Interval Estimation Part II

16205-ch6

Interval Estimation

Part IINotes of STAT6205 by Dr. Fan

6205-ch6 2

Overview• Sections 6.4 to 6.6• Introduction• Confidence Intervals for variances

o One varianceo Ratio of two variances

• Confidence Intervals for proportionso One proportiono Difference of two proportions

• Sample size

6205-ch6 3

Confidence Intervals for One Variance s^2

• Condition: normal populationFind the two optimal divisors (a, b) from Table X

1))1(

(2

2

bSn

aP

6205-ch6 4

Example

Suppose the heights of female students on campus has a normal distribution. In a random sample of 21 students, we find that sample mean is 66 inches and sample S.D. is 3 inches. Find a 95% C.I. for variance.

6205-ch6 5

F DistributionDefinition:

Q: What is the distribution of X^-1 where X ~ F(n,m)?

6205-ch6 6

Confidence Intervals for 22

21 /

6205-ch6 7

6205-ch6 8

6205-ch6 9

Proportions

BPS - 5th Ed.Chapter 19 10

• The proportion of a population that has some outcome (“success”) is p.

• The proportion of successes in a sample is measured by the sample proportion:

sample the in nsobservatio of number totalsample the in successes of numberp̂

“p-hat”

Inference about a Proportion

Simple Conditions

BPS - 5th Ed.Chapter 19 11

Inference about a Proportion

Sampling Distribution

BPS - 5th Ed.Chapter 19 12

Building a Confidence IntervalPopulation Proportion

BPS - 5th Ed.Chapter 19 13

Standard Error

BPS - 5th Ed.Chapter 19 14

Since the population proportion p is unknown, the standard deviation of the sample proportion will need to be estimated by substituting for p.

n

ppSE

ˆˆ

1

ˆ p

Confidence Interval

BPS - 5th Ed.Chapter 19 15

10

6205-ch6 16

Confidence Interval for One Proportion

• Example:

Suppose we want to estimate p, the proportion of people who favor Proposition A. When a SRS of size 81 is taken, 23 respond in favor of Prop A. Calculate a 90% CI for p.

6205-ch6 17

Confidence Interval for Difference of Two Proportions

• Condition: Two “large” independent samples

• Therefore, the (1-a) CI for p1-p2 is

6205-ch6 18

Is the prevalence of a disease higher in LA

or SF?

Suppose we too SRSs of size 1500 and 1400 from LA and SF respectively to answer this question. We find that in LA, 115 individuals have the disease and in SF, 189 individuals have the disease. Find a 95% CI for PLA –PSF. What can you conclude?

6205-ch6 19

Sample Size• How big of a sample size will we need to ensure

the precision of our interval?

• Required sample size for estimating a mean

• Required sample size for estimating a proportion