Definition 95% Confidence IntervalA Definition of a Student’s t Random VariableConfidence LevelDesired Confidence LevelPubHlth 540 – Fall 2011 6. Estimation Page 1 of 72 Nature Population/ Sample Observation/ Data Relationships/ Modeling Analysis/ Synthesis Unit 6. Estimation “Use at least twelve observations in constructing a confidence interval” - Gerald van Belle What is the mean of the blood pressures of all the students at the Amherst Regional High School? It’s too much work to measure the blood pressure of every individual student in this population. So we will make a guess based on the blood pressures of a sample of students. This is estimation. Estimation involves using a statistic calculated for our sample as our estimate of the population quantity of interest. In this example, where the population mean is of interest, possible choices might be (1) the sample mean blood pressure; (2) the sample median blood pressure; (3) the average of the smallest and largest blood pressure; and so on. The point is, there is no one choice for estimation. What we mean by a “good” choice of estimator is one focus of this unit. The other focus of this unit is confidence interval estimation. A confidence interval is a single estimate together with a “safety net”, which can also be thought of as a “margin of error”PubHlth 540 – Fall 2011 6. Estimation Page 2 of 72 Nature Population/ Sample Observation/ Data Relationships/ Modeling Analysis/ Synthesis Table of Contents Topic 1. Unit Roadmap …………………………………………………………. 2. Learning Objectives …………………………………………………… 3. Introduction ……………………..…………………..…..………..…… a. Goals of Estimation ……………………………………………. b. Notation and Definitions ………………………………………. c. How to Interpret a Confidence Interval ………………………… 4. Preliminaries: Some Useful Probability Distributions ………………. a. Introduction to the Student t- Distribution ……………………. b. Introduction to the Chi Square Distribution ………………….. c. Introduction to the F-Distribution ……………………………. d. Sums and Differences of Independent Normal Random Vars .. 5. Normal Distribution: One Group ………………………………… a. Confidence Interval for μ, σ2 Known ……………………..….. b. Confidence Interval for μ, σ2 Unknown ……………………… c. Confidence Interval for σ2 …………………………………. 6. Normal Distribution: Paired Data ……………………………….. a. Confidence Interval for μDIFFERENCE ……………………………... b. Confidence Interval for σ2 DIFFERENCE ……………………………. 7. Normal Distribution: Two Independent Groups: …………………. a. Confidence Interval for [μ1 - μ2] ………………………………… b. Confidence Interval for 2212σσ ……….………………….……. 8. Binomial Distribution: One Group ……………………………….. a. Confidence Interval for π ……….…………………………….…. 9. Binomial Distribution: Two Independent Groups ……………….. a. Confidence Interval for [12ππ−] …………………………….….. 3458101320202428313333384144454849495760606464 Appendices i. Derivation of Confidence Interval for μ – Single Normal with σ2 Known ………………………………………………….. ii. Derivation of Confidence Interval for σ2 – Single Normal ………. iii. SE of a Binomial Proportion ………………..…………………..… 677072PubHlth 540 – Fall 2011 6. Estimation Page 3 of 72 Nature Population/ Sample Observation/ Data Relationships/ Modeling Analysis/ Synthesis 1. Unit Roadmap Nature/ Populations Sample Observation/ Data Unit 6. Estimation Relationships Modeling Analysis/ Synthesis Recall that numbers that are calculated from the entirety of a population are called population parameters. They are represented by Greek letters such as μ (population mean) and 2σ (population variance). Often, it is not feasible to calculate the value of a population parameter. Numbers that we calculate from a sample are called statistics. They are represented by Roman letters such as X (sample mean) and 2S(sample variance). In our introduction to sampling distributions, we learned that a sample statistic such as X is a random variable in its own right. This can be understood by imagining that there are infinitely many replications of our study so that there are infinitely many X . Putting this together, if a sample statistic such as X is to be used as an estimate of a population parameter such as μ , the incorporation of a measure of its variability (in this case the standard error of X ) allows us to construct a “margin of error” about X as our estimate of μ . The result is a confidence interval estimate..PubHlth 540 – Fall 2011 6. Estimation Page 4 of 72 Nature Population/ Sample Observation/ Data Relationships/ Modeling Analysis/ Synthesis 2. Learning Objectives When you have finished this unit, you should be able to:  Explain that there is possibly more than one way to estimate a population parameter based on data in a sample.  Explain the criteria of unbiased and minimum variance in the selection of a “good” estimator.  Define the Student t, chi square, and F probability distribution models.  Explain that the sum and difference of independent random variables that are distributed normal are also distributed normal .  Interpret a confidence interval.  Calculate point and confidence interval estimates of the mean and variance of a single normal distribution.  Calculate point and confidence interval estimates of the mean and variance of a single normal distribution in the paired data setting.  Calculate point and confidence