MS&E 226 \"Small\" Data Problem Set 1 Due: October 2, 2017, 5:00 PM (electronic submission on Gradescope) NOTE: For any code that you write, please submit both your code and the (relevant) output produced along with your write-up. Problem 1. Our grading policy is described in detail in the syllabus. Some key points: Each part of each problem will be graded as follows: - You will receive zero points if you do not attempt it. - If you attempt it, but there are either substantial methodological errors or major conceptual misunderstandings of the material, you will receive 2 points. - If you attempt it and there are no substantial methodological errors or major conceptual misunderstandings of the material, you will receive 3 points. You can discuss the assignments among yourselves, but everybody must turn in his/her own written solutions in his/her own words. If you do a substantial subset of the work on your problem set with others, document on each assignment the other students that you worked with. As noted in the syllabus, problem sets and project submissions are submitted and graded through Gradescope. To ensure this process is smooth, there are a few things to keep in mind: - You are required to tag your answers correctly. The graders will ignore any part of your solution that is not tagged. Note that this means you also have to correctly tag your code. Allow enough time prior to submission to ensure you are able to tag correctly. - In order to grade code you submitted, we need to be able to copy your code. Make sure this is possible (e.g., do not upload screenshot images of your code). We will deduct points if we cannot check your code. - If you believe we have made a mistake grading your work, you should submit a regrade request through Gradescope. This sends your request directly to the grader on that particular question. You have to submit your regrade request within 14 days of the grades of that particular problem set or project part being published. - If you decide to do the mini-project in a team of two (see below), only one team member should submit the project to Gradescope and then add the other team member to the submission. Copy \"I have read and understood the grading policy\" to get credit for this problem. Problem 2. The blue M&M candy was introduced in 1995. Before then, the color mix in a bag of plain M&Ms was as follows: 30% brown, 20% yellow, 20% red, 10% green, 10% orange, 10% tan. After the change, it was 24% blue , 20% green, 16% orange, 14% yellow, 13% red, 13% brown. A friend of mine has two bags of M&Ms, and he tells me that one is from 1994 and one from 1996. He won't tell me which is which, but he gives me one M&M drawn uniformly at random from each bag. One is yellow and one is green. What is the probability that the yellow M&M came from the 1994 bag? Problem 3. Let A, B, C be n n matrices. Show that their product is associative, that is: (AB)C = A(BC). Now let x 6= 0 be a n 1 column vector. Let D= xxT . xT x Use the above associativity property to compute D2 . Problem 4. Before getting started, work your way through chapters 1-4 of R for Data Science (R4DS), which you can find at http://r4ds.had.co.nz/. Over the course of the quarter, the remaining chapters will prove useful as well, so feel free to work your way through the entire book at your own pace, or use it as a reference guide. Obtain the electric.company dataset from the following location: http://www.stat.columbia.edu/gelman/arm/examples/ARM_Data.zip This data is from an educational experiment performed around 1970 on a set of elementary school classes. The treatment in this experiment was exposure to a new educational television show called The Electric Company. In each of four grades, the classes were split into treated and control groups. At the end of the school year, students in all the classes were given a reading test, and the average test score within each class was recorded. In every class, the treatment group was provided with one of the two options (the column named Supplement. in the data): Supplement (S) or Replacement (R). Supplement (S) meant that the Electric Company TV show was supplemented with what the students were taught in class for the treatment group. Replacement (R) meant that the Electric Company TV show was prescribed as a replacement to what students were taught in the class for the treatment group. Do the following: a) Plot treated.Posttest vs. control.Posttest for Supplement and Replacement cases for all grades and cities. Furthermore, include the line y = x in both plots. Color the points by grade and use a different marker for each city. Make sure the plot has a title and the legend has proper labels. That is, create one figure with two subplots using the techniques discussed in R4DS. b) Find the sample average of treated.Pretest - control.Pretest and treated.Posttest - control.Posttest for both Supplement and Replacement cases. (Hint: Though not required, sections 5.5 and 5.6 of R4DS might be useful for this.) c) What observations can you make? Can you argue whether the Replacement (R) strategy was better or worse on average than the Supplement (S) strategy according to this data