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Chapter 7 Estimation

Chapter 7 Estimation. Instructor: Xijin Ge SDSU Dept. of Math/Stat. Sampling situation: how many hours do SDSU students spend in social networking sites?. Population:11,400 students in SDSU. Random Sample #1:. 100 students. Evaluating estimators by repeated sampling and estimation.

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Chapter 7 Estimation

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  1. Chapter 7 Estimation Instructor: Xijin Ge SDSU Dept. of Math/Stat

  2. Sampling situation: how many hours do SDSU students spend in social networking sites? Population:11,400 students in SDSU Random Sample #1: 100 students

  3. Evaluating estimators by repeated sampling and estimation Population:11,400 students in SDSU Random Sample #1 Sample mean Random Sample #2 Random Sample #3

  4. An Experiment • Suppose r.v. X is a number we got from rolling an honest die • p.d.f. • This is the exact value of the mean that we derived theoretically.

  5. Point estimator Point estimate Estimation of the mean • Pretend that we don’t know the exact value of the mean and want to evaluate it through experiments. • Rolling die 30 times and calculate average • 56 students did the experiments

  6. Histogram of all point estimates x = scan(“data.txt”) hist(x,xlim=c(2.5,4.5)) Unbiased: centered at the right spot.

  7. Defining “Unbiased” estimator An estimator is an unbiased estimator for a parameter if and only if To test if the statistic is an unbiased estimator we need to repeat the sampling and estimation many, many times. If the average of the estimated values approaches the true/theoretical value, then it is unbiased.

  8. Proof: Theorem 7.1.1

  9. Are unbiased estimations accurate? We sampled 100 students and Can we guarantee that the true mean is close to 1.5hr?

  10. Desirable properties of point estimator • Unbiased, and • Small variance for large sample sizes.

  11. Proof. P58, rules of variance • 10 minutes group quiz • (total 5 points) • Give answer (1 point) • Prove it (2 points) • Discuss it (2 points)

  12. Proof. P58, rules of variance

  13. For larger sample size, the difference between repeated estimation is smaller. Population:11,400 students in SDSU Random Sample #1 Sample mean Random Sample #2 Random Sample #3

  14. Discussions on Theorem 7.1.2 • Sample means based on small sample may differ significantly from actual population mean. • Sample mean based on a large sample can be expected to lie reasonable close to actual population mean. • Standard deviation of This is called standard error of the mean.

  15. Th. 7.1.3. Estimation of variance Proof in Appendix C. Note S is not an unbiased estimator of σ.

  16. Distribution of X Properties of Moment Generating Functions:

  17. X is normally distributed !

  18. Expected value of your learning outcome • What is an unbiased estimator? • Sample mean is an unbiased estimator of population mean. • Variance of sample mean decrease linearly with the increase of sample size. • Unbiased estimation of variance is S^2 • X is normally distributed!

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