependent samples

Form confidence intervals for the difference between
two independent population means (standard deviations
known or unknown)

Compute confidence interval limits for the difference
between two independent population proportions

Determine the required sample size to estimate a mean
or proportion within a specified margin of error
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-2
Estimation: Additional Topics
Chapter Topics
Confidence Intervals
Population
Means,
Dependent
Samples
Population
Means,
Independent
Samples
Population
Proportions
Examples:
Same group
before vs. after
treatment
Group 1 vs.
independent
Group 2
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Proportion 1 vs.
Proportion 2
Sample Size
Determination
Large
Populations
Finite
Populations
Ch. 8-3
8.1
Dependent Samples
Tests Means of 2 Related Populations
Dependent
samples



Paired or matched samples
Repeated measures (before/after)
Use difference between paired values:
di = xi - yi


Eliminates Variation Among Subjects
Assumptions:
 Both Populations Are Normally Distributed
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-4
Mean Difference
The ith paired difference is di , where
Dependent
samples
di = xi - yi
The point estimate for
the population mean
paired difference is d :
The sample
standard
deviation is:
n
d
d
i 1
i
n
n
Sd 
2
(d

d
)
 i
i1
n 1
n is the number of matched pairs in the sample
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-5
Confidence Interval for
Mean Difference
Dependent
samples
The confidence interval for difference
between population means, μd , is
d  t n1,α/2
Sd
Sd
 μd  d  t n1,α/2
n
n
Where
n = the sample size
(number of matched pairs in the paired sample)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-6
Confidence Interval for
Mean Difference
(continued)
Dependent
samples

The margin of error is
ME  t n1,α/2

sd
n
tn-1,/2 is the value from the Student’s t
distribution with (n – 1) degrees of freedom
for which
α
P(t n1  t n1,α/2 ) 
2
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-7
Paired Samples Example
Dependent
samples
Person
1
2
3
4
5
6
Six people sign up for a
weight loss program. You
collect the following data:

Weight:
Before (x)
After (y)
136
205
157
138
175
166
125
195
150
140
165
160
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Difference, di
11
10
7
-2
10
6
42
di

d = n
= 7.0
Sd 
2
(d

d
)
 i
n 1
 4.82
Ch. 8-8
Paired Samples Example
(continued)
Dependent
samples

For a 95% confidence level, the appropriate t value is
tn-1,/2 = t5,.025 = 2.571

The 95% confidence interval for the difference between
means, μd , is
d  t n1,α/2
7  (2.571)
Sd
S
 μd  d  t n1,α/2 d
n
n
4.82
4.82
 μd  7  (2.571)
6
6
 1.94  μd  12.06
Since this interval contains zero, we cannot be 95% confident, given this
limited data, that the weight loss program helps people lose weight
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-9
Difference Between Two Means:
Independent Samples
8.2
Population means,
independent
samples

Different data sources
 Unrelated
 Independent


Goal: Form a confidence interval
for the difference between two
population means, μx – μy
Sample selected from one population has no effect on the
sample selected from the other population
The point estimate is the difference between the two
sample means:
x–y
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-10
Difference Between Two Means:
Independent Samples
(continued)
Population means,
independent
samples
σx2 and σy2 known
Confidence interval uses z/2
σx2 and σy2 unknown
σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Confidence interval uses a value
from the Student’s t distribution
Ch. 8-11
σx2 and σy2 Known
Population means,
independent
samples
σx2 and σy2 known
Assumptions:
*
σx2 and σy2 unknown
 Samples are randomly and
independently drawn
 both population distributions
are normal
 Population variances are
known
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-12
σx2 and σy2 Known
(continued)
When σx and σy are known and
both populations are normal, the
variance of X – Y is
Population means,
independent
samples
2
σx2 and σy2 known
σx2 and σy2 unknown
σ 2X Y
*
2
σy
σx


nx
ny
…and the random variable
Z
(x  y)  (μX  μY )
2
σ 2x σ y

nX nY
has a standard normal distribution
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-13
Confidence Interval,
σx2 and σy2 Known
Population means,
independent
samples
σx2 and σy2 known
*
σx2 and σy2 unknown
(x  y)  z α/2
The confidence interval for
μx – μy is:
σ 2X σ 2Y
σ 2X σ 2Y

