Statistics for Business and EconomicsStatistics for Business and Economics 7thEdition Chapter Goals...
Transcript of Statistics for Business and EconomicsStatistics for Business and Economics 7thEdition Chapter Goals...
Chapter 8
Estimation: Additional Topics
ECON 509, by Dr. M. Zainal Ch. 8-1
Statistics for Business and Economics
7th Edition
Chapter GoalsAfter 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
Ch. 8-2ECON 509, by Dr. M. Zainal
Estimation: Additional Topics
Ch. 8-3ECON 509, by Dr. M. Zainal
Chapter Topics
Population Means,
Independent Samples
Population Means,
Dependent Samples
Sample Size Determination
Group 1 vs. independent
Group 2
Same group before vs. after
treatment
Finite Populations
Examples:
Population Proportions
Proportion 1 vs. Proportion 2
Large Populations
Confidence Intervals
Dependent Samples
Tests Means of 2 Related Populations Paired or matched samples Repeated measures (before/after) Use difference between paired values:
Eliminates Variation Among Subjects Assumptions:
Both Populations Are Normally Distributed
Ch. 8-4ECON 509, by Dr. M. Zainal
Dependent samples
di = xi - yi
Mean Difference
The ith paired difference is di , where
Ch. 8-5ECON 509, by Dr. M. Zainal
di = xi - yi
The point estimate for the population mean paired difference is d : n
dd
n
1ii
n is the number of matched pairs in the sample
1n
)d(dS
n
1i
2i
d
The sample standard deviation is:
Dependent samples
Confidence Interval forMean Difference
The confidence interval for difference between population means, μd , is
Ch. 8-6ECON 509, by Dr. M. Zainal
Where n = the sample size
(number of matched pairs in the paired sample)
nStdμ
nStd d
α/21,ndd
α/21,n
Dependent samples
Confidence Interval forMean Difference
The margin of error is
tn-1,/2 is the value from the Student’s t distribution with (n – 1) degrees of freedom for which
Ch. 8-7ECON 509, by Dr. M. Zainal
(continued)
2α)tP(t α/21,n1n
nstME d
α/21,n
Dependent samples
Six people sign up for a weight loss program. You collect the following data:
ECON 509, by Dr. M. Zainal Ch. 8-8
Paired Samples Example
Weight:Person Before (x) After (y) Difference, di
1 136 125 112 205 195 103 157 150 7 4 138 140 - 25 175 165 106 166 160 6
42
d = din
4.821n
)d(dS
2i
d
= 7.0
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
Ch. 8-9ECON 509, by Dr. M. Zainal
12.06μ1.946
4.82(2.571)7μ6
4.82(2.571)7
nStdμ
nStd
d
d
dα/21,nd
dα/21,n
Paired Samples Example(continued)
Since this interval contains zero, we cannot be 95% confident, given this limited data, that the weight loss program helps people lose weight
Dependent samples
Difference Between Two Means:Independent Samples
Different data sources Unrelated Independent
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:
Ch. 8-10ECON 509, by Dr. M. Zainal
Population means, independent
samples
Goal: Form a confidence interval for the difference between two population means, μx – μy
x – y
Difference Between Two Means:Independent Samples
Ch. 8-11ECON 509, by Dr. M. Zainal
Population means, independent
samples
Confidence interval uses z/2
Confidence interval uses a value from the Student’s t distribution
