Confidence Intervals(Chapter 8)
• Confidence Intervals for numerical data:– Standard deviation known– Standard deviation unknown
• Confidence Intervals for categorical data
Estimation Process: Example
• We are interested in knowing the average household income in a certain county.
• A sample with 144 observations yields a sample mean X=$72,000.
• It is also “known” that in this county, =$24,000• How can we get a “good” estimate for the true
average household income ? Or:• How far away (“how bad”) can X be as an
estimate for ?
Estimation Process
Mean, , is unknown
Population Random Sample
Mean X = 50
Sample
I am 95% confident that is between 40 & 60.
Point Estimates
Estimate Population
Parameters …
with SampleStatistics
Mean
Proportion
Variance
Difference
p
2
1 2
X
SP
2S
1 2X X
Interval Estimates
• Provides range of values– Take into consideration variation in sample
statistics from sample to sample– Based on observation from 1 sample– Give information about closeness to unknown
population parameters– Stated in terms of level of confidence
• Never 100% sure
Confidence Interval Estimates
Mean
Unknown
ConfidenceIntervals
Proportion
Known
Confidence Interval for( Known)
• Assumptions– Population standard deviation is known– Population is normally distributed– If population is not normal, use large sample
• Confidence interval estimate
/ 2 / 2X Z X Zn n
General Formula
The general formula for a confidence interval is:
Point Estimate ± (Critical Value)(Standard Error)
Where:•Point Estimate is the sample statistic estimating the population parameter of interest
•Critical Value is a table value based on the sampling distribution of the point estimate and the desired confidence level
•Standard Error is the standard deviation of the point estimate
Point Estimate ± Margin of Error
Elements of Confidence Interval Estimation
• Level of confidence– Confidence in which the interval will contain
the unknown population parameter
• Precision (range)– Closeness to the unknown parameter
• Cost– Cost required to obtain a sample of size n
Level of Confidence
• Denoted by• A relative frequency interpretation
– In the long run, of all the confidence intervals that can be constructed will contain the unknown parameter
• A specific interval will either contain or not contain the parameter– No probability involved in a specific interval
100 1 %
100 1 %
Interval and Level of Confidence
Confidence Intervals
Intervals extend from
to
of intervals constructed contain ;
do not.
_Sampling Distribution of the Mean
XX Z
X/ 2
/ 2
XX
1
XX Z
100 1 %
100 %
/ 2 XZ / 2 XZ
Factors Affecting Margin of error (Precision)
• Data variation– Measured by
• Sample size–
• Level of confidence–
Intervals Extend from
© 1984-1994 T/Maker Co.
X - Z to X + Z xx
Xn
100 1 %
Determining Sample Size (Cost)
Too Big:
• Requires too much resources
Too small:
• Won’t do the job
Determining 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.
Round Up
2 22 2
2 2
1.645 45219.2 220
Error 5
Zn
Do You Ever Truly Know σ?
• Probably not!
• In virtually all real world business situations, σ is not known.
• If there is a situation where σ is known then µ is also known (since to calculate σ you need to know µ.)
• If you truly know µ there would be no need to gather a sample to estimate it.
• Assumptions– Population standard deviation is unknown– Population is normally distributed– If population is not normal, use large sample
• Use Student’s t Distribution• Confidence Interval Estimate
–
Confidence Interval for( Unknown)
/ 2, 1 / 2, 1n n
S SX t X t
n n
Student’s t Distribution
Zt
0
t (df = 5)
t (df = 13)Bell-ShapedSymmetric
‘Fatter’ Tails
Standard Normal
Example
A random sample of 25 has 50 and 8.
Set up a 95% confidence interval estimate for
n X S
/ 2, 1 / 2, 1
8 850 2.0639 50 2.0639
25 2546.69 53.30
n n
S SX t X t
n n
Confidence Interval Estimate for Proportion
• Assumptions– Two categorical outcomes
– Population follows binomial distribution
– Normal approximation can be used if and
– Confidence interval estimate
–
5np 1 5n p
/ 2 / 2
1 1S S S SS S
p p p pp Z p p Z
n n
Example
A random sample of 400 Voters showed 32 preferred Candidate A. Set up a 95% confidence interval estimate for p.
/ /
1 1
.08 1 .08 .08 1 .08.08 1.96 .08 1.96
400 400.053 .107
s s s ss s
p p p pp Z p p Z
n n
p
p
Determining Sample Size for Proportion
Out of a population of 1,000, we randomly selected 100 of which 30 were defective. What sample size is needed to be within ± 5% with 90% confidence?
Round Up
2 2
2 2
1 1.645 0.3 0.7
Error 0.05227.3 228
Z p pn
Excel Tutorial
Constructing Confidence Intervals using Excel:
• Tutorial
•Excel spreadsheet