UNIT 8: INTRODUCTION TO INTERVAL ESTIMATION 8.1

STATISTICAL METHODS FOR BUSINESS
UNIT 8: INTRODUCTION TO INTERVAL ESTIMATION
8.1.- Introduction to interval estimation
8.2.- Confidence intervals. Construction and
characteristics
8.3.- Confidence intervals for the mean
8.4.- Confidence intervals for the proportion
8.5.- Confidence intervals for the variance
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UNIT 8. GOALS
• Describe advantages and disadvantages of point
estimates and interval estimates.
•Interpret the characteristics of precision and confidence in
estimations.
• Build confidence intervals for the population mean.
• Determine sample size for the mean.
• Build confidence intervals for the population variance and
proportion.
• Determine sample size for the proportion.
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Types of estimates
GOAL
Approximate population parameters which are unknown
The estimates of the parameters can be expressed as:
A) Point Estimate
Disadvantage: there is no measure of how good the estimate is
B) Interval Estimate
Specify a range within which the parameter is estimated to lie
Margin of error
Point estimate
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Relationship between precision and confidence level
Precision
Estimation of p
Maximum
p$
(
p$1
(
0
Confidence
Minimum
0%
)
p$2
)
1
Minimum
Maximum
100%
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Building Confidence Intervals
POPULATION X
Simple random
sample
(X1, X2, ...,Xn)
θ is an unknown
paramenter
P(T1 ≤ θ ≤ T2 ) = 1− α
Estimator
T(X1, X2, ...,Xn)
(
)
T1
T2
P(a ≤ dT ≤ b) = 1 − α
Pivotal quantity
dT(X1, X2, ...,Xn, θ)
Confidence level
1-α
(
a
)
b
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Example. Building a confidence interval for the
population mean
POPULATION X≈
≈N(µ
µ,σ
σ)
σ Is known
Simple random
sample
(X1, X2, ...,Xn)
Estimator
µ is unknown
σ
σ 

 X − 1.96 n , X + 1.96 n 


n
X=
∑X
i =1
)
X
n
Pivotal quantity
dX =
(
i
X−µ
≈ N(0,1)
σ
n
P(−1.96 ≤ d N ( 0,1) ≤ 1.96) = 0.95
Confidence level
1 - α = 0.95
(
-1.96
)
1.96
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Interval Precision
Interval precision is measured through the width of the interval, A.
In case of a symmetric interval, precision can be then evaluated
through the margin of error, ε=A/2
WidthA=T2-T1
(
T
1
Factors influencing precision
)T
Margin of error ε
2
• Confidence level
• Information about the population (probability model and
parameters)
• Sample information (size, sampling methods, estimator)
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Interpretation of confidence level (1-α)
Parameter
“If independent samples are
taken repeatedly from the
same population, and a
confidence interval
calculated for each sample,
THEN
a (1-α)% of the intervals
will include the unknown
population parameter”.
(
)
CI1
(
)
CI2
(
CI3
CI does not include the parameter
(
)
CI4
(
)
CI5
Statistics Glossary
http://www.stats.gla.ac.uk/
)
(
)
....
CI does not include the parameter
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Confidence interval for population mean
NORMAL POPULATION
σ is known
POPULATION IS NOT NORMAL
σ is known and n is large
dX =
X−µ
≈ N(0,1)
σ
n
Standard Normal Distribution N(0,1)
1- α
Obtaning value k
P( −k ≤ N(0,1) ≤ k ) = 1 − α
(
T1
)
X
T2
Confidence interval for
population mean
with a confidence level 1 - α
P(T1 ≤ µ ≤ T2 ) = 1− α

σ
σ
,X + k 
X − k
n
n

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Confidence interval for population mean
dX =
NORMAL POPULATION
σ is unknown
X−µ
≈ t n −1
S
n
Student's t-distribution
1- α
Obtaining value k
P ( −k ≤ t n − 1 ≤ k ) = 1 − α
(
)
T1
X
P(T1 ≤ µ ≤ T2 ) = 1− α
T2
Confidence interval for
population mean
with a confidence level 1 - α

S
S
,X + k 
X − k
n
n

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CI for µ when the distribution of the
population is unknown
Tchebysheff's inequality allows to build confidence intervals when the
distribution of the population is not known, σ is known and n is small
(
P d X − E(d X ) < kσ d
dX =
X−µ
σ
n
X
)
1
≥ 1− 2 = 1− α
k
E(d X ) = 0
k=
1
α
Var (d X ) = 1

1 σ
1 σ 
 ≥ 1 − α
P X −
<µ<X+
α n
α n

Confidence interval for
population mean
with a confidence level 1 - α

1 σ
1 σ 
, X+
X −

α n
α n

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Confidence intervals for population mean
Situation
X is Normal, σ is known
X is not Normal, σ is
known and n is large
X is Normal, σ is
unknown
Confidence
Interval

σ
σ
,X + k 
X − k
n
n

Confidence
Level
1−α
k is found in N(0,1) tables
S
S 

,X + k
X − k

n
n

1−α
k is found is Student’s t-distribution (d.f.= n-1)
X follows an unknown
distribution, σ is known
and n is small

1 σ
1 σ
,X +
X −

α
n
α
n


≥1−α
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Sample size for estimating µ
GOAL
To estimate µ, given a certain confidence level (1α) and certain margin of error ε
X ≈ N(µ, σ)

σ
σ
,X + k 
X − k
n
n

Sample size
ε=k
σ
n
 kσ 
n=

 ε 
2
X is not known

1 σ
1 σ
,X +
X −

α
n
α
n


1 σ
ε=
α n
σ2
n=
αε 2
• Sample size increases with confidence level
• Sample size increases with the value of the standard deviation
• Sample size increases as the margin of error ε decreases
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Confidence interval for population proportions
dp̂ =
n is large
p̂ − p
p(1 − p)
n
≈ N(0,1)
p̂(1 − p̂)
n−1
Standard Normal Distribution
N(0,1)
Obtaning value k
1- α
P( −k ≤ N(0,1) ≤ k ) = 1 − α
(
)
T1
p$
P(T1 ≤ p ≤ T2 ) = 1− α
T2
Confidence interval for
population proportion
with a confidence level 1 - α

p̂(1−p̂)
p̂(1−p̂) 
p̂−k

,p̂+k
n−1
n−1 


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Sample size for estimating p
Goal
To estimate p, given a certain confidence level
(1-α
α) and certain margin of error ε

p(1 − p)
p(1 − p) 
, pˆ + k
 pˆ − k

n
n 

Sample size
p(1 − p)
ε=k
n
n = k2
p(1 − p)
ε2
Since p is unknown, there are to basic options to approximate p(1-p):
• Carry out a pilot survey in order to a get a preliminar estimation of p
• The upper limit of p(1-p) is assumed, p(1-p)= 0.25
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Confidence interval for population variance
NORMAL
POPULATION
dS2
(n − 1)S 2
2
χ
=
≈
n −1
σ2
Chi-square distribution
Obtaining values k1 and k2
P(k1 ≤ χn2−1 ≤ k2 ) = 1− α
1- α
k1
P(χ
2
n −1
α
≤ k1) =
2
k2
P(χ
2
n −1
α
≥ k2 ) =
2
Confidence interval for population variance
with a confidence level 1 - α
(n−1)S2 (n−1)S2 
,


k
k
2
1


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