R. Exercise. There are If! parts to the following question. 1. R is not technically needed to solve this Problem Set since the questions may be solved analytically. However, make sure you are also able to do them in R in order to be ready for the rest of the course. In R, create a dataset {dataframe} with lll observations. Add the illowing elements and use them to answer the questions below. {a} Create a variable t containing numbers from 1 to IDD {sequentially}. What is the mean of t? Report your answer to the nearest one decimal place {for example: 10.7}. {b} Create a variable we will call Cr, which has value 3 for all 10!} observations. What is the variance of ca? Report your answer as an integer value. {o} Create another variable .5, , a random normal variable with mean I] and standard de- viation 1. Then create rt, a random uniform variable over [0,1]. Elsi] refers to the 'expectation', 'expected value" or 'mean' of rt.W'hat is E [m]? Report your answer rounded to the nearest one decimal point {for example 1'17}. {d} What is the variance of st? Hint: the variance of a random uniform variable over [(1, b] is equal to 11-2-[5 a}? Enter as a ratio of two integers [for example g}. {e} Create an outcome variable 1;; dened as y; = or +r +15; where ,6} = 2. What is E [3.1] '? Report your answer as an integer value. {f} Estimate 3. Suppose the standard CLS 95% condence interval of E is {1.61, 1'34} {Note that your condence interval will depend on the exact draws of E3, and so it might be slightly different}. 'True or False: We can reject the hypothesis that ,3 = 1 with 95% condence. i. True ii. False {g} Create 113 as a random normal variable with mean 1'] and standard deviation 1. Create a new variable 1 {replace the old variable q], a random normal variable with mean I] and standard deviation 1. Generate :33 as qt = 1'; + Er? +v3. What is E [qt]? Hint: E [X\"] of a random uniorm variable between D and 1 is n+4. Please report your answer as an integer value. (h) Generate outcome variable & defined as z = o + Art + 7qt + et, where B = 2, y = 3. What is E [z,]? Report your answer as an integer value. (i) Estimate / from the (misspecified) model: Assume that the standard OLS 95% confidence interval is (5.832,10.014). Which point estimate derives this confidence interval? Hint: The confidence intervals are symmetric around the point estimate. Use three places after the decimal point. (j) Now suppose that the point estimate is 7.8 and the standard error is 1.04. In this instance, what is the upper bound of the standard OLS 95% confidence interval (recall the critical value for a standard OLS 95% confidence interval is 1.96)? Use two places after the decimal point