1. ELEC 303 – Random Signals Lecture 18 – Statistics, Confidence Intervals Dr. Farinaz Koushanfar ECE Dept., Rice University Nov 10, 2009

2. Statistics

3. Example

4. Reduction of Cholesterol Level

5. Example (Cont’d)

6. Sample Mean

7. Sample Median

8. Sample Median (Cont’d)

9. Sample Mean vs. Sample Median

10. Percentile

11. Location of Data

12. Variability

13. Averages

14. Sample Variance

15. Statistics

16. Standard Deviation

17. Sample Range

18. Interquartile Range

19. Averaging?

20. Data Handling

21. Dot Plots

22. Histogram

23. Example

24. Histogram (Cont’d)

26. Confidence interval Consider an estimator for unknown  We fix a confidence level, 1-  For every  replace the single point estimator with a lower estimate and upper one s.t. We call, a 1-  confidence interval

27. Confidence interval - example Observations Xi’s are i.i.d normal with unknown mean  and known variance  /n Let  =0.05 Find the 95% confidence interval

28. Confidence interval (CI) Wrong: the true parameter lies in the CI with 95% probability…. Correct: Suppose that  is fixed We construct the CI many times, using the same statistical procedure Obtain a collection of n observations and construct the corresponding CI for each About 95% of these CIs will include 

29. A note on Central Limit Theorem (CLT) Let X 1, X 2, X 3,... X n be a sequence of n independent and identically distributed RVs with finite expectation µ and variance σ 2 > 0 CLT: as the sample size n increases, PDF of the sample average of the RVs approaches N(µ,σ 2 /n) irrespective of the shape of the original distribution

30. CLT A probability density functionDensity of a sum of two variables Density of a sum of three variablesDensity of a sum of four variables

31. CLT Let the sum of n random variables be S n, given by S n = X 1 +... + X n. Then, defining a new RV The distribution of Z n converges towards the N(0,1) as n approaches  (this is convergence in distribution),written as In terms of the CDFs

32. Confidence interval approximation Suppose that the observations X i are i.i.d with mean  and variance  that are unknown Estimate the mean and (unbiased) variance We may estimate the variance  /n of the sample mean by the above estimate For any given , we may use the CLT to approximate the confidence interval in this case From the normal table:

33. Confidence interval approximation Two different approximations in effect: – Treating the sum as if it is a normal RV – The true variance is replaces by the estimated variance from the sample Even in the special case where the X i ’s are i.i.d normal, the variance is an estimate and the RV T n (below) is not normally distributed