Congratulations, you have the correct answers for the multiple choice questions. In the next set of videos, we will continue with new questions that can be tackled by econometric methods. You will see that each question may need an extension of the methods we discussed before. The first question that we have might stand out for anyone who wonders whether it's worthwhile to organize the Olympic games. No doubt that you will know that such an organization can become very costly. But before one is assigned to such an organization, you need to make a budget plan. It is no secret that these budget plans are usually, if not always, quite different than the effective money spent ones the games are over. Let us have a look at the next table which contains the percentage cost overrun for Olympic summer and winter games for the periods 1968 to 2016 when they were available. When the data are put in a picture like this, then it seems that the percentages on average are lower after observation 10 which would correspond around 1999 here, even when we would dismiss the 720 percent of Montreal 1976. Now let's have a look at the question whether there are any differences before and after 1999. Call the variable with observation before '99 y. Now we have that the average of y is 229.5. Call the variable with observations after 1999 x with an average of 74.89. What we now need is a t-test, but now for two different populations. Consider the observations y_1 to y_n and the observations x_1 to x_n, and their sample averages as follows, which are to be associated with the population means Mu_1 and Mu_2. Therefore, the Olympic Games data here N is equal to 10 and M is equal to nine. The question is now whether the null hypothesis holds against the alternative hypothesis. To test this, we can use the following expression. We now need the standard deviation of this difference. There are two options. One is that it can be assumed that the variance is Sigma_1 squared and Sigma_2 squared are the same. The other obviously is that they are not. The sample variances are S_1 squared and S_2 squared. In case it can be assumed that Sigma_1 squared is equal to Sigma_2 squared, the pooled standard deviation of the difference y bar minus x bar is like this. With N plus M sufficiently large, we have the following. When it should be assumed that Sigma_1 squared is unequal to Sigma_2 squared, then the test statistic reads like this. Back to the numbers on the Olympic Games. We already had the averages of y and x. Next, we can calculate that S_1 squared is 37000 and S_2 squared is 7000. When we pool the variances, we get, and then the t-test value becomes like this. The value of 2.196 is beyond the two standard deviations of a standard normal distribution. So we can conclude that the differences are significant indeed. Let us see if we would draw the same conclusion if we assume that the variances are different. We then get this. Again, this t-test value is significant. As an alternative strategy, suppose we would use the simple regression with a dummy variable that is zero for the first 10 observations and one for the last nine observations. That is, this expression. Then we get the t-test value for b is minus 2.197 which is again significant at the five percent level. So what we see here is that different methods all lead to the same conclusion. In conclusion, to answer the question yes, the costs are less overrun for more recent Olympic games. Now you may wonder whether the differences between the pre '99 and a post '99 games could be attributed to just the Montreal games in 1976. So what happens to the computations if you delete those summer games? Try yourselves.
