Fluent Essays

Please keep the statistics assignment by September 20th 2021 at 5:00 PM ## NAME: ## NetID: ## ## Instructions: ## ## Complete the following R program by ## filling in lines labled __FILL IN___ ## ## Each ___FILL IN____ should be only one line but you ## may add additional lines of code if you think it is ## necessary. ## n_sim = 3000 my_viewsA = 60 my_viewsB = 31 my_viewsC = 41 all_views = my_viewsA + my_viewsB + my_viewsC n_impsA = 160 n_impsB = 120 n_impsC = 150 all_imps = n_impsA + n_impsB + n_impsC my_CTRs = c(my_viewsA/n_impsA, my_viewsB/n_impsB, my_viewsC/n_impsC) my_deviation = max(my_CTRs) - mean(my_CTRs) all_results = rep(0, n_sim) for (cur_sim in 1:n_sim){ pool = c(rep(1, all_views), rep(0, all_imps - all_views)) impsA = sample(pool, n_impsA, replace=FALSE) viewsA = sum(impsA) pool_BC = ____FILL IN______ impsB = ___FILL IN_____ viewsB = ____FILL IN_____ viewsC = ___FILL IN____ CTRs = c(viewsA/n_impsA, viewsB/n_impsB, viewsC/n_impsC) deviation = ____FILL IN ___ all_results[cur_sim] = ___FILL IN_____ } print("Simulated p-value:") print( ___FILL IN____ ) ## NAME: ## NetID: ## ## Instructions: ## ## Complete the following R program by ## filling in lines labled __FILL IN___ ## ## Each ___FILL IN____ should be only one line but you ## may add additional lines of code if you think it is ## necessary. ## my_wins = 4 my_ties = 2 my_loss = 1 n_games = my_wins + my_ties + my_loss my_point = 1*my_wins + 0*my_ties + (-1)*my_loss n_sim = 2000 all_results = rep(0, n_sim) for (cur_sim in 1:n_sim){ outcome = sample(c(-1, 0, 1), n_games, replace=TRUE) all_results[cur_sim] = ____FILL IN_____ } print("Simulated p-value:") print( ___FILL IN____ ) 16:960:486 Fall 2021 Homework 1 Due: Wednesday, Sept 22nd, Midnight. Instructions: For the written components, write or type your answers clearly on a piece of paper, take a picture/scan, and submit it on Canvas. Please make sure your photo/scan is read-able. You may also write your answer on a tablet and submit a pdf file. For the code components, download the relevant R files and follow the instructions to complete the code. You completed code should run without error and display an output at the end. 1 Problem 1: matrix algebra 1.1 Part a (written) We warm up with some matrix multiplications. Let Y =   y11 y12y21 y22 y31 y32   and X =   x11 x12x21 x22 x31 x32  . First, write down Y >, that is, the transpose of the first matrix Y Next, calculate the following matrix multiplication: Y X> =   y11 y12y21 y22 y31 y32  ( x11 x21 x31 x12 x22 x32 ) =?? 1.2 Part b (written) Let u = (u1, u2, . . . , un) and v = (v1, v2, . . . , vm) be two vectors (vectors are columns by default). Then, the outer product of u and v is a n × m matrix of the form uv> =   u1 u2 . . . un   (v1, v2, . . . , vm) =   u1v1 u1v2 . . . u1vm u2v1 u2v2 . . . u2vm . . . unv1 unv2 . . . unvm   , where we view u as an n × 1 matrix and v> as a 1 × m matrix. 1 Now, take the Y and X matrices from part 1 and calculate the following sums of outer products of the columns of X and Y : Y∗1X > ∗1 + Y∗2X > ∗2 =?? Compare your answer with part a. 2 Problem 2 You are playing rock, paper, scissor with a friend. Your friend claims that they have a secret strategy and, in the 7 games that you played together, they won 4 games, tied 2 games, and lost 1 game. You want to test the null hypothesis that they are playing at random – in which case each game results in a win, tie, or loss with 1/3 probability. 2.1 Part a (code) You use the following test statistic: you give 1 point for each win, 0 point for each tie, and -1 point for each loss. Download the file hw1 p2a.R and follow the instructions within. Include the completed file as part of your submission. 2.2 Part b (code) You believe that a win is much more impressive than a tie and so you modify the test statistic: you give 2 points for each win, 0 points for each tie, and -1 point for each loss. Copy your completed hw1 p2a.R into a new file called hw1 p2b.R and modify the code to implement the new test statistic. Include hw1 p2b.R in your submission. 3 Problem 3 Consider an extension of A/B test where you compare 3 different video titles. You observe the following: 1. Title A was shown 160 times (160 impressions) and received 60 clicks/views. (observed CTR = 0.375) 2. Title B was shown 120 times and received 31 clicks/views. (observed CTR = 0.26) 3. Title C was shown 150 times and received 41 clicks/views. (observed CTR = 0.27) You believe title A is the best title but you want to make sure that this result is not due to chance. You test the null hypothesis that all three titles have the same underlying CTR. 2 You simulate the null hypothesis by randomly permuting all the impressions, which is equivalent to generating a new set of impressions for title A by drawing without replacement, then generating a new set of impressions for title B by drawing without replacement from the remaining impressions, and then giving the rest to title C. You use the test statistic of (max observed CTR) − (mean observed CTR). 3.1 Part a (code) Download the file hw1 p3a.R and follow the instructions within. Include the completed file as part of your submission. 3.2 Part b (written) Now suppose title A has 1600 impressions, title B has 1200 impressions, and title C has 1500 impressions but the clicks/views are still 60, 31, and 41 respectively. Compare the new p-value with that of part a – is it larger, smaller, or about the same? Why is this the case? Give an intuitive justification. 3