 μX  μY  (x  y)  z α/2

nx ny
nx ny
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-14
σx2 and σy2 Unknown,
Assumed Equal
Assumptions:
Population means,
independent
samples
 Samples are randomly and
independently drawn
σx2 and σy2 known
 Populations are normally
distributed
σx2 and σy2 unknown
σx2 and σy2
assumed equal
*
 Population variances are
unknown but assumed equal
σx2 and σy2
assumed unequal
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-15
σx2 and σy2 Unknown,
Assumed Equal
(continued)
Forming interval
estimates:
Population means,
independent
samples
 The population variances
are assumed equal, so use
the two sample standard
deviations and pool them to
estimate σ
σx2 and σy2 known
σx2 and σy2 unknown
σx2 and σy2
assumed equal
*
σx2 and σy2
assumed unequal
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
 use a t value with
(nx + ny – 2) degrees of
freedom
Ch. 8-16
σx2 and σy2 Unknown,
Assumed Equal
(continued)
Population means,
independent
samples
The pooled variance is
σx2 and σy2 known
σx2 and σy2 unknown
σx2 and σy2
assumed equal
*
sp2 
(n x  1)s2x  (n y  1)s2y
nx  ny  2
σx2 and σy2
assumed unequal
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Ch. 8-17
Confidence Interval,
σx2 and σy2 Unknown, Equal
σx2 and σy2 unknown
σx2 and σy2
assumed equal
*
The confidence interval for
μ1 – μ2 is:
sp2
sp2
σx2 and σy2
assumed unequal
(x  y)  t nx ny 2,α/2
Where
sp2 
sp2
nx

ny
 μX  μY  (x  y)  t nx ny 2,α/2
nx

sp2
ny
(n x  1)s2x  (n y  1)s2y
nx  ny  2
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-18
Pooled Variance Example
You are testing two computer processors for speed.
Form a confidence interval for the difference in CPU
speed. You collect the following speed data (in Mhz):
CPUx
Number Tested
17
Sample mean
3004
Sample std dev
74
CPUy
14
2538
56
Assume both populations are
normal with equal variances,
and use 95% confidence
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-19
Calculating the Pooled Variance
The pooled variance is:
2
2




n

1
S

n

1
S

17  1742  14  1562
x
x
y
y
2
S 

p
(n x  1)  (ny  1)
(17 - 1)  (14  1)
 4427.03
The t value for a 95% confidence interval is:
tnx ny 2 , α/2  t 29 , 0.025  2.045
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-20
Calculating the Confidence Limits

The 95% confidence interval is
(x  y)  t nx ny 2,α/2
(3004  2538)  (2.054)
sp2
nx

sp2
ny
 μX  μY  (x  y)  t nx ny 2,α/2
sp2
nx

sp2
ny
4427.03 4427.03
4427.03 4427.03

 μX  μY  (3004  2538)  (2.054)

17
14
17
14
416.69  μX  μY  515.31
We are 95% confident that the mean difference in
CPU speed is between 416.69 and 515.31 Mhz.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-21
σx2 and σy2 Unknown,
Assumed Unequal
Assumptions:
Population means,
independent
samples
 Samples are randomly and
independently drawn
σx2 and σy2 known
 Populations are normally
distributed
σx2 and σy2 unknown
 Population variances are
unknown and assumed
unequal
σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal
*
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-22
σx2 and σy2 Unknown,
Assumed Unequal
(continued)
Forming interval estimates:
Population means,
independent
samples
 The population variances are
assumed unequal, so a pooled
variance is not appropriate
σx2 and σy2 known
 use a t value with  degrees
of freedom, where
σx2 and σy2 unknown
2
σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal
*
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
 s2x
s2y 
( )  ( )
n y 
 n x
v
2
2
2 2


