Download Confidence Intervals

Confidence Intervals
Ginger Holmes Rowell, Ph. D.
April 10, 2002
Estimation & Inference
• Point Estimators
* one value estimates a population
parameter
Examples
• Sample Mean estimates _______
• Proportion of sample
estimates ____________
Estimation & Inference
• Interval Estimators
* a range of values estimates a
population parameter
Example: Confidence Intervals
• For estimating
–Population Means
–Population Proportions
Confidence Intervals
• Two Features
* Interval Estimate of the population
parameter
* Level of Confidence associated
with the estimate
Confidence Intervals
Confidence Interval =
Point Estimator +/- Margin of Error
Confidence Intervals
• Example for Means:
Consider the average number
of students in a teacher’s class
• Form you hypothesis:
* What do you think?
Example for Means
• Collect the data:
• Class Size Data
29, 27, 30, 28, 28, 30, 29, 21, 22,
29, 26, 31, 28, 25, 34, 23, 24, 29,
34, 27, 30, 25, 25, 30, 30, 27, 29,
29, 29, 28
Sample Size = 30
Class Size Data
• Sample Size = 30
• Sample Mean = 27.9 students
• Sample Standard Deviation = 3
people
• Data are collected from a
Normal population with
standard deviation of 3 students
The Empirical Rule
• IF the data are symmetrical and
mound shaped, then approximately
• 68% of the data fall within 1
standard deviation from the mean
• 95% of the data fall within 2
standard deviations from the mean
• Almost 100% of the data fall within
3 standard deviation from the mean
The Empirical Rule
Forming a 95% CI
• An approximation for 95%
confidence in an estimate of the
average class size
Forming a 95% CI
sample mean +/- 2*SE
* SE = Standard Error
* = Standard Deviation of the Mean
= Standard Deviation/Square root of
sample size
Forming a 95% CI
• sample mean +/- 2*SE
=27.9 +/- 2*(3/sqrt(30))
=27.9 +/- 1.1
=(27.9 - 1.1, 27.9 + 1.1)
=(26.8, 29.0)
Interpretation
• We are 95% confident that the
true population average class
size is between 26.8 and 29
students.
General Interpretation: 95% CI
• If we sample repeatedly, then
each time we would get a
different sample mean and thus
the confidence interval would
change.
General Interpretation: 95% CI
• If we did this 100 times, we
would expect 95 of the intervals
to contain the true population
mean and 5% of the intervals
would not.
General Interpretation: 95% CI
• The only problem is that you
don’t know if your confidence
interval from your sample is one
of the 95 that is correct, or if it is
one of the 5 that is wrong.
Using Your TI-83 for CI
• Press STAT>TESTS>Zinterval
* Input: Stat
* sigma = 3
* x-bar = 27.9
* n = 30
* C-Level = 95%
* select Calculate
Using Your TI-83 for CI
• (26,826, 28.974)
• x-bar=27.9
• n=30
Using Your TI-83 for CI
• Now find the following
confidence intervals for the
same example
* 99%
* 90%
* 80%
CI’s for population average
class size
• 99% - (26.489, 29.311)
• 95% - (26,826, 28.974)
• 90% - (26.999, 28.801)
• 80% - (27.198, 28.602)
• Conclusions about confidence
level and width of confidence
interval?
What is the effect of sample
size on CI width?
• What is your intuition?
* Try a 95% CI with a sample size
of 100
–(27.312, 28.488)
* The 95% CI with n=30 is
–(26,826, 28.974)
Applications
• Time to cook a pizza
• Time to cook microwave
popcorn
• Very important in the medical
profession
• ...
Questions
• ???