1. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Ch. 15 Probability Errors in Sampling 1.Sample Selection and Probability Error 2.The Expected Value and Standard Error 3.Using the Normal Curve 4.The Correction Factor

2. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 2/19 INDEX 1 Sample Selection and Probability Error 2 The Expected Value and Standard Error 3 Using the Normal Curve 4 The Correction Factor

3. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 3/19 1. Sample Selection and Probability Error Example After coding each student into different number from 1 to 4,738, select 100 numbers at random by using random number generator.  Survey about sense of marriage4,738 freshmen  Survey about sense of marriage from 4,738 freshmen selecting 100 students (100 samples) selecting 100 students (100 samples)  at random without replacement ☞ no bias

4. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/19 Representation of sample Does the Selected samples represents well? (gender ratio of the population: male student 64%) 250 random samples were drawn from the respondents. The sample size was 100. The number of man in each sample is shown below. Among them the number of getting exactly 64 male students is only 18. 63 59 64 70 71 66 71 61 65 65 74 65 62 68 65 69 74 65 62 62 59 59 64 68 57 67 60 60 60 65 65 58 72 66 65 66 65 68 68 63 57 59 61 63 67 63 55 61 69 69 59 60 64 66 59 65 67 63 56 58 61 67 63 61 65 61 62 65 64 69 67 65 65 68 66 59 65 62 63 67 68 51 63 62 68 65 69 63 59 61 68 69 75 67 65 62 63 60 65 67 64 71 66 65 60 65 64 65 68 59 66 71 64 66 57 59 60 61 61 61 63 60 70 73 66 66 70 66 67 63 63 60 60 56 62 68 71 64 66 57 58 66 66 62 73 70 62 60 64 72 66 65 72 63 62 66 61 64 59 58 58 68 67 74 63 66 60 66 62 67 65 64 58 66 59 68 64 65 66 64 63 70 58 55 54 70 58 60 74 63 63 73 65 61 66 60 62 70 66 66 63 58 60 64 58 61 59 54 61 55 64 65 64 66 58 68 66 62 69 65 67 69 62 61 65 68 73 68 58 58 62 72 65 69 61 65 72 64 70 67 63 67 62 66 63 58 64 67 69 72 1. Sample Selection and Probability Error

5. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/19 As we select samples the probability 64% of drawing male students is not changed, but the probability of being realized in the sample can be changed by the probability error.  the sample size is 100, histogram of the number of male student (using 250 samples) For the samples selected at random, the percentage of male student goes up or down by chance. 1. Sample Selection and Probability Error Representation of sample 0 5 10 15 49525558616467707376 The number of men %

6. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/19 Sample size and probability error  If we increase the sample size? The lager the sample size, the less the probability error. (law of averages) (% in the sample) = (% in the population) + (probability error) As the sample size goes up, the sample ratio goes similar to the population ratio. Empirical histogram ( R =250) Empirical histogram ( R =250) 1. Sample Selection and Probability Error

7. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 7/19 Sample Selecting  How can we get as many as 250 samples of which the size is 400 from 4,738? Large circle represents 4,738 people and each shadowed figure represents a sample containing 400 people. Though there are some overlaps, any two of them are not the same. Drawing 400-size samples from 4,738 people  There are 4738 C 400 ways. 1. Sample Selection and Probability Error

8. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/19 INDEX 1234 Sample Selection and Probability Error The Expected Value and Standard Error Using the Normal Curve The Correction Factor

9. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 9/19 2. The Expected Value & SE Expected value & SE  With a simple random sample, the expected value for the sample percentage equals the population percentage.  However, the sample percentage will not be exactly equal to its expected value-it will be off by a probability error. ☞ How big is this error likely to be? SE in probability sample represents the standardized size of the probability error.

10. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/19 Calculating SE ① Find the SE for the number of men in the sample Making a Box model: There should be only 1 ’ s (male) and 0 ’ s (female) in the box. The number of men in the sample is like the some of 100 draws from the box. (a simple random sample drawn without replacement) SD of box = SE for the sum of 100 draws = (law of square roots) ② Convert to percent, relative to the size of the sample SE for percentage (%) = (SE for number)/(sample size)  100% = 4.8/100  100%=4.8%  The SE for the percentage of men in a sample of 100 would be 4.8% 2. The Expected Value & SE

11. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/19 Expexted value & SE  What happens as the sample gets bigger? ☞ Law of square roots! ☞ Law of square roots!  The SE for the sample number goes up like the square root of the sample size.  The SE for the percentage goes down like the square root of the sample size. 2. The Expected Value & SE

12. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 12/19 INDEX 1234 Sample Selection and Probability Error The Expected Value and Standard Error Using the Normal Curve The Correction Factor

13. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 13/19 3. Using the Normal Curve Example 1 A company has 1,000,000 clients. Taking a simple random sample of 400 of them as a part of market research study. 20% of the clients earn over \40,000,000 per year. ① expected value of the percentage ② the SE of the percentage SD of box = SE for the sum = SE for the percentage =(8/400)  100=2% Over than \40 million =1, others=0 Expected value of box =0.2, sum=400  0.2=80, Expected value of percentage = (80/400)  100=20% The percentage of high earners in the sample will be around 20%, give or take 2% or so. When drawing at random from a box of 0’s and 1’s, the expected value for the percentage of 1’s among the draws equals the percentage of 1’s in the box.

14. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 14/19 Example 2 Estimate the probability that between 18% and 22% of the persons in the sample earn more than \40 million a year The expected value for the percentage is 20% and the SE is 2% now convert 2 standard units. Then probability of shaded area is about 68%. 3. Using the Normal Curve

15. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 15/19 INDEX 1234 Sample Selection and Probability Error The Expected Value and Standard Error Using the Normal Curve The Correction Factor

16. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 16/19 4. The Correction Factor Accuracy of the estimation of percentage Estimating the percentage of voters who are Democratic 1.2 million eligible voters in New Mexico, and about 12.5 million in the state of Texas. Simple random sampling of 2500 voters from each state without replacement.  For which poll is the probability error likely to be smaller? The New Mexico Poll is sampling one voter out of 500 while Texas poll is only sampling one voter out of 5,000. However, it does seem that New Mexico poll should be more accurate then the Texas poll. But, in fact, the accuracies are about the same. Because the two samples of 2500 provide with similar amount of information. When estimating percentages, it is the absolute size of the sample which determines accuracy, not the size relative to the population. This is true if the sample is only a small part of the population, which is the usual case.

17. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 17/19 The size of population and accuracy Democrats(1=50%) others (0=50%); Two boxes: NM, TX Drawing 2,500 at random from each box with the with replacement: Because the ratio of 1’s equal to 50% and the sizes of sample are same, the SE of the Democrats equal. There is no difference between Drawing from the two boxes. At each trial, the probability of Drawing 0 or 1 is 50 to 50, and the box size does not matter. The number of tickets drawn is Much smaller than the number In the box. So the ratio is almost The same even if we do not replace drawing with replacementdrawing without replacement  If the population size is large enough, it has nothing to do with the accuracy of the estimate 4. The Correction Factor

18. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 18/19 drawing with replacement & drawing without replacement When drawing without replacement, the box gets a bit smaller, reducing the variability slightly. So, the SE for drawing without replacement is a little less then the SE for drawing with replacement. (SE when drawing without replacement)= (SE when drawing with replacement)  (correction factor) Correction factor = (SE when drawing without replacement)= (SE when drawing with replacement)  (correction factor) Correction factor = * n = # of drawn cards = 2,500 (fixed) 4. The Correction Factor Number of cards in the box Correction factor 5,000 0.70718 10,000 0.86607 100,000 0.98743 500,000 0.99750 1,000,000 0.99875 12,500,000 0.99990

19. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 19/19 The correction factor  When the number of tickets in the box is large relative to the number of draws, the correction factor is nearly 1 can be ignored.  It is the absolute size of the sample which determines accuracy, through the SE for drawing with replacement. The size of the population of the population does not really matter. 4. The Correction Factor