interval estimation part ii
DESCRIPTION
Interval Estimation Part II. Notes of STAT6205 by Dr. Fan. Overview. Sections 6.4 to 6.6 Introduction Confidence Intervals for variances One variance Ratio of two variances Confidence Intervals for proportions One proportion Difference of two proportions Sample size. - PowerPoint PPT PresentationTRANSCRIPT
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Interval Estimation
Part IINotes of STAT6205 by Dr. Fan
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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
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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
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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.
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F DistributionDefinition:
Q: What is the distribution of X^-1 where X ~ F(n,m)?
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Confidence Intervals for 22
21 /
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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
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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.
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Confidence Interval for Difference of Two Proportions
• Condition: Two “large” independent samples
• Therefore, the (1-a) CI for p1-p2 is
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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?
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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