s
 sx 
  /(n x  1)   y  /(n y  1)
n 
 nx 
 y
Ch. 8-23
Confidence Interval,
σx2 and σy2 Unknown, Unequal
σx2 and σy2 unknown
σx2 and σy2
assumed equal
σx2 and σy2
assumed unequal
(x  y)  t ,α/2
*
The confidence interval for
μ1 – μ2 is:
2
2
s2x s y
s2x s y

 μX  μY  (x  y)  t ,α/2

nx ny
nx ny
Where
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v
 s2x
s2y 
( )  ( )
n y 
 n x
2
2
 s2 
 s2x 
  /(n x  1)   y  /(n y  1)
n 
 nx 
 y
2
Ch. 8-24
8.3
Population
proportions
Two Population Proportions
Goal: Form a confidence interval for
the difference between two
population proportions, Px – Py
Assumptions:
Both sample sizes are large (generally at
least 40 observations in each sample)
The point estimate for
the difference is
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
pˆ x  pˆ y
Ch. 8-25
Two Population Proportions
(continued)
Population
proportions

The random variable
Z
(pˆ x  pˆ y )  (p x  p y )
pˆ x (1 pˆ x ) pˆ y (1 pˆ y )

nx
ny
is approximately normally distributed
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-26
Confidence Interval for
Two Population Proportions
Population
proportions
The confidence limits for
Px – Py are:
(pˆ x  pˆ y )  Z / 2
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
pˆ x (1 pˆ x ) pˆ y (1 pˆ y )

nx
ny
Ch. 8-27
Example:
Two Population Proportions
Form a 90% confidence interval for the
difference between the proportion of
men and the proportion of women who
have college degrees.

In a random sample, 26 of 50 men and
28 of 40 women had an earned college
degree
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-28
Example:
Two Population Proportions
(continued)
Men:
ˆp x  26  0.52
50
Women:
ˆp y  28  0.70
40
pˆ x (1 pˆ x ) pˆ y (1 pˆ y )
0.52(0.48) 0.70(0.30)



 0.1012
nx
ny
50
40
For 90% confidence, Z/2 = 1.645
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-29
Example:
Two Population Proportions
(continued)
The confidence limits are:
(pˆ x  pˆ y )  Z α/2
pˆ x (1  pˆ x ) pˆ y (1  pˆ y )

nx
ny
 (.52  .70)  1.645 (0.1012)
so the confidence interval is
-0.3465 < Px – Py < -0.0135
Since this interval does not contain zero we are 90% confident that the
two proportions are not equal
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-30
8.4
Sample Size Determination
Determining
Sample Size
Large
Populations
For the
Mean
For the
Proportion
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Finite
Populations
For the
Mean
For the
Proportion
Ch. 8-31
Margin of Error

The required sample size can be found to reach
a desired margin of error (ME) with a specified
level of confidence (1 - )

The margin of error is also called sampling error

the amount of imprecision in the estimate of the
population parameter

the amount added and subtracted to the point
estimate to form the confidence interval
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-32
Sample Size Determination
Large
Populations
For the
Mean
Margin of Error
(sampling error)
x  z α/2
σ
n
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ME  z α/2
σ
n
Ch. 8-33
Sample Size Determination
Large
Populations
(continued)
For the
Mean
ME  z α/2
σ
n
Now solve
for n to get
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
z σ
n
2
ME
2
α/2
2
Ch. 8-34
Sample Size Determination
(continued)

To determine the required sample size for the
mean, you must know:

