1. Interval Estimation Part II
Notes of STAT6205 by Dr. Fan
6205-ch6

2. 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
6205-ch6

3. Confidence Intervals for One Variance s^2
Condition: normal population
Find the two optimal divisors (a, b) from Table X
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 Distribution
Definition: Q: What is the distribution of X^-1 where X ~ F(n,m)?
6205-ch6

6. Confidence Intervals for
6205-ch6

7. 6205-ch6

8. 6205-ch6

9. 6205-ch6

10. Proportions
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:
“p-hat”
Chapter 19
BPS - 5th Ed.

11. Inference about a Proportion Simple Conditions
Chapter 19
BPS - 5th Ed.

12. Inference about a Proportion Sampling Distribution
Chapter 19
BPS - 5th Ed.

13. Building a Confidence Interval Population Proportion
Chapter 19
BPS - 5th Ed.

14. Standard Error
Since the population proportion p is unknown, the standard deviation of the sample proportion will need to be estimated by substituting for p.
Chapter 19
BPS - 5th Ed.

15. Confidence Interval 10
Chapter 19
BPS - 5th Ed.

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
6205-ch6