confidence interval for single population mean and proportion - Copy.ppt

notes

1. Confidence Interval Estimation:
Single Sample Means and Proportions

2. Chapter Topics
•Confidence Interval Estimation for the Mean
(Known)
•Confidence Interval Estimation for the Mean
(Unknown)
•Confidence Interval Estimation for the
Proportion
•The Situation of Finite Populations
•Sample Size Estimation

3. Mean, , is
unknown
PopulationRandom Sample
I am 95%
confident that 
is between 40 &
60.
Mean
X = 50
Estimation Process
Sample

4. Estimate Population
Parameter...
with Sample
Statistic
Mean 
Proportion p p
s
Variance s
2
Population Parameters
Estimated

2
Difference - 
1 2
x - x
1 2
X
_
__

5. •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
Confidence Interval Estimation

6. Confidence Interval
Sample
Statistic
Confidence Limit
(Lower)
Confidence Limit
(Upper)
A Probability That the Population Parameter
Falls Somewhere Within the Interval.
Elements of Confidence
Interval Estimation

7. Parameter =
Statistic ± Its Error
© 1984-1994 T/Maker Co.
Confidence Limits for
Population Mean
X Error
= Error = X
XX
X
Z





x
Z
X
ZX 
Error
Error
X

8. 90% Samples
95% Samples

x
_
Confidence Intervals
xx
..  64516451
xx
 96.196.1 
xx ..  582582 
99% Samples
n
ZXZX
X

 
X
_

9. •Probability that the unknown population
parameter is in the confidence interval in
100 trials.
•Denoted 100(1 - ) % = level of
confidence e.g. 90%, 95%, 99%

Is Probability That the Parameter Is Not
Within the Interval in 100 trials (NOT THIS
TRIAL ALONE!)
Level of Confidence is an
EXPECTED RELATIONSHIP

10. Confidence Intervals
Intervals
Extend from
100(1 - ) %
of Intervals
Contain .
% Do Not.
1 -
/2/2
X
_

x
_
Intervals &
Level of Confidence
Sampling
Distribution of
the Mean
to
X
ZX 
X
ZX 

X

11. •Data Variation
measured by 
•Sample Size
•Level of Confidence
(1 - )
Intervals Extend from
© 1984-1994 T/Maker Co.
Factors Affecting
Interval Width
X - Z to X + Z 
xx
n/
XX


12. Mean
Confidence
Intervals
Proportion
Finite
Population Known
Confidence Interval Estimates

13. •Assumptions
Population Standard Deviation Is Known
Population Is Normally Distributed
If Not Normal, use large samples
•Confidence Interval Estimate
Confidence Intervals (Known - this
is hardly ever true)
n
ZX
/


2

n
ZX
/


2

14. Example
Suppose we wish to test using 95%
confidence interval, the mean age at
which patients with hypertension are
diagnosed. A random sample of 12
subjects diagnosed with hypertension
had the following ages recorded:
32.8 40.0 41.0 42.0 45.5 47.0 48.5 50.0
51.0 52.0 54.0 59.2

15. Suppose the population standard deviation in
age at diagnosis of all hypertensives is known to
be 7.2 years (i.e. =7.2).
The 95% CI for the true age a diagnosis is
given by
n
ZX



2/


16. Mean
Confidence
Intervals
Proportion
Finite
Population Known
Confidence Interval Estimates

17. •Assumptions
Population Standard Deviation Is Unknown
Sample size must be large enough for central limit
theorem or Population Must Be Normally Distributed
•Use Student’s t Distribution
•Confidence Interval Estimate
Confidence Intervals (Unknown)
n
S
tX
n,/

 12

n
S
tX
n,/

 12

18. Z
t
0
t (df = 5)
Standard
Normal
t (df = 13)Bell-Shaped
Symmetric
‘Fatter’ Tails
Student’s t Distribution

19. •Number of Observations that Are Free
to Vary After Sample Mean Has Been
Calculated
•Example
Mean of 3 Numbers Is 2
X
1 = 1 (or Any Number)
X
2
= 2 (or Any Number)
X
3
= 3 (Cannot Vary)
Mean = 2
degrees of freedom =
n -1
= 3 -1
= 2
Degrees of Freedom (df)

20. Upper Tail Area
df.25.10.05
11.0003.0786.314
20.8171.8862.920
30.7651.6382.353
t0
Assume: n = 3 df
= n - 1 = 2
 = .10
/2 =.05
2.920
t Values
 / 2
.05
Student’s t Table

21. A random sample of n = 25 has = 50 and
s = 8. Set up a 95% confidence interval
estimate for .
. .4669 5330
X
Example: Interval Estimation
Unknown
n
S
tX
n,/
 12 
n
S
tX
n,/
 12
25
8
0639250 .

25
8
0639250 .

22. Mean
Confidence
Intervals
Proportion
Finite
Population Known
Confidence Interval Estimates

23. •Assumptions
Two Categorical Outcomes
Population Follows Binomial Distribution
Normal Approximation Can Be Used

n·p 5 & n·(1 - p)  5
•Confidence Interval Estimate
Confidence Interval Estimate
Proportion
n
)p(p
Zp
ss
/s



1
2 p
n
)p(p
Zp
ss
/s



1
2

24. A random sample of 400 Voters showed 32
preferred Candidate A. Set up a 95%
confidence interval estimate for p.
p .053 .107
Example: Estimating Proportion
n
)p(p
Zp
ss
/s



1
2
p
n
)p(p
Zp
ss
/s



1
2
400
08108
96108
).(.
..


400
08108
96108
).(.
..

p

25. Sample Size
Too Big:
•Requires too
much resources
Too Small:
•Won’t do
the job

26. Does the CI Contain the True
Mean?
Click to try a couple

27. 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
Z
Error
   
22
2
2 2
2
164545
5
2192220
 .
.
Example: Sample Size
for Mean
Round Up

28. What sample size is needed to be within ± 5 with
90% confidence? Out of a population of 1,000,
we randomly selected 100 of which 30 were
defective.
Example: Sample Size
for Proportion
Round Up
3227
05
703064511
2
2
2
2
.
.
))(.(..
error
)p(pZ
n 


228

29. Chapter Summary
•Discussed Confidence Interval Estimation for
the Mean(Known)
•Discussed Confidence Interval Estimation for
the Mean(Unknown)
•Addressed Confidence Interval Estimation for
theProportion
•Determined Sample Size