Download Chap. 8: Estimation of Population Parameters: Confidence Intervals

Statistics for Business and
Economics
Estimation of Population Parameters:
Confidence Intervals
Chapter 8
Learning Objectives

1. State What Is Estimated

2. Distinguish Point & Interval Estimates

3. Explain Interval Estimates


4. Compute Confidence Interval
Estimates for Population Mean &
Proportion
5. Compute Sample Size
Thinking Challenge

Suppose you’re
interested in the
average amount of
money that
students in this
class (the
population) have
on them. How
would you find
out?
Statistical Methods
Statistical
Methods
Descriptive
Statistics
Inferential
Statistics
Estimation
Hypothesis
Testing
Estimation Process
Population
Mean, mX, is
unknown
Random Sample
Mean
`X = 50
I am 95%
confident that
mX is between
40 & 60.
Unknown Population Parameters
Are Estimated
Estimate Population
Parameter...
Mean
mx
Proportion
Variance
Differences
p
with Sample
Statistic
`x
ps
2
2
sx
s
m1 - m2
`x1 -`x2
Estimation Methods
Estimation
Point
Estimation
Confidence
Interval
Interval
Estimation
Bootstrapping
Prediction
Interval
Point Estimation

1. Provides Single Value



Based on Observations from 1 Sample
2. Gives No Information about How Close
Value Is to the Unknown Population
Parameter
3. Sample Mean`X = 3 Is Point Estimate
of Unknown Population Mean
Estimation Methods
Estimation
Point
Estimation
Confidence
Interval
Interval
Estimation
Bootstrapping
Prediction
Interval
Interval Estimation

1. Provides Range of Values


2. Gives Information about Closeness to
Unknown Population Parameter


Based on Observations from 1 Sample
Stated in terms of Probability
 Knowing Exact Closeness Requires Knowing
Unknown Population Parameter
3. e.g., Unknown Population Mean Lies
Between 50 & 70 with 95% Confidence
Key Elements of
Interval Estimation
A Probability That the Population Parameter Falls
Somewhere Within the Interval.
Confidence Interval
Confidence Limit
(Lower)
Sample Statistic
(Point Estimate)
Confidence Limit
(Upper)
Confidence Limits
for Population Mean
Parameter =
Statistic ± Error
© 1984-1994 T/Maker Co.
(1)
m x  X  Error
(2)
Error  X - m x or X  m x
X - mx
(3)
Z
(4)
Error  Zs x
(5)
m x  X  Zs x
sx

Error
sx
Many Samples Have Same
Confidence Interval
`X = mx ± Zs`x
sx_
mx-2.58s`x
mx-1.65s`x
mx-1.96s`x
mx mx+1.65s`x
90% Samples
95% Samples
99% Samples
`X
mx+2.58s`x
mx+1.96s`x
Level of Confidence


1. Probability that the Unknown
Population Parameter Falls Within Interval
2. Denoted (1 - a) %


a Is Probability That Parameter Is Not Within
Interval
3. Typical Values Are 99%, 95%, 90%
Intervals &
Level of Confidence
Sampling
Distribution
of Mean
_
a/2
sx
1-a
a/2
m`x = mx
_
X
(1 - a) % of
Intervals
Contain mX .
Intervals
Extend from
`X - Zs`X to
`X + Zs`X
a % Do Not.
Large Number of Intervals
Factors Affecting
Interval Width

1. Data Dispersion


2. Sample Size


Measured by sX
Intervals Extend from
`X - Zs`X to`X + Zs`X
s`X = sX / n
3. Level of Confidence
(1 - a)

Affects Z
© 1984-1994 T/Maker Co.
Confidence Interval Estimates
Confidence
Intervals
Mean
sx Known
Proportion
sx Unknown
Variance
Finite
Population
Confidence Interval Estimate
Mean (sX Known)

1. Assumptions




Population Standard Deviation Is Known
Population Is Normally Distributed
If Not Normal, Can Be Approximated by
Normal Distribution (n  30)
2. Confidence Interval Estimate
sX
sX
X - Za / 2 
 m X  X  Za / 2 
n
n
Estimation Example
Mean (sX Known)
mean of a random sample of n = 25
is`X = 50. Set up a 95% confidence
interval estimate for mX if sX = 10.
sX
sX
X - Za / 2 
 m X  X  Za / 2 
n
n
10
10
50 - 196
. 
. 
 m X  50  196
25
25
46.08  m X  53.92
The
Thinking Challenge

You’re a Q/C inspector for
Gallo. The sX for 2-liter
bottles is .05 liters. A
random sample of 100
bottles showed`X =
1.99 liters. What is the
90% confidence interval
estimate of the true
mean amount in 2-liter
bottles?
2
liter
© 1984-1994 T/Maker Co.
Confidence Interval
Solution*
X - Za / 2 
199
. - 1645
.

sX
n
.05
100
 m X  X  Za / 2 
sX
n
.  1645
.
 m X  199

1982
.
.
 m X  1998
.05
100
Confidence Interval Estimates
Confidence
Intervals
Mean
sx Known
Proportion
sx Unknown
Variance
Finite
Population
Confidence Interval Estimate
Mean (sX Unknown)

