1 SAMPLE MEAN and its distribution

2 CENTRAL LIMIT THEOREM: If sufficiently large sample is taken from population with any distribution with mean  and standard deviation , then sample mean has sample normal distribution N( ,  2 /n) It means that: sample mean is a good estimate of population mean  with increasing sample size n, standard error SE is lower and estimate of population mean is more reliable

3 SAMPLE MEAN and its distribution

ESTIMATORS 4 point interval

Properties of Point Estimators 5 UNBIASEDNESS CONSISTENCY EFFICIENCY

Properties of Point Estimators 6 UNBIASEDNESS An estimator is unbiased if, based on repeated sampling from the population, the average value of the estimator equals the population parameter. In other words, for an unbiased estimator, the expected value of the point estimator equals the population parameter.

Properties of Point Estimators 7 UNBIASEDNESS

8 Properties of Point Estimators individual sample estimates true value of population parameter

9 ZÁKLADNÍ VLASTNOSTI BODOVÝCH ODHADŮ y – sample estimates M - „average“ of sample estimates bias of estimates true value of population parameter

10 Properties of Point Estimators CONSISTENCY An estimator is consistent if it approaches the unknown population parameter being estimated as the sample size grows larger Consistency implies that we will get the inference right if we take a large enough sample. For instance, the sample mean collapses to the population mean (X ̅ → μ) as the sample size approaches infinity (n → ∞). An unbiased estimator is consistent if its standard deviation, or its standard error, collapses to zero as the sample size increases.

11 Properties of Point Estimators CONSISTENCY

12 Properties of Point Estimators EFFICIENCY An unbiased estimator is efficient if its standard error is lower than that of other unbiased estimators

13 Properties of Point Estimators unbiased estimator with large variability (unefficient) unbiased estimator with small variability (efficient)

14 POINT ESTIMATES Point estimate of population mean: Point estimate of population variance: bias correction

15 POINT ESTIMATES sample population  this distance is unknown (we do not know the exact value of  so we can not quatify reliability of our estimate

16 INTERVAL ESTIMATES Confidence interval for parametr  with confidence level  (0,1) is limited by statistics T 1 a T 2 :. point estimate of unknown population mean  computed from sample data– we do not know anything about his distance from real population mean T1T1 T2T2 interval estimate of unknown population mean  - we suppose, that with probability P =1-  population mean  is anywhere in this interval of number line

17 CONFIDENCE LEVEL  IN INTERVAL ESTIMATES  these intervals include real value of population mean (they are „correct“), there will be at least (1-  ).100 % these „correct“ estimates this interval does not include real value of population mean (it is „incorrect“), there will be at most (100  ) % of these „incorrect“ estimates

18 TWO-SIDED INTERVAL ESTIMATES T1T1 T2T2 P = 1 -  = 1 – (  1 +  2 )  1 =  /2  2 =  /2  T   1 a  2 represent statistical risk, that real population parameter is outside of interval (outside the limits T 1 a T 2

19 ONE-SIDED INTERVAL ESTIMATES LEFT-SIDED ESTIMATERIGHT-SIDED ESTIMATE

20 COMPARISON OF TWO- AND ONE- SIDED INTERVAL ESTIMATES T1T1 two-sided interval estimate P = 1 -   /2  T T2T2 one-sided interval estimate P = 1 -  T1T1 

21 CONFIDENCE INTERVAL (CI) OF POPULATION MEAN   small sample (less then 30 measurements) t  /2,n-1 quantil of Student ‘s t-distribution with (n-1) degrees of freedom and  /2 confidence level lower limit of CIupper limit of CI

22 CONFIDENCE INTERVAL (CI) OF POPULATION MEAN   large sample (over 30 data points) z  /2 quantile of standardised normal distribution lower limit of CIupper limit of CI instead of  (population SD) there is possible to use sample estimate S

23 CONFIDENCE INTERVAL (CI) OF POPULATION STAND. DEVIATION   for small samples

24 CONFIDENCE INTERVAL (CI) OF POPULATION STAND. DEVIATION   for large samples