Download Confidence Interval Estimation

Confidence Interval
Estimation
Dr. Geoffrey M. Ducanes
UP School of Economics
Session Outline
1. Confidence Intervals for the Mean
a. Large Samples
b. Small Samples
2. Confidence Intervals for Proportion
3. Confidence Intervals for the Standard Deviation
Introduction
• This morning you learned about point estimators.
• Assuming representative sample
• Sample mean is point estimator for population mean.
• Sample proportion is point estimator for population proportion.
• Sample standard deviation is point estimator for population standard
deviation.
• Problem with using point estimate is that it is almost never
equal to the exact parameter of the population.
• [Illustration using Excel]
Introduction
• Might be better to specify an interval of values and a
statement of how confident you are that the interval contains
the population parameter.
• Confidence Interval:
Point estimate ± Margin of Error
How to estimate a confidence interval?
1. Compute the point estimate
2. Compute the margin of error
3. Construct the interval endpoints using the point
estimate and the margin of error
Confidence Interval for the Mean
(large sample or  known)
1. Find the sample statistics n
and 𝑥.
2. Specify , if known. Otherwise
if n  30, use the sample
standard deviation s as
estimate for .
3. Find the critical value 𝑧𝑐 that
corresponds to the given level
of significance.
4. Find the margin of error E.
5. Find the left and right
endpoints and form the
confidence interval.
𝑥
𝑥=
𝑛
𝑠=
𝑥−𝑥
𝑛−1
2
Use the standard Normal
Table or Excel.
𝐸 = 𝑧𝑐
𝜎
𝑛
Interval: 𝑥 - E <  < 𝑥 + E
Question
• From the formula, why is the effect of a larger sample size on
the width of the interval?
Example
• Ilocos Norte HH income and expenditure (CI data – EPDP
Training)
Source: Caldwell, S. 2010. Statistics Unplugged
Confidence Interval for the Mean
(small sample, unknown)
•
•
In many situations, population standard deviation is
unknown and large sample is not available.
Follow same procedure but use t-distribution
Confidence Interval for the Mean
(small sample, unknown)
1. Find the sample statistics n, 𝑥,
and s.
2. Identify the degrees of
freedom, the level of
confidence c, and the critical
value 𝑡𝑐 .
3. Find the margin of error E.
4. Find the left and right
endpoints and form the
confidence interval.
𝑥
𝑥=
𝑛
𝑠=
𝑥−𝑥
𝑛−1
2
d.f. = n1
Use the t Table or Excel.
𝐸 = 𝑡𝑐
𝑠
𝑛
Interval: 𝑥 - E <  < 𝑥 + E
Example
• Batanes food expenditure and housing and utilities
expenditure (CI data – EPDP Training)
Confidence Interval for a Population
Proportion p
•
•
In cases where binomial distribution can be approximated
by a normal distribution (np  5 and nq  5),
Construction of confidence interval for p similar to
constructing confidence interval for a population mean.
Point estimate ± Margin of error
𝑝±E
•
E=𝑧𝑐
•
•
𝑝𝑞
𝑛
Confidence Interval for the Population
Proportion p
1.
2.
3.
4.
5.
6.
Identify the sample statistics x
and n, where x is the number of
success.
Find the point estimate 𝑝.
Verify that the sampling
distribution of 𝑝 can be
approximated by a normal
distribution
Find the critical value 𝑧𝑐 that
corresponds to the given level
of confidence c.
Find the margin of error E.
Find the left and right endpoints
and form the confidence
interval.
n𝑝  5, n𝑞  5
Use the standard Normal
Table or Excel.
𝐸 = 𝑧𝑐
𝑝𝑞
𝑛
Interval: 𝑝- E <  < 𝑝+ E
Example
• Metro Manila households with at least one aircon, at least
one car (CI data – EPDP Training)
Confidence Interval for the population
standard deviation 
•
•
•
Uses Chi-Square Distribution
Confidence Interval for 
(𝑛−1)𝑠 2
2
𝜒𝑅
<<
(𝑛−1)𝑠 2
𝜒𝐿2
Confidence Interval for the population
standard deviation 
1. Verify that the population has
a normal distribution.
2. Identify the sample statistic n
and the degrees of freedom.
3. Find the point estimate for the
population variance 𝑠 2 .
4. Find the critical values 𝜒𝑅2 and
𝜒𝐿2 that correspond to the
given level of confidence c.
5. Compute the left and right
endpoints to form the
confidence interval.
d.f. = n -1
𝑠=
𝑥−𝑥
𝑛−1
2
Use Chi-Square Table or
Excel.
(𝑛−1)𝑠 2
2
𝜒𝑅
<<
(𝑛−1)𝑠 2
𝜒𝐿2
Example
• Ilocos Norte HH income and expenditure (CI data – EPDP
Training)
References
• Larson, R. and B. Farber. 2012. Elementary
Statistics: Picturing the World (5th Edition).
Prentice Hall
• Caldwell, S. 2010. Statistics Unplugged (3rd
Edition). Wadsworth CENGAGE Learning.
• Danao, R. 2013. Introduction to Statistics and
Econometrics. Diliman: UP Press