1. Assumptions


Population Standard Deviation Is Unknown
Population Must Be Normally Distributed

2. Use Student’s t Distribution

3. Confidence Interval Estimate
X - t a / 2, n -1 
S
n
 m X  X  t a / 2, n -1 
S
n
Student’s t Distribution
Standard
Normal
Bell-Shaped
t (df = 13)
Symmetric
t (df = 5)
‘Fatter’ Tails
0
Z
t
Student’s t Table
Upper Tail Area
df
.25
.10
.05
a/2
Assume:
n=3
df = n - 1 = 2
a = .10
a/2 =.05
1 1.000 3.078 6.314
2 0.817 1.886 2.920
.05
3 0.765 1.638 2.353
t Values
0
2.920
t
Degrees of Freedom (df)


1.
Number of Observations that Are
Free to Vary After Sample Statistic Has
Been Calculated
degrees of freedom
2.

Example
Sum of 3 Numbers Is 6
X1 = 1 (or Any Number)
X2 = 2 (or Any Number)
X3 = 3 (Cannot Vary)
Sum = 6
= n -1
= 3 -1
=2
Estimation Example
Mean (sX Unknown)
random sample of n = 25 has`X = 50 &
S = 8. Set up a 95% confidence interval
estimate for mX.
S
S
X - t a / 2, n -1 
 m X  X  t a / 2, n -1 
n
n
8
8
50 - 2.0639 
 m X  50  2.0639 
25
25
46.69  m X  53.30
A
Thinking Challenge


You’re a time study
analyst in manufacturing.
You’ve recorded the
following task times
(min.): 3.6, 4.2, 4.0,
3.5, 3.8, 3.1.
What is the 90%
confidence interval
estimate of the
population mean task
time?
Confidence Interval Solution*
`X = 3.7

S = 3.8987

n = 6, df = n -1 = 6 -1 = 5

S / n = 3.8987 / 6 = 1.592

t.05,5 = 2.0150
3.7 - (2.015)(1.592)  mX  3.7 +
(2.015)(1.592)


.492  mX  6.908
Confidence Interval Estimates
Confidence
Intervals
Mean
sx Known
Proportion
sx Unknown
Variance
Finite
Population
Estimation for
Finite Populations

1. Assumptions

Sample Is Large Relative to Population



n / N > .05
2. Use Finite Population Correction Factor
3. Confidence Interval (Mean, sX
Unknown)
X - t a / 2, n -1 
S
n

N-n
N -1
 m X  X  t a / 2, n -1 
S
n

N-n
N -1
Confidence Interval Estimates
Confidence
Intervals
Mean
sx Known
Proportion
sx Unknown
Variance
Finite
Population
Confidence Interval Estimate
Proportion

1. Assumptions



Two Categorical Outcomes
Population Follows Binomial Distribution
Normal Approximation Can Be Used


n·p  5 & n·(1 - p)  5
2. Confidence Interval Estimate
ps - Z 
ps  (1 - ps )
n
 p  ps  Z 
ps  (1 - ps )
n
Estimation Example
Proportion
A
random sample of 400 graduates showed
32 went to grad school. Set up a 95%
confidence interval estimate for p.
ps - Z a / 2 
.08 - 196
. 
ps  (1 - ps )
n
.08  (1-.08 )
400
 p  ps  Z a / 2 
. 
 p  .08  196
.053  p  .107
ps  (1 - ps )
n
.08  (1-.08 )
400
Thinking Challenge

You’re a production
manager for a
newspaper. You want to
find the % defective. Of
200 newspapers, 35 had
defects. What is the
90% confidence interval
estimate of the
population proportion
defective?

Confidence Interval
Solution*
n·p  5
n·(1 - p)  5
ps  (1 - ps )
ps  (1 - ps )
ps - Z a / 2 
 p  ps  Z a / 2 
n
n
.175  (.825)
.175  (.825)
.175 - 1645
.
.

 p  .175  1645

200
200
.1308  p  .2192
Estimation Methods
Estimation
Point
Estimation
Confidence
Interval
Interval
Estimation
Bootstrapping
Prediction
Interval
Bootstrapping Method

1. Used If Population Is Not Normal

2. Requires Computer

3. Steps





Take Initial Sample
Sample Repeatedly from Initial Sample
Compute Sample Statistic
Form Resampling Distribution
Limits Are Values That Cut Off Smallest & Largest
a/2 %
Estimation Methods
Estimation
Point
Estimation
Confidence
Interval
Interval
Estimation
Bootstrapping
Prediction
Interval
Prediction Interval



1. Used to Estimate Future Individual X
Value
2. Not Used to Estimate Unknown
Population Parameter
3. Prediction Interval Estimate
X - t a / 2, n -1  S  1 
1
n
 X f  X  t a / 2, n -1  S  1 
1
n
Finding Sample Sizes
(1)
(2)
(3)
Z
X - mx
sx

Error
sx
Error  Zs x  Z
n
2
Z sx
2
Error
2
sx
n
I don’t want to
sample too much
or too little!
Sample Size Example
What
sample size is needed to be 90%
confident of being correct within  5? A
pilot study suggested that the standard
deviation is 45.
n
2
Z sx
2
Error
2
1645
.
45)
(
)(

 219.2 @ 220
(5)
2
2
2
Thinking Challenge

You work in Human
Resources at Merrill Lynch.
You plan to survey
employees to find their
average medical expenses.
You want to be 95%
confident that the sample
mean is within ± $50.
A pilot study showed that
sX was about $400. What
sample size do you use?
Sample Size
Solution*
Z sx
2
n
2
Error 2
196
. ) (400 )
(

(50)
2
2
2
 245.86 @ 246
Conclusion

1. Stated What Is Estimated

2. Distinguished Point & Interval Estimates

3. Explained Interval Estimates


4. Computed Confidence Interval
Estimates for Population Mean &
Proportion
5. Computed Sample Size
End of Chapter
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