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SAMPLE MEAN and its distribution

SAMPLE MEAN and its distribution. SAMPLE MEAN and its distribution. 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:

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SAMPLE MEAN and its distribution

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  1. SAMPLE MEAN and its distribution

  2. SAMPLE MEAN and its distribution 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 http://onlinestatbook.com/stat_sim/sampling_dist/index.html

  4. ESTIMATORS • point • interval

  5. Properties of Point Estimators • UNBIASEDNESS • CONSISTENCY • EFFICIENCY

  6. Properties of Point Estimators • UNBIASEDNESS • An estimator isunbiased 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.

  7. Properties of Point Estimators UNBIASEDNESS

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

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

  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 (X̅ → 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 small variability (efficient) unbiased estimator with large variability (unefficient)

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

  15. population this distance isunknown (we do not knowtheexactvalueofm) , sowecan not quatifyreliabilityofourestimate sample POINT ESTIMATES

  16. point estimate of unknown population mean  computed from sample data– we do not know anything about his distance from real population mean T2 T1 interval estimate of unknown population mean  - we suppose, that with probability P =1- population mean  is anywhere in this interval of number line INTERVAL ESTIMATES Confidence interval for parametr  with confidence level(0,1) is limited by statistics T1 a T2:.

  17. 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 CONFIDENCE LEVEL  IN INTERVAL ESTIMATES

  18. 1= /2 P = 1 -  = 1 – (1 + 2) 2= /2  T1 T2 T TWO-SIDED INTERVAL ESTIMATES 1 a 2 represent statistical risk, that real population parameter is outside of interval (outside the limits T1 a T2 

  19. ONE-SIDED INTERVAL ESTIMATES LEFT-SIDED ESTIMATE RIGHT-SIDED ESTIMATE

  20. /2  T one-sided interval estimate P = 1 -  T1 /2  T1 T2 two-sided interval estimate P = 1 -  COMPARISON OF TWO- AND ONE-SIDED INTERVAL ESTIMATES

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

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

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

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

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