σx2 and σy
2
assumed equal
σx2 and σy
2 known
σx2 and σy
2 unknown
σx2 and σy
2
assumed unequal
(continued)
σx2 and σy
2 Known
Ch. 8-12ECON 509, by Dr. M. Zainal
Population means, independent
samples
Assumptions:
Samples are randomly andindependently drawn
both population distributionsare normal
Population variances areknown
*σx2 and σy
2 known
σx2 and σy
2 unknown
σx2 and σy
2 Known
Ch. 8-13ECON 509, by Dr. M. Zainal
Population means, independent
samples
…and the random variable
has a standard normal distribution
When σx and σy are known and both populations are normal, the
variance of X – Y is
y
2y
x
2x2
YX nσ
nσσ
(continued)
*
Y
2y
X
2x
YX
nσ
nσ
)μ(μ)yx(Z
σx2 and σy
2 known
σx2 and σy
2 unknown
Confidence Interval, σx
2 and σy2 Known
Ch. 8-14ECON 509, by Dr. M. Zainal
Population means, independent
samples
The confidence interval for μx – μy is:*
y
2Y
x
2X
α/2YXy
2Y
x
2X
α/2 nσ
nσz)yx(μμ
nσ
nσz)yx(
σx2 and σy
2 known
σx2 and σy
2 unknown
σx2 and σy
2 Unknown,Assumed Equal
Ch. 8-15ECON 509, by Dr. M. Zainal
Population means, independent
samples
Assumptions: Samples are randomly and
independently drawn
Populations are normallydistributed
Population variances areunknown but assumed equal*σx
2 and σy2
assumed equal
σx2 and σy
2 known
σx2 and σy
2 unknown
σx2 and σy
2
assumed unequal
σx2 and σy
2 Unknown,Assumed Equal
Ch. 8-16ECON 509, by Dr. M. Zainal
Population means, independent
samples
(continued)
Forming interval estimates:
The population variances are assumed equal, so usethe two sample standard deviations and pool them toestimate σ
use a t value with (nx + ny – 2) degrees offreedom
*σx2 and σy
2
assumed equal
σx2 and σy
2 known
σx2 and σy
2 unknown
σx2 and σy
2
assumed unequal
σx2 and σy
2 Unknown,Assumed Equal
Ch. 8-17ECON 509, by Dr. M. Zainal
Population means, independent
samples
The pooled variance is
(continued)
* 2nn1)s(n1)s(n
syx
2yy
2xx2
p
σx2 and σy
2
assumed equal
σx2 and σy
2 known
σx2 and σy
2 unknown
σx2 and σy
2
assumed unequal
Confidence Interval, σx
2 and σy2 Unknown, Equal
Ch. 8-18ECON 509, by Dr. M. Zainal
The confidence interval forμ1 – μ2 is:
Where
*σx2 and σy
2
assumed equal
σx2 and σy
2 unknown
σx2 and σy
2
assumed unequal
y
2p
x
2p
α/22,nnYXy
2p
x
2p
α/22,nn ns
ns
t)yx(μμns
ns
t)yx(yxyx
2nn1)s(n1)s(n
syx
2yy
2xx2
p
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 CPUyNumber Tested 17 14Sample mean 3004 2538Sample std dev 74 56
Ch. 8-19ECON 509, by Dr. M. Zainal
Assume both populations are normal with equal variances, and use 95% confidence
Calculating the Pooled Variance
Ch. 8-20ECON 509, by Dr. M. Zainal
4427.031)141)-(17
56114741171)n(n
S1nS1nS
22
y
2yy
2xx2
p
(()1x
The pooled variance is:
The t value for a 95% confidence interval is:
2.045tt 0.025 , 29α/2 , 2nn yx
Calculating the Confidence Limits
The 95% confidence interval is
Ch. 8-21ECON 509, by Dr. M. Zainal
y
2p
x
2p
α/22,nnYXy
2p
x
2p
α/22,nn ns
ns
t)yx(μμns
ns
t)yx(yxyx
144427.03
174427.03(2.054)2538)(3004μμ
144427.03
174427.03(2.054)2538)(3004 YX
515.31μμ416.69 YX
We are 95% confident that the mean difference in CPU speed is between 416.69 and 515.31 Mhz.
