1. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement8-1 Additional real-life examples (proportions) Supplement 8: Additional real-life examples (proportions) *The part of biasedness (including the proof) is a result of a correspondence between Dr. Ka-fu Wong and YueShen Zhou. The example was drawn from a clip sent over by Nipun Sharma. Use it at your own risks. Comments, if any, should be sent to kafuwong@econ.hku.hk.

2. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement8-2 Mean and variance (1) Suppose a proportion p of the population is female. An observation is randomly drawn from the population. Code x = 1 if the drawn observation is female. Code x = 0 if the drawn observation is male. What is the population mean and variance of this random variable X? E(X) = (1)Prob(x=1) + (0)Prob(x=0) = (1)p + (0)(1-p) = p Var(X) = (1-p) 2 Prob(x=1) + (0-p) 2 Prob(x=0) = (1-p) 2 p + p 2 (1-p) = (1-p) p [1-p + p] = (1-p)p

3. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement8-3 Mean and variance (2) Suppose a proportion p of the population is female. A sample of n observations is randomly drawn with replacement from the population. Code x = 1 if a drawn observation is female, 0 otherwise. What is the population mean and variance of m = (x 1 +…+x n )/n? E(m) = E[(x 1 +…+x n )/n] = [E(x 1 ) + E(x 2 ) + … + E(x n )]/n = E(X) = p Var(m) = Var[(x 1 +…+x n )/n] = Var[(x 1 +…+x n )]/n 2 = [Var (x 1 )+…+Var(x n )]/n 2 = Var(X)/n = (1-p)p/n

4. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement8-4 Central limit theorem of proportion Let x 1,…,x n be a iid sample from a population with p proportion of success. (Failure coded as 0 and success as 1.) ∑x i /n is simply the proportion of success and hence the simple average of the outcomes from the n trials. ∑x i /n will be approximately normal according to CLT.

5. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement8-5 Do you think Chinese officials spent too much government money on the following? Base on a poll of 18,000 persons.

6. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement8-6 Can we construct the 95% confidence intervals for the population proportion? Standard error= [p(1-p)/n] 1/2 npstd error lower limitupper limit 180000.9580.0014951030.955070.9609303 180000.8620.0025707330.8569610.8670385 180000.860.0025862890.8549310.865069 180000.850.0026614530.8447840.8552164 180000.8070.002941570.8012350.8127654 180000.8020.0029701850.7961790.8078215 180000.6790.0034797750.672180.6858202 180000.50.003726780.4926960.5073044

7. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement8-7 Why would some books used a different formula? The population proportion is unknown. Thus, an estimate of the variance of sample proportion will be When n is large, the difference between two estimators of sample variance are negligible. This is why some books use n, some use (n-1).

8. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement8-8 Why would some books used a different formula? When n is large, the difference between two estimators of sample variance are negligible. This is why some books use n, some use (n-1). is a biased estimator for is an unbiased estimator for

9. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement8-9 Proof: A biased estimator

10. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement8-10 Proof: An unbiased estimator

11. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Supplement8-11 - END - Supplement 8: Additional real-life examples