Hypothesis Testing

1. Hypothesis Testing Fertility of American Women (aged 15-44), 1980-2006 Births per 1,000 women Births per 1,000 women UT UT NH NH SOURCE: www.census.gov/population/www/socdemo/fertility.html#hist yr = 0, …, 26 ^

2. Hypothesis Testing of Suppose you are asked to empirically investigate the charge that obstetricians are guilty of inducing demand by performing unnecessary cesarean sections. Empirical Model (linear-log) s = 1, …, 47 t = 1, …, 13 (1970 to 82) Panel data

4. Empirical Model (linear-log) s = 1, …, 47 t = 1, …, 13 (1970 to 82) Gruber and Owings (1996) OLS regression ^ where _____ CRate = 0.119 11.9% of deliveries are by cesarean section Standard error of The hypothesis test of answers the question of whether is negative enough to be convincing evidence that the true coefficient is negative rather than simply being negative due to chance. But first, let’s interpret what the estimated coefficient means, assuming temporarily that it is statistically significant.

5. What is the predicted effect of a 10% decrease in the fertility rate? ^ ^ point Gruber and Owings interpretation of “A fall in the fertility rate of 10 percent is associated with an increase in the likelihood of cesarean delivery of 0.97 percentage points.” (p. 113)

6. Set H0 and H1 Does the empirical evidence convincingly knock H0 down? Testing for demand inducement by obstetricians (OBs): Gruber and Owings use economics to tell a story of why this might be true. Two types of potential errors

7. Hypothesis Testing of ___ ___

8. Normally, a negative sign for is not sufficient to convince us to reject the null hypothesis and, in this case, conclude that OBs are guilty of inducing demand by carrying out unnecessary c-sections. Instead, has to be sufficient smaller than zero for us to be relatively confident that the true is negative. In other words, has to be smaller than some critical negative value, call it , for us to be willing to reject the null hypothesis. Decision Rule Choose by setting Type I error Reject true H0 Suppose Prob (Type I error) = significance level of the test

9. Given how would you illustrate the probability of making a Type II error? Type II error: accept a false H0. Suppose Note: There is no way to measure Type II error without knowing the true value of

10. As the significance level of the test becomes more stringent, what happens to the prob(Type I error) and prob(Type II error)? Given that changing the critical value decreases the probability of making one type of error while increasing the probability of making the other type of error, how should we set ? Setting . Think about the cost of making each type of error. • unnecessary c-sections

11. Typical practice set the significance level so that the probability of making at type I error is small. Prob(Type I error ) is typically set at either .10, .05 or .01 But you should always question whether the typical practice is appropriate. How do you find One possibility: solve for where Too cumbersome, requiring that we solve difficult problems for every new hypothesis.

12. How do you find Easier process: transform the test statistic into a standard normal one. Define z statistic as: But if H0 is true, then , so Standard normal distribution Much easier a table of the standard normal distribution can be used to find the critical value for a variety of hypothesis tests.

13. BUT how do we calculate when is unknown? Standard deviation of Solution substitute , which is the standard error of , for t statistic t distribution k+1= # parameters n = Sample size The shape of the t distribution depends on the number of degrees of freedom. It is a little fatter than the standard normal distribution due to the increased variation of estimating

14. Decision Rule The critical value, tC , depends on