1. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Ch. 20 Test for Difference in Average between Two Groups 1.Standard Error for Average Difference 2.Comparing Two Averages 3.Comparing Average Difference between two groups under the controlled experiment 4.When to use a multi-sample Z -test?

2. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 2/15 INDEX 1 Standard Error for Average Difference 2 Comparing Two Averages 3 Comparing Average Difference between two groups under the controlled experiment 4 When to use a multi-sample Z-test?

3. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 3/15 1. Standard Error for Average Difference S.E. for Difference in Groups Box A Aver. 120 S.D. 60 Box B Aver. 80 S.D. 40 -If Two Groups were Independent, S.E. of Average difference is - a, b are S.E. of each average  Random Replacement Sampling Aver. of Box A from 400times drawing out = 120  3 Aver. of Box A from 100times drawing out = 80  4

4. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/15 Ex 2 1.Standard Error for Average Difference Number of Each number drawn out is 20  4 S.E. for Difference in number between card 1 drawn out and card 5 would be ☞ As one certain card being drawn out more, other cards being drawn out less Number of drawn card is not Stochastic Independent each other 100 times replacement sampling from a box containing cards have number on from 1 to 5 each other 12435

5. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/15 INDEX 1 Standard Error for Average Difference 2 Comparing Two Averages 3 Comparing Average Difference between two groups under the controlled experiment 4 When to use a multi-sample Z-test?

6. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/15 2. Comparing Two Averages Comparing Two Averages. Ex) Every year, 1,000 of high school freshmen are drawn to have a math test. Below is a result of test in 1998 and 2001. Aver. in 1988 = 64 S.D. = 11 Aver. in 2001 = 66 S.D. = 10 ‘Difference in Average is by a real factor or by chance?’

7. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 7/15 Comparing Two Averages 2. Comparing Two Averages 61 50 73 97 88 35 Box of 1988 100 50 73 32 40 66 Box of 2001 1,000 times Random Replacement Sampling Two Samples are independent each other ☞ Make two Box Model

8. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/15 Multi-Sample - test 2. Comparing Two Average H 0 : Mathematical Ability hasn ’ t Improved Observed Difference – Expected Difference S.E. of observed difference Observed Difference = 66-64 = 2 Observed Difference = 66-64 = 2 Expected Difference = 0 Expected Difference = 0 S.E. of observed difference S.E. of observed difference S.E. in 1988 S.E. in 1988 S.E. in 2001 S.E. in 2001 S.E. for Aver. Difference S.E. for Aver. Difference Reject!

9. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 9/15 INDEX 1 Standard Error for Average Difference 2 Comparing Two Averages 3 Comparing Average Difference between two groups under the controlled experiment 4 When to use a multi-sample Z-test?

10. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/15 Randomly controlled Experiment Ex) Vitamin C ’ s prevention effect to common cold Ex) Vitamin C ’ s prevention effect to common cold Treatment Group: Randomly Draw 100 people from 200 people were given 2000mg of vitamin everyday Control Group: Residual 100 people were given same amount of fake pill. After Experiment, Counting number of catching cold Treatment Group: Average = 2.3 times Standard Deviation = 3.1 times Control Group: Average = 2.6 times Standard Deviation = 2.9 times 3. Comparing Average Difference between two groups under the controlled experiment ‘ Is Difference in average statistical significant? ’

11. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/15 Randomly controlled Experiment 3. Comparing Average Difference between two groups under the controlled experiment Is it alright to use the formula formula in the case of Non- replacement Sampling and dependent? Difference in Two Averages = -0.3 Difference in Two Averages = -0.3 S.E of observed difference S.E of observed difference S.E. of Treatment S.E. of Treatment S.E. of Control S.E. of Control 평 S.E. for difference in Average 평 S.E. for difference in Average ? Observed Difference – Expected Difference S.E. of observed difference

12. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 12/15 Calculation of S.E. 3. Comparing Average Difference between two groups under the controlled experiment With the Data from randomly controlled experiment, We can use the 무작위로 formula to calculate S.E. for difference. Two Problems Non-Replacement Sampling Two Aver. are dependent each other Decreasing S.E. Increasing S.E. Offset

13. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 13/15 INDEX 1 Standard Error for Average Difference 2 Comparing Two Averages 3 Comparing Average Difference between two groups under the controlled experiment 4 When to use a multi-sample Z-test?

14. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 14/15 4. When to use a multi-sample Z-test? Using a multi-sample Z-test A multi-sample Z-test is used to test significance of difference in Average between Two groups. If Two Groups were not independent, The 두 표본이 formula can’t be applied. But, in randomly controlled experiment, the 무작위로 formula can be applied, even though two groups are dependent each other.

15. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 15/15 4. When to use a multi-sample Z-test? Using a multi-sample Z-test  A multi-sample Z-test is used to test significance of difference in Average between Two samples.  If Samples were not independent, The 두 표본이 formula can ’ t be applied to calculate the S.E. of Difference.  But, in randomly controlled experiment, the 무작위로 formula can be applied, even though two groups are dependent each other.