1. Chapter Topics Confidence Interval Estimation for the Mean (s Known)
(s Unknown)
Confidence Interval Estimation for the
Proportion
Sample Size Estimation

2. I am 95% confident that m is between 40 & 60.
Estimation Process
Population
Random Sample
I am 95% confident that m is between 40 & 60.
Mean
X = 50
Mean, m, is unknown
Sample

3. Population Parameters Estimated
Estimate Population
with Sample
Parameter...
Statistic _
Mean m X
Proportion p p s 2
Variance s 2 s _ _
Difference
m - m
x - x 1 2 1 2

4. 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

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

6. 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

7. Sampling Distribution of the Mean
Intervals &
Level of Confidence
Sampling Distribution of the Mean s _ x a /2 a /2
1 - a _ X
Notice that the interval width is determined by 1-a in the sampling distribution.
Intervals Extend from
(1 - a) % of Intervals Contain m.
a % Do Not. to
Confidence Intervals

8. Factors Affecting Interval Width
Data Variation
measured by s
Sample Size
Level of Confidence (1 - a)
Intervals Extend from
X - Zs to X + Z s x x
© T/Maker Co.

9. Confidence Interval Estimates
Intervals
Mean
Proportion s s
Known
Unknown

10. Confidence Intervals (s Known)
Assumptions
Population Standard Deviation Is Known
Population Is Normally Distributed
If Not Normal, use large samples
Confidence Interval Estimate

11. Confidence Intervals (s Unknown)
Assumptions
Population Standard Deviation Is Unknown
Population Must Be Normally Distributed
Use Student’s t Distribution
Confidence Interval Estimate

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

13. 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 X = 1 (or Any Number) X = 2 (or Any Number) X = 3 (Cannot Vary) Mean = 2
degrees of freedom = n -1 = 3 -1 = 2

14. Student’s t Table .05 2 t a / 2 .05 2.920 t Values Upper Tail Area df
Assume: n = df = n - 1 = 2
a = a/2 =.05
Upper Tail Area df
.25
.10
.05
Confidence intervals use a/2, so divide a! 1
1.000
3.078
6.314 2
0.817
1.886
2.920
.05 3
0.765
1.638
2.353 t
2.920
t Values

15. Example: Interval Estimation
s Unknown
A random sample of n = 25 has = 50 and
s = 8. Set up a 95% confidence interval estimate for m. . m . 46 69 53 30

16. Confidence Interval Estimate Proportion
Assumptions
Two Categorical Outcomes
Population Follows Binomial Distribution
Normal Approximation Can Be Used
n·p ³ & n·(1 - p) ³ 5
Confidence Interval Estimate

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

18. 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. 2 2 2 2 Z s 1 .
645 45 = n = =
219 . 2 @
220 2 2
Error 5
Round Up