Quantitative Methods in Social Sciences (E774): Review Session V

1. Quantitative Methods inSocial Sciences (E774): Review Session V Dany Jaimovich October 28, 2009

2. Plan for today • A1, PP1 • Things in context • Finally… statistical inference! • Point estimation • Confidence Intervals • Dealing with proportions • t-tests • A bit of STATA

3. Review • Last RS we concentrate in probability distribution of the POPULATION • We calculated z-score and start doing some predictions about the probability of random events. • We needed to know the population PARAMETERS

4. Review • Now we are going to concentrate en “real world” and work with samples instead of population. • Then, we are moving from parameters to statics and, here the important thing, we are going to be doing STATISTICAL INFERENCE.

5. INFERENCE • When doing inference, you always will have a point estimation and a confidence interval (CI) around it. • Unbiased estimator: is centered around the parameter. • Efficient estimator: minimum possible standard errors.

6. INFERENCE • Confidence intervals (CI) depend on the researcher: 90%, 95%, 99%... Or we can call them margin of error: 10%, 5%, 1% • To calculate CI, we need to remember the two tails questions: Is x equal to certain value? Critical Z value at 95%=1.96 • For the case of small sample, we will see t-test later…

7. INFERENCE • The confidence interval at the 95% will be given by: • If we want CI at 99%:

8. INFERENCE • EXERCISE 1: A development agency implemented a program aimed to improve agricultural production in country HHH. a country-wide survey reveals σ=500 in the results. We implemented a survey to see the effects in a particular region, interviewing 50 farmers, finding an average increase of 1000kg. • What are the expected results in this region at 95% confidence? • With 1% margin of error?

9. INFERENCE • EXERCISE 1: Not convinced with the result, a new survey was conducted. Now 100 farmers were interviewed, and again the mean was 1000. • Will CI change? Why? • If necessary, calculate new CI. • Is the program working?? Are we sure?? • What if we don’t know σ?

10. INFERENCE • SPECIAL CASE: PROPORTIONS • When the variable of analysis is a proportion, there are some differences: • Standard errors= • The CI for proportions is: =

11. INFERENCE • EXERCISE 2: The development agency also included a question of satisfaction with the aid program. In the first survey, 28 declared to be happy with program. In the second, 60 declared to be happy. • Do farmers like the program?

12. INFERENCE • When we work with samples instead of populations, normality assumption might not be optimal. • The alternative is to use a distribution that “adapts” to the sample size: when sample is too small accept values that are far from the mean • This is the Student t-distribution.

13. INFERENCE • For the t-distribution change with the “degrees of freedom” (df):

14. INFERENCE • The df are: (number of observations – variables to be explained). When doing t-test for the mean, df=n-1. • When using t-distribution, an estimator of the population variance (σ) is used, called “σ-hat”:

15. INFERENCE • Similarly, a new test of significance will replace the z-score… t-score or t-test: • If the sample if big (over 100) the use of s or “σ hat” for the test does not make big difference. • (exercises with t-Table next week)

16. INFERENCE

17. INFERENCE • STATA example: • cii n mean sd • cii 49 1000 3000, level(95) • ttesti n mean sd test-value , level (95) • ttesti 10 100 20 90 , level (95)