The desired level of confidence (1 - ), which
determines the z/2 value

The acceptable margin of error (sampling error), ME

The population standard deviation, σ
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-35
Required Sample Size Example
If  = 45, what sample size is needed to
estimate the mean within ± 5 with 90%
confidence?
z σ
(1.645) (45)
n

 219.19
2
2
ME
5
2
α/2
2
2
2
So the required sample size is n = 220
(Always round up)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-36
Sample Size Determination:
Population Proportion
Large
Populations
For the
Proportion
pˆ  z α/2
pˆ (1 pˆ )
n
ME  z α/2
pˆ (1  pˆ )
n
Margin of Error
(sampling error)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-37
Sample Size Determination:
Population Proportion
(continued)
Large
Populations
For the
Proportion
ME  z α/2
pˆ (1  pˆ )
n
pˆ (1 pˆ ) cannot
be larger than
0.25, when p̂ =
0.5
Substitute
0.25 for pˆ (1 pˆ )
and solve for
n to get
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
0.25 z
n
2
ME
2
α/2
Ch. 8-38
Sample Size Determination:
Population Proportion
(continued)



The sample and population proportions, p̂ and P, are
generally not known (since no sample has been taken
yet)
P(1 – P) = 0.25 generates the largest possible margin
of error (so guarantees that the resulting sample size
will meet the desired level of confidence)
To determine the required sample size for the
proportion, you must know:

The desired level of confidence (1 - ), which determines the
critical z/2 value

The acceptable sampling error (margin of error), ME

Estimate P(1 – P) = 0.25
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-39
Required Sample Size Example:
Population Proportion
How large a sample would be necessary
to estimate the true proportion defective in
a large population within ±3%, with 95%
confidence?
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-40
Required Sample Size Example
(continued)
Solution:
For 95% confidence, use z0.025 = 1.96
ME = 0.03
Estimate P(1 – P) = 0.25
0.25 z
n
2
ME
2
α/2
2
(0.25)(1.9 6)

 1067.11
2
(0.03)
So use n = 1068
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-41
8.5
Sample Size Determination:
Finite Populations
Finite
Populations
For the
Mean
1. Calculate the required
sample size n0 using the
prior formula:
z 2α/2 σ 2
n0 
ME 2
A finite population
correction factor is added:
σ Nn 
Var(X) 


n  N 1 
2
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
2. Then adjust for the finite
population:
n0N
n
n0  (N - 1)
Ch. 8-42
Sample Size Determination:
Finite Populations
Finite
Populations
For the
Proportion
A finite population
correction factor is added:
P(1- P)  N  n 
ˆ
Var(p) 


n
 N 1 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
1. Solve for n:
NP(1  P)
n
(N  1)σ p2ˆ  P(1 P)
2. The largest possible value
for this expression
(if P = 0.25) is:
0.25(1  P)
n
(N  1)σ p2ˆ  0.25
3. A 95% confidence interval
will extend ±1.96 σpˆ from
the sample proportion
Ch. 8-43
Example: Sample Size to
Estimate Population Proportion
(continued)
σpˆ
How large a sample would be necessary to
estimate within ±5% the true proportion of
college graduates in a population of 850
people with 95% confidence?
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-44
Required Sample Size Example
(continued)
Solution:
 For 95% confidence, use z0.025 = 1.96
 ME = 0.05
1.96 σpˆ  0.05  σpˆ  0.02551
nmax
0.25N
(0.25)(850 )


 264.8
2
2
(N  1)σ pˆ  0.25 (849)(0.02 551)  0.25
So use n = 265
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-45
Chapter Summary

Compared two dependent samples (paired samples)


Compared two independent samples





Formed confidence intervals for the paired difference
Formed confidence intervals for the difference between two
means, population variance known, using z
Formed confidence intervals for the differences between two
means, population variance unknown, using t
Formed confidence intervals for the differences between two
population proportions
Formed confidence intervals for the population variance
using the chi-square distribution
Determined required sample size to meet confidence
and margin of error requirements
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 8-46