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

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

Confidence Intervals for One Variance s^2 Condition: normal population Find the two optimal divisors (a, b) from Table X 6205-ch6

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

F Distribution Definition: Q: What is the distribution of X^-1 where X ~ F(n,m)? 6205-ch6

Confidence Intervals for 6205-ch6

6205-ch6

6205-ch6

6205-ch6

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.

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

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

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

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.

Confidence Interval 10 Chapter 19 BPS - 5th Ed.

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

Confidence Interval for Difference of Two Proportions Condition: Two “large” independent samples Therefore, the (1-a) CI for p1-p2 is 6205-ch6

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

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