σx2 and σy
2 Unknown,Assumed Unequal
Ch. 8-22ECON 509, by Dr. M. Zainal
Population means, independent
samples
Assumptions: Samples are randomly and
independently drawn
Populations are normallydistributed
Population variances areunknown and assumedunequal
*σx
2 and σy2
assumed equal
σx2 and σy
2 known
σx2 and σy
2 unknown
σx2 and σy
2
assumed unequal
σx2 and σy
2 Unknown,Assumed Unequal
Ch. 8-23ECON 509, by Dr. M. Zainal
Population means, independent
samples
(continued)
Forming interval estimates:
The population variances areassumed unequal, so a pooledvariance is not appropriate
use a t value with degreesof freedom, where
σx2 and σy
2 known
σx2 and σy
2 unknown
*σx
2 and σy2
assumed equal
σx2 and σy
2
assumed unequal1)/(n
ns
1)/(nns
)ns
()ns(
y
2
y
2y
x
2
x
2x
2
y
2y
x
2x
v
Confidence Interval, σx
2 and σy2 Unknown, Unequal
Ch. 8-24ECON 509, by Dr. M. Zainal
The confidence interval forμ1 – μ2 is:
*σx
2 and σy2
assumed equal
σx2 and σy
2 unknown
σx2 and σy
2
assumed unequal
y
2y
x
2x
α/2,YXy
2y
x
2x
α/2, ns
nst)yx(μμ
ns
nst)yx(
1)/(nns
1)/(nns
)ns
()ns(
y
2
y
2y
x
2
x
2x
2
y
2y
x
2x
vWhere
Two Population Proportions
Ch. 8-25ECON 509, by Dr. M. Zainal
Goal: Form a confidence interval for the difference between two population proportions, Px – Py
The point estimate for the difference is
Population proportions
Assumptions:Both sample sizes are large (generally at least 40 observations in each sample)
yx pp ˆˆ
Two Population Proportions
The random variable
is approximately normally distributed
Ch. 8-26ECON 509, by Dr. M. Zainal
Population proportions
(continued)
y
yy
x
xx
yxyx
n)p(1p
n)p(1p
)p(p)pp(Z
ˆˆˆˆ
ˆˆ
Confidence Interval forTwo Population Proportions
Ch. 8-27ECON 509, by Dr. M. Zainal
Population proportions
The confidence limits for Px – Py are:
y
yy
x
xxyx n
)p(1pn
)p(1pZ )pp(ˆˆˆˆˆˆ
2/
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
Ch. 8-28ECON 509, by Dr. M. Zainal
Example: Two Population Proportions
Men:
Women:
Ch. 8-29ECON 509, by Dr. M. Zainal
0.101240
0.70(0.30)50
0.52(0.48)n
)p(1pn
)p(1py
yy
x
xx
ˆˆˆˆ
0.525026px ˆ
0.704028py ˆ
(continued)
For 90% confidence, Z/2 = 1.645
Example: Two Population Proportions
The confidence limits are:
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
Ch. 8-30ECON 509, by Dr. M. Zainal
(continued)
(0.1012)1.645.70)(.52
n)p(1p
n)p(1pZ)pp(
y
yy
x
xxα/2yx
ˆˆˆˆˆˆ
Sample Size Determination
Ch. 8-31ECON 509, by Dr. M. Zainal
For the Mean
DeterminingSample Size
For theProportion
Large Populations
Finite Populations
For the Mean
For theProportion
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
Ch. 8-32ECON 509, by Dr. M. Zainal
Sample Size Determination
Ch. 8-33ECON 509, by Dr. M. Zainal
nσzx α/2
nσzME α/2
Margin of Error (sampling error)
For the Mean
Large Populations
Sample Size Determination
Ch. 8-34ECON 509, by Dr. M. Zainal
nσzME α/2
(continued)
2
22α/2
MEσzn Now solve
for n to get
For the Mean
Large Populations
Sample Size Determination
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
Ch. 8-35ECON 509, by Dr. M. Zainal
(continued)
Required Sample Size Example
If = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence?
Ch. 8-36ECON 509, by Dr. M. Zainal
(Always round up)
219.195
(45)(1.645)ME
σzn 2
22
2
22α/2
So the required sample size is n = 220
Sample Size Determination:Population Proportion
Ch. 8-37ECON 509, by Dr. M. Zainal
n)p(1pzp α/2
ˆˆˆ
n)p(1pzME α/2
ˆˆ
Margin of Error (sampling error)
For theProportion
Large Populations
Sample Size Determination:Population Proportion
Ch. 8-38ECON 509, by Dr. M. Zainal
2
2α/2
MEz 0.25n
Substitute 0.25 for and solve for n to get
(continued)
n)p(1pzME α/2
ˆˆ
cannot be larger than 0.25, when = 0.5
)p(1p ˆˆ
p̂)p(1p ˆˆ
For theProportion
Large Populations
Sample Size Determination:Population Proportion
The sample and population proportions, 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
Ch. 8-39ECON 509, by Dr. M. Zainal
(continued)
p̂
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?
Ch. 8-40ECON 509, by Dr. M. Zainal
Required Sample Size Example
Ch. 8-41ECON 509, by Dr. M. Zainal
Solution:For 95% confidence, use z0.025 = 1.96ME = 0.03Estimate P(1 – P) = 0.25
So use n = 1068
(continued)
1067.11(0.03)
6)(0.25)(1.9ME
z 0.25n 2
2
2
2α/2
Chapter Summary
Compared two dependent samples (paired samples) Formed confidence intervals for the paired difference
Compared two independent samples 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
Ch. 8-42ECON 509, by Dr. M. Zainal