CHAPTER 9: CONFIDENCE
INTERVAL
E370 Spring 2016
So far
• Sample mean and sample proportion are both random
variables
• Sample mean: approximated by either normal distribution
or t-distribution. (Condition: n>=30)
• Sample proportion: normal distribution. (Condition:
n*pi>=5 and n*(1-pi)>=5)
• Central limit theorem
Main Goals for this chapter
1. Be able to construct confidence intervals for population
mean and population proportion. (use the distributions
of sample statistics)
2. Be able to interpret confidence intervals for population
mean and proportion
3. Be able to calculate sample size given margin of error.
Some concepts:
Terminology
Level of Confidence
The area under the curve inside the CI, a
probability equal to 1 − 𝛼
𝛼
The area under the curve outside the CI, a
probability equal to 𝛼. It is used to determine the
level of confidence
Point Estimate
The center of the CI. For this class, either a sample
mean or a sample proportion.
Margin of Error
The distance from the point estimate to either limit:
half the width of the CI
Task 1: How to construct confidence
intervals
Population mean
Population standard
deviation is known
Population standard
deviation is unknown.
Sample standard
deviation is known
Population
proportion
When 𝑛𝑝 ≥ 5 and
𝑛(1 − 𝑝) ≥ 5
𝝈
𝑿±𝒁𝜶×( )
𝒏
𝟐
𝒔
𝑿 ± 𝒕 (𝜶,𝒅𝒇) × ( )
𝒏
𝟐
𝒑 × (𝟏 − 𝒑)
𝑝±𝒁𝜶×
𝒏
𝟐
Steps to construct confidence intervals
• 1. Point estimates: either sample mean (𝑥) or sample
proportion (p).
• 2. Choose a level of confidence (1 − 𝛼): to get z-score
(𝑧(𝛼) ) or t-score (𝑡(𝛼,𝑑𝑓) ).
2
2
• 3. Sample (𝑠) or population (𝜎) standard deviation.
• 4. Combine the above two to get margin of error (e).
• 5. Get lower and upper bound of the confidence interval.
Task 2: How to calculate sample size
given margin of error
When estimating a
population mean
When estimating a
population proportion
𝒁 𝜶/2 × 𝝈 2
n=(
)
e
𝒁 𝜶/2 × 𝑝(1 − 𝑝)
n=(
)
e
2
Task 3: How to interpret confidence
intervals for population mean and
proportions
• Example of interpretation:
• We are 90% confident that the mean waiting time for all pharmacy
customers is between 17.92 minutes and 21.65 minutes.
• With 95% confidence, we believe that the true proportion of IU
students who are football fans is between 5.63% and 18.37%.