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