Sampling Distribution of and Simulation Methods. Ontario Public Sector Salaries. Strange Sample? Lecture 11. Reading: Sections

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1 Sampling Distribution of and Simulation Methods Lecture 11 Reading: Sections Ontario Public Sector Salaries Public Sector Salary Disclosure Act, 1996 Requires organizations that receive public funding from the Province of Ontario to disclose annually the names, positions, salaries and total taxable benefits of employees paid $1, or more in a calendar year E.g. Government of Ontario, Crown Agencies, Municipalities, Hospitals, Boards of Public Health, School Boards, Universities, Colleges, Hydro One, Ontario Power Generation, etc. 213 disclosure of 212 salaries: 2 Strange Sample? For all ON public sector employees w/ salaries of $1K+, mean is $127.5K and s.d. $39.6K Are these numbers parameters or statistics? Shape of the salary distribution? (2 explanations) Random sample of 1, Ontario public sector employees has a mean salary of $131.2K Why is different than? How likely is such a big sample mean if claim true? i.e , 39.6, 1? 3 Lecture 11, Page 1 of 8

2 Ontario Public Sector Employees Making $1K or more (88545 employees) Laura Formusa, President & CEO, Hydro One, $1.4M Thomas Mitchell, President & CEO, Ontario Power Generation, $1.72M Salary ($1,s) 4 STATA Summary of Population. su salary, detail; salary Percentiles Smallest 1% % % Obs % Sum of Wgt % Mean Largest Std. Dev % % Variance % Skewness % Kurtosis Mean and Variance of But what is the shape of the sampling distribution of? 6 Lecture 11, Page 2 of 8

3 1% Condition / 1% Rule Derivation of assumes that each observation ( ) is independent of others For this to be true, must sample with replacement OR sample without replacement from a population that is infinitely large In contrast, real applications involve sampling without replacement from a finite population BUT if sample < 1% of population, assumption true enough: can use theoretical result 7 Recall Parking Permit Ex (Lec. 1) Probability Population.8; Permits (X) Probability.3.2 Sampling Dist. n = Sample Mean (X-bar) Work to find and not needed. Why is work needed? 8 Shape of sampling distribution of? Central Limit Theorem (CLT): For a random sample from any population the sampling distribution of the sample mean () is approximately Normal for a sufficiently large sample size Rough rule of thumb: sample size 3 < 3 sufficient for modestly non-normal populations If population is Normal, then n 1 ok > 3 necessary for extreme departures 9 Lecture 11, Page 3 of 8

4 1 Sampling Distribution of Sample Mean n = 1, mu = , sigma/root(n) = X-bar Is sampling error a plausible explanation for as big as 131.2? 11 Sampling Error: Plausible Explanation? , , / What How if sample does size 5? compare to 127.5, 131.2? (A) 39.6,5 They are equal? (B) They are roughly equal What (C) serious problem is may much we face bigger in trying to find this probability? (D) is slightly smaller (E) is much smaller 12 Lecture 11, Page 4 of 8

5 Ontario Public Sector (8769 employees) Salary ($1,s) Ontario Public Sector (8769 employees) Original dist. is so skewed (remember truncation too) that this non-linear trans. only helps a bit ln(salary ($1,s)) Fortune 5 Companies, 213 (5 companies) 8.e-6 6.e-6 4.e-6 2.e Employees Fortune 5 Companies, 213 (5 companies) Ln(Employees) 13 Monte Carlo Simulation Monte Carlo Simulation: A problem solving method where a computer generates many random samples and you make an inference based on patterns in outcomes Simulation is most useful when theoretical results (e.g. CLT) do not apply and the problem is too big for an analytic approach It allows us to find sampling distributions with a high degree of accuracy 14 Recall Central Limit Theorem The CLT says the sampling distribution of the sample mean is Bell shaped no matter what the shape of the population so long as the sample size is sufficiently large What is sufficiently large? Is a rule of thumb always correct or is it just a rough guide? What factors affect how large is sufficiently large? 15 Lecture 11, Page 5 of 8

6 n = 5: Sufficiently Large? Monte Carlo simulation: many samples of 5 ON public employees (in each sample, n = 5) # simulation draws (# samples drawn) = very big Simulation error: Chance difference between simulated probability and true probability Drive it to zero by doing many draws For each sample compute the sample mean Summarize distribution of : graphically (histogram) and numerically (Stata summary) n = 5; simulation draws = 5 Is a sample size of 5 sufficiently large such that the sampling distribution of the sample mean is Normal (i.e. CLT kicks in)? X-bar 17 X-bar Percentiles Smallest 1% % % Obs 5 25% Sum of Wgt. 5 5% Mean Largest Std. Dev % % Variance % Skewness % Kurtosis Is the simulation giving the correct mean and Std. Dev.? 18 Lecture 11, Page 6 of 8

7 Simulated Sampling Distribution (n = 5) X-bar ON Pub. Sec. (88545 employees) Salary ($1,s) A Random Sample of 5 employees Salary ($1,s) 19 salary Percentiles Smallest 1% % % Obs 5 25% Sum of Wgt. 5 5% Mean Largest Std. Dev % % Variance % Skewness % Kurtosis n = 5; simulation draws = Median 21 Lecture 11, Page 7 of 8

8 Median Percentiles Smallest 1% % % Obs 5 25% Sum of Wgt. 5 5% Mean Largest Std. Dev % % Variance % Skewness % Kurtosis How to interpret ? How to interpret ? 22 Lecture 11, Page 8 of 8