You have commissioned a study to find out whether a smaller class size improves student performance in their first year of middle school.

Question 1 [20 points]

 You have commissioned a study to find out whether a smaller class size improves student performance in their first year of middle school. 

1.    [5 points]

If you could conduct an experiment to determine whether and how smaller class size affects student performance in their first year of middle school, what would you do? Explain the design of your experiment and motivate your strategy. 

2.    [7 points]

Suppose now that you cannot run an experiment, but you can collect data on 680 first- year middle school students in Los Angeles. For each student, you can observe the size of their class and a standardized test score taken at the end of their first year of middle school.

What type of data would have at your disposal (e.g., cross-section, time-series, panel)? Intuitively, what type of correlation would you expect between class size and test score? Would this correlation be informative about the causal effect of class size on student performance? Explain. 

3.    [8 points]

Using your data, you separate students into two groups, one with class size lower or equal to 15 students and one with class size greater than 15. The standardized test score ranges from 0 to 50. You compute the mean score for these two groups, which are reported in the table below:

Group                         # Students       Average Test Score       Std. Error

1 - Class Size ≤ 15        274                             26.70                     0.31

2 - Class Size > 15        406                             25.30                     0.21

Assume that the sampling method for each of these two groups is random sampling and that the two samples are independent.

Indicate with S1 and S2 the test score of students in group 1 (class size of 15 or less) and 2 (class size larger than 15), respectively.

Assume that E (S1) = µ1 and V ar (S1) = σ2 and that E (S2) = µ2 and V ar (S2) = σ2.

Under these assumptions, what is the expected value of (S2 − S1)? How would you esti- mate it? Provide an estimate of this using the statistics reported in the table above. What is the variance of (S2 − S1)? Provide an estimate of this using the statistics reported in the table above (Hint:  the standard error of the sample mean y¯ is equal to the standard deviation of y divided the square root of the sample size. It is useful to read this brief note http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1255808/).

 Consider the following bivariate linear regression model 

yi = β0 + β1xi1 + ui     where the subscript i indicates individual i.  Assume that you have a random sample of N individuals for whom you observe y and x1. Also assume

1.    [7 points]

Starting from equation (2) and using the Law of Iterated Expectations, show that Cov(u, x1) = 0.

3.    [20 points]

Consider the data set “beauty.dta” available on Blackboard in the folder Assignment

1.     The two main variables of interest are the log-wage of each individual (lwage) and a measures of physical attractiveness (looks).  Each individual in the sample was ranked by an interviewer for physical attractiveness, using five categories (1=homely, 2=quite plain, 3=average, 4=good looking, and 5=strikingly beautiful or handsome).

Using Stata, compute the fraction of the sample in each of the physical attractiveness categories. What is the mode of the distribution?

You notice that there are very few people at the two extremes (looks = 1 and looks = 5). So, generate a new categorical variable that puts individuals into one of three groups: below average (looks = 1 or looks = 2), average (looks = 3), and above average (looks = 4 and looks = 5). Create this new variable using the command recode and label its values appropriately. Now, tabulate the new categorical variable for men and women separately. Is there any noticeable difference between the two distributions?

Run a regression of lwage on a constant and looks separately for men and women. Using equation above and the regression results, find out what the average log-wage is for men and women. Make sure you get the same answer when computing average log-wage for men and women using the command summarize.

Referring to the regression results, what type of correlation do you find between physical attractiveness and log-wage? Is it different between men and women? Would you say that the results represent evidence that physical attractiveness matters for how much people are paid? Explain briefly.