Download Probability and Statistics Outline Statistical Methods Estimation

Probability and Statistics
LECTURE 7
INTERVAL ESTIMATION
Outline
1.
State What Is Estimated
2.
Distinguish Point & Interval Estimates
3.
Explain Interval Estimates
4.
Compute Confidence Interval Estimates
for Population Mean & Proportion
Adapted from http://www.prenhall.com/mcclave
7-1
7-2
Statistical Methods
Estimation Process
Population


Mean, , is
unknown

Random Sample
I am 95%
confident that
 is between
40 & 60.
Mean
X= 50
 
Sample


7-3
Parameter vs sample statistic
• Parameter is numerical descriptive
measure of population
• Sample statistic is a numerical
descriptive measure of sample
• Example
 
7-4
Unknown Population
Parameters Are Estimated
Estimate Population
Parameter...
Mean

Proportion
p
Variance

Differences
7-5
7-6
2
1 -  2
with Sample
Statistic
x
p^
s
2
x1 -x2
Estimation Methods
Estimation
Point
Estimation
Interval
Estimation
7-7
Point Estimation
1. Provides Single Value
2. Example:
3. Disadvantage: Gives No Information
about How Close Value Is to the
Unknown Population Parameter
7-8
Key Elements of
Interval Estimation
Interval Estimation
1. Provides Range of Values
2. Gives Information about Closeness to
Unknown Population Parameter

A probability that the interval contains the
population parameter.
Sample statistic
Confidence interval (point estimate)
Stated in terms of Probability
 Knowing Exact Closeness Requires Knowing
Unknown Population Parameter
3. Example: Unknown Population Mean Lies
Between 50 & 70 with 95% Confidence
7-9
Confidence
limit (lower)
Confidence
limit (upper)
7 - 10
Confidence Interval (CI)
Estimates
Confidence Interval for
Population Mean ( Known)
1. Assumptions
Confidence
Intervals


Mean
Known
Proportion
 Unknown

Population Standard Deviation Is Known
Population Is Normally Distributed
If Population Is Not Normal, Large Sample
Size (so that the CLT holds) Is Required
(In This Case The Sampling Distribution
Of
Is Approximately Normal)
2. Confidence Interval Estimate
7 - 11
7 - 12
Proof of the Confidence
Interval (CI) formula
Proof of the Confidence
Interval (CI) formula
This proof is optional
Start with sampling distribution of
sample mean. Recall that under some
conditions, this distribution is normal or
approx. normal
Convert to Z distribution
1-α
α/2
x
1-α
α/2
Z
7 - 13
7 - 14
Proof of the Confidence
Interval (CI) formula
Let’s define a new notation:
• Zα/2 is the Z value such that the area to
its right is equal to α/2
• So we have figure:
Proof of the Confidence
Interval (CI) formula
P(-Zα/2 < Z < Zα/2) = ?
Replace
Rearranging the expression, we should
obtain an probability equation for the CI
Z
7 - 15
-Zα/2 0
Zα/2
7 - 16
Confidence Level
1. Probability that the Interval contains
Unknown Population Parameter
Z and 
/2
/2
1-
2. Denoted 1 - 

Is Probability That Interval does Not
contain Parameter
3. Typical Values Are 99%, 95%, 90%
7 - 17

0
As  is the probability that the interval does not
contain , we construct an interval that places
area /2 in each tail.
is the z value such
that the area /2 lies to its right.
7 - 18
Z and 
How to look up Table 1
/2
/2
1-

0
If we know , we can find
Appendix of your textbook.
using Table 1 in
Example:  = 0.1 → /2 = 0.05 → P(Z <
Look up table 1, and we find
= 1.645
) = 0.95
7 - 19
7 - 20
Factors Affecting
Interval Width
Z and 
Note that we actually find 2 values for
Z by looking up table 1:
With P = 0.9495, we have Z = 1.64
With P = 0.9505, we have Z = 1.65.
Taking the average of the two values,
we have
= 1.645.
1. Data variability
measured by 
2. Sample Size
Intervals Extend from
X - ZX toX + ZX
3. Level of Confidence
(1 - )

Affects Z
© 1984-1994 T/Maker Co.
7 - 21
7 - 22
Intervals &
Confidence Level
Sampling
Distribution
of Mean
Intervals
extend from
X - ZX to
X + ZX
Estimation Example
Mean ( Known)
The mean of a random sample of n = 25
isX = 50. Set up a 95% confidence
interval estimate for  if  = 10. Assume
the population is normally distributed.
100(1 - ) %
of intervals
contain .
100 % do
Large number of intervals not.
7 - 23
7 - 24
Estimation Example
Solution
Confidence Interval for
Population Mean ( Unknown)
1. Assumptions


Population Standard Deviation Is Unknown
Population is Normally Distributed
2. Use Student’s t Distribution
3. Confidence Interval Estimate
We are 95% confident that the population
mean lies between 46.08 and 53.92
7 - 25
7 - 26
Degrees of Freedom (df)
T statistic
If we are sampling from a normal distribution, the
t statistic has a sampling distribution very
much like that of the z statistic: moundshaped, symmetric, with mean 0.
The main difference between the sampling
distributions of t and z is that the t statistic is
more variable than the z.
7 - 27
1. Number of Values that Are Free to Vary
when Calculating a Sample Statistic.
2. Degrees of freedom of the t statistic (for
one-sample case) is n1.
3. Example
7 - 28
Degrees of Freedom (df)
Example
7 - 29
Degrees of Freedom (df)
The smaller the number of degrees of
freedom associated with the t statistic,
the more spread out the sampling
distribution of T will be.
7 - 30
Student’s t Table (Table 2
in Appendix of textbook)
Student’s t Distribution
/2
Standard
Normal
Bell-Shaped
t (df = 13)
Assume:
n=3
df = n - 1 = 2
 = .10
/2 =.05
Symmetric
t (df = 5)
‘Fatter’ Tails
.05
Z
t
0
t values
7 - 32
7 - 31
Estimation Example
Mean ( Unknown)
Student’s t Table
/2
Assume:
n=3
df = n - 1 = 2
 = .10
/2 =.05
A random sample of n = 25 hasx = 50 &
s = 8. Set up a 95% confidence interval
estimate for . Assume normal population.
.05
t values
2.920
7 - 33
7 - 34
Estimation Example
Solution
Confidence Interval Estimates
Confidence
Intervals
Mean
Known
7 - 35
7 - 36
Proportion
 Unknown
Confidence Interval
Proportion
Estimation Example
Proportion
1. Assumptions



A random sample of 400 graduates
showed 32 went to grad school. Set up a
95% confidence interval estimate for p.
Two Categorical Outcomes
Population Follows Binomial Distribution
Normal Approximation To The Sampling
Distribution Of
Can Be Used

and
2. Confidence Interval Estimate
7 - 37
7 - 38
Estimation Example
Solution
Precision of estimate
•
•
•
The smaller the width of CI, the higher the
precision of estimation.
What are the ways to make the estimate
more precise?
In practice, if we choose to increase
sample size to increase precision, we
should also take into account the costs of
study.
7 - 39
7 - 40
CI for Finite populations
In cases where n/N > 0.05, the formulas
for CI must be adjusted (as the
standard error of estimate changes)
Please consult your textbook or
reference books if you are interested in
the adjusted formulas. Knowing these
formulas is not required for the
test/exam.
7 - 41
Conclusion
1.
State What Is Estimated
2.
Distinguish Point & Interval Estimates
3.
Explain Interval Estimates
4.
Compute Confidence Interval Estimates
for Population Mean & Proportion
7 - 42