Transcript Document

Chapter Topics
•Confidence Interval Estimation for the Mean
(s Known)
•Confidence Interval Estimation for the Mean
(s Unknown)
•Confidence Interval Estimation for the
Proportion
•Sample Size Estimation
Estimation Process
Population
Mean, m, is
unknown
Sample
Random Sample
Mean
X = 50
I am 95%
confident that m
is between 40 &
60.
Population Parameters
Estimated
Estimate Population
Parameter...
Mean
m
Proportion
Variance
Difference
with Sample
Statistic
_
X
p
s
2
m - m
1
ps
s
2
2
_
_
x - x
1
2
Confidence Interval Estimation
• Provides Range of Values
–
Based on Observations from 1 Sample
• Gives Information about Closeness
to Unknown Population Parameter
• Stated in terms of Probability
Never 100% Sure
Elements of Confidence
Interval Estimation
A Probability That the Population Parameter
Falls Somewhere Within the Interval.
Sample
Confidence Interval
Statistic
Confidence Limit
(Lower)
Confidence Limit
(Upper)
Level of Confidence
•
•
•
Probability that the unknown
population parameter falls within the
interval
•
Denoted (1 - a) % = level of confidence
e.g. 90%, 95%, 99%
 a Is Probability That the Parameter Is Not
Within the Interval
Intervals &
Level of Confidence
Sampling
Distribution of
the Mean
a/2
Intervals
Extend from
s_
x
1-a
mX  m
a/2
_
X
(1 - a) % of
Intervals
Contain m.
X  ZsX
to
a % Do Not.
X  ZsX
Confidence Intervals
Factors Affecting
Interval Width
•
•
Data Variation
measured by s
•
Sample Size
sX  sX / n
•
Level of Confidence
Intervals Extend from
X - Zs
x
to X + Z s
x
(1 - a)
© 1984-1994 T/Maker Co.
Confidence Interval Estimates
Confidence
Intervals
Mean
s Known
Proportion
s Unknown
Confidence Intervals (s Known)
•
•
Assumptions
–
Population Standard Deviation Is Known
–
Population Is Normally Distributed
–
If Not Normal, use large samples
Confidence Interval Estimate
s  m 
X  Za / 2 
n
s
X  Za / 2 
n
Confidence Intervals (s Unknown)
•
Assumptions
–
–
Population Standard Deviation Is Unknown
Population Must Be Normally Distributed
•
Use Student’s t Distribution
•
Confidence Interval Estimate
S
S  m  X t
X  ta / 2 ,n1 
a / 2 ,n1 
n
n
Student’s t Distribution
Standard
Normal
Bell-Shaped
Symmetric
‘Fatter’ Tails
t (df = 13)
t (df = 5)
0
Z
t
Degrees of Freedom (df)
•
•
•
Number of Observations that Are Free
to Vary After Sample Mean Has
Been
Calculated
•
Example
–
Mean of 3 Numbers Is 2
X1 = 1 (or Any Number)
X2 = 2 (or Any Number)
X3 = 3 (Cannot Vary)
Mean = 2
degrees of freedom =
n -1
= 3 -1
=2
Student’s t Table
a/2
Upper Tail Area
df
.25
.10
.05
Assume: n = 3
=n-1=2
df
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
0
t Values
2.920
t
Example: Interval Estimation
s Unknown
•A random sample of n = 25 hasX = 50 and
•s = 8. Set up a 95% confidence interval
estimate for m.
S
S
X  ta / 2 ,n1 
 m  X  ta / 2 ,n1 
n
n
50  2 . 0639 
8
25
m
46 . 69
 m 
50  2 . 0639 
53 . 30
8
25
Confidence Interval Estimate
Proportion
•
Assumptions
– Two Categorical Outcomes
– Population Follows Binomial Distribution
– Normal Approximation Can Be Used
–
•
n·p  5
&
n·(1 - p)  5
Confidence Interval Estimate
ps ( 1  ps )
ps  Za / 2 
n
 p
ps ( 1  ps )
ps  Za / 2 
n
Sample Size
Too Big:
•Requires too
much resources
Too Small:
•Won’t do
the job
Example: Sample Size
for Mean
•What sample size is needed to be 90%
confident of being correct within ± 5? A
pilot study suggested that the standard
deviation is 45.
Z s
2
n
2
Error
2

1645
.
5
2
2
45
2
 219.2 @ 220
Round Up