Spring2022Exam2Solution

NAME: PUID: 3 Question 1: Confidence intervals [18 points] A chemist is interested in assessing the amount of calcium carbonate produced at the conclusion of a particular chemical reaction. She performs 100 reactions and records an average yield of 50 grams with a standard deviation of 5 grams. [ Note: Use the provided z-table for this problem.] (a) [9 points] Using the table above, calculate the 80% two-sided confidence interval for the average calcium carbonate yield (i.e., calculate the upper and lower bounds of the 80% confidence interval). ?𝐼 80 = (?̅ − (? 80 ⋅ ? √ ? ) , ?̅ + (? 80 ⋅ ? √ ? )) ?𝐼 80 = (50 − (1.28 ⋅ 5 √100 ) , 50 + (1.28 ⋅ 5 √100 )) ?𝐼 80 = (50 − (1.28 ⋅ 1 2 ) , 50 + (1.28 ⋅ 1 2 )) ?𝐼 80 = (50 − 0.64, 50 + 0.64) ?𝐼 80 = (49.36, 50.64)

NAME: PUID: 6 Question 3: Linear regression (Regularization, and cross validation) [16 points] (a) [6 points] The following plots illustrate different regression models fitted to the same data set. Label each plot as either overfitting, proper fitting, or underfitting the data. (b) [4 points] Suppose we train the model y ̂(?) = −12? 1 2 + 8? 1 + 25? 2 − 19 using linear regression without regularization from a data set. Which of the following models are a possible outcome if we run the regression algorithm with regularization over the same data set? Select all that apply. a. ? ̂(?) = −1.5? 1 + 0.4? 2 + 10 b. ? ̂(?) = −5? 1 2 + 8? 1 + 7? 2 − 11 c. ? ̂(?) = 20? 1 2 + 40.5? 1 − 28? 2 + 22.6 d. ? ̂(?) = −100? 1 2 − 9? 1 − 34? 2 + 42 Proper fitting Overfitting Under fitting

NAME: PUID: 9 (c) [4 points] Complete the following word-doc matrix (co-occurrence matrix) based on (b) with word-based 1-gram. dog red hit heavy doc1 1 1 0 0 doc2 2 0 1 0 doc3 1 0 0 1 (d) Use the following word-doc matrix to answer the following questions. blinder machine learning block chain money Document A 1 1 2 0 0 0 Document B 5 5 5 10 0 0 Document C 0 3 0 0 2 3 Document D 10 10 2 0 4 0 i. [2 points] In which document is the term frequency score for the word “learning” the highest? ii. [2 points] Which words have the highest inverse document frequency score? Document A “block” and “money”

NAME: PUID: 12 (b) [4 points] Given the following means and standard deviations, form the normalized feature matrix 𝑋 ????𝑎?𝑖𝑧?? . 𝑀??? 𝑥 = 5, 𝑆??????? ??𝑣𝑖??𝑖?? 𝑥 = 5 𝑀??? 𝑥 2 = 300, 𝑆??????? ??𝑣𝑖??𝑖?? 𝑥 2 = 100 (c) [3 points] Using the normalized feature matrix 𝑋 ????𝑎?𝑖𝑧?? , write down the least squares equation for finding the optimal parameter vector 𝛽 = [? 1 , ? 2 ,?] 𝑇 , without solving. All components of the equation except the parameter vector should be in numerical form. 𝑋 ????𝑎?𝑖𝑧?? = [ −2 1 1 1 3 1 6 5 1 ] 𝛽 = ([ −2 1 6 1 3 5 1 1 1 ][ −2 1 1 1 3 1 6 5 1 ]) −1 [ −2 1 6 1 3 5 1 1 1 ] ( 1020 1040 1010 ) OR ([ −2 1 6 1 3 5 1 1 1 ][ −2 1 1 1 3 1 6 5 1 ]) 𝛽 = [ −2 1 6 1 3 5 1 1 1 ] ( 1020 1040 1010 )

NAME: PUID: 15 Question 6: Objects and classes [15 points] Write a class called Student so that the following piece of code can be used to calculate the letter grades of each student and the class average. Assume that names and exam1 , exam2 and final are already defined previously as a list of names of students and lists of their grades for exam 1, exam 2 and the final, respectively for the students enrolled in Cobra for Data Science class. All exams are out of 100 points. Suppose students pass or fail the class with threshold of 40 (i.e., students need a score of 40 or more to pass). Assume each exam is equally weighted and there are no other assignments in the class. NOTE: Make sure that no new instance or class variables are introduced by any of the lines below , i.e., all the necessary class and instance variables should be predefined to receive full credit. In addition, note that calc_point_grade() should return the average score of exam1 , exam2 and final , whereas calc_letter_grade() should retur n either ‘Pass’ or ‘Fail’ depending on whether the score was greater than or equal to 40 or less than 40, respectively. roster = [Student(i) for i in names] for i,j in enumerate(roster): j.exam1 = exam1[i] j.exam2 = exam2[i] j.final = final[i] for i in roster: i.calc_point_grade() i.calc_letter_grade() grade_total = 0 for i in roster: grade_total = i + grade_total Student.class_avg = grade_total / len(roster) [Write your solution in the box on the next page.]