1. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist 1.Introduction 2.Confidence Intervals 3.Interpreting a Confidence Intervals Ch. 16 The Accuracy of Percentages

2. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 2/11 INDEX 1Introduction 2 Confidence Intervals 3 Interpreting a Confidence Intervals

3. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 3/11 1. Introduction SE for the sample percentage EX) Estimating the percentage of Democrats  Population: a district with 100,000 eligible voters  Sample: a simple random sample of 2,500 voters  In sample percentage: 53% (1328 out of 2500) The difference between the estimated Democrats and the population percentage? - Sample percentage is different from the population percentage due to the probability error. - SE explains the difference of between the two.

4. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/11 1. Introduction Calculation of SE Is it likely to be off by as much as 3 percentage points?  Population: A box including 100,000 ballots of 1 votes from each eligible voters. (Democrat=1, Otherwise=0)  Sample: drawing 2,500 ballots at random  SD of box =  SE of the # of eligible voters who is Democrats in sample=  SE of sample percentage=25/2500=1% (3% points is 3 SEs) When sampling from a 0-1 box whose composition is unknown, the SD of the box can be estimated by substituting the fraction of 0 ’ s and 1 ’ s in the sample for the unknown fractions in the Box. The estimate is good when the sample is reasonably large

5. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/11 INDEX 1Introduction 2 Confidence Intervals 3 Interpreting a Confidence Intervals

6. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/11 2. 신뢰구간 confidence level & confidence interval  What happens with a cutoff at 2 SEs? 51%53%55% 2SE Sample percentage  The interval ‘ sample percentage  1  SE ’ is 68%-confidence interval for the population percentage  The interval ‘ sample percentage  2  SE ’ is 95%-confidence interval for the population percentage  The interval ‘ sample percentage  3  SE ’ is 99.7%-confidence interval for the population percentage Sample percentage = population percentage + probability error  in population Democrats percentage = 53%  1%  (51%, 55%): confidence interval of confidence level 95%

7. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 7/11 2. 신뢰구간 [Example 1] Estimating Democrats Percentage  Population : a town with 25,000 electoral voters  Sample : drawing at random 1,600 electoral voters  Democrats percentage in sample: 57% (917 out of 1,600)  SD of box =  SE of in sample Democrats =  SE of in sample Democrats percentage = 20/1,600=1.25(%)  95% confidence interval of in population Democrats percentage = 57%  2  1.25%

8. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/11 2. 신뢰구간 Interpreting a confidence level  confidence levels are often quoted as being “ about ” so much. SEs have been estimated from the data. Since it is based on central limit theorem, large samples justify it.  A sample percentage near 0% or 100% suggests that the box is lopsided and a very large number of draws will be needed before the normal approximation takes over.  If the sample percentage is near 50%, the normal approximation  Should be satisfactory when there are only a hundred draws of so The normal approximation has been used. If the sample size gets larger, the more accurate the confidence level becomes

9. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 9/11 INDEX 1Introduction 2 Confidence Intervals 3 Interpreting a Confidence Intervals Interpreting a Confidence Intervals

10. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/11 3. Interpreting a Confidence Intervals Interpreting a Confidence Intervals  The probability of population percentage of Democrats to be between 54.5% and 59.5% is 95%? Among all possible samples, 95% does include the population percentage in the confidence interval of ‘ sample percentage  2SE ’, but not 5%  Confidence intervals changes from sample to sample.  The center and length of the confidence intervals change. [Example 1]the confidence interval for the percentage of Democrats is

11. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/11 The 95%-confidence interval of red marbles in the box is shown for 100 different samples. The interval covers the population percentage, marked by a vertical line. The ratio of red marbles in the box = 80% Each takes a simple random sample of 2,500 marbles The 95%-confidence interval of red marbles in the box is shown for 100 different samples. The interval covers the population percentage, marked by a vertical line. The ratio of red marbles in the box = 80% Each takes a simple random sample of 2,500 marbles The confidence interval changes from sample to sample. 94 of the samples covers the population percentage. 3. Interpreting a Confidence Intervals Interpreting a Confidence Intervals