Problem 4. Let A be an n x m matrix, and let 5 E lg". Recall that a vector 2' E R\" is said to be a solution to the system of linear equations A5 = b if A5 = b. (a) Let us consider the homogeneous system of linear equations A5 = (l. (i) Let 2,13 E R\" be solutions of the system of linear equations A5 = 5. Prove that 2' + 13 is also a solution of the system of linear equations Af = fl. (ii) Let 2' E R\" be a solution of the system of linear equations Azi' = fl", and let 1' E R be a scalar. Prove that r?! is also a solution of the system of linear equations A5 = (l (b) Now consider the non-homogeneous system of linear equations A5 = (7, where 3% q (i) Suppose that 2,13 E R\" are solutions of the system of linear equations A5? = b. Is it true that 2' + if is also a solution of the system of linear equations A5 = b? Test scores for a history class had a mean of 79 with a standard deviation of 4.5. Test scores for a physics class had a mean of 69 with a standard deviation of 3.7. Suppose a student gets 85 on the history test and 74 on the physics test. Calculate the z-score for each test and then decide which exam the student did relatively better and why. history z-score: 67.4, physics z-score: 55.3. Student did relatively better on the history since its z-score is higher than the physics z-score history z-score: -133, physics z-score: 1.35. Student did relatively better on the history since its z-score is higher than the physics z-score history z-score: -133, physics z-score: -1.35. Student did relatively better on the physics since its z-score is higher than the history z-score history z-score: 1.33, physics z-score: 1.35. Student did relatively better on the physics since its z-score is higher than the history z-score The grades below are from a random Statistics student. Each category has a percentage associated which represents how much of the category contributes toward the final grade. Find the student's final numerical average for the course. (Leave answer as a percentage and round to the nearest whole number) Category Percentage Exams: 78 409% Quizzes: 92 30% Labs: 70 20% Homework: 81 15% Extra Credit: 86 109%2 1. Standard Normal Distribution: Table for cumulative areas from the LEFT of common z-scores (Shading everything to LEFT] x-score -3 -2 -1 0 2 3 Area 0.001 0.023 0.159 0.500 0.841 0.977 0.999 For the following questions, sketch the normal curve with the proper shading and provide the area to 3 decimal places: What is the area to the LEFT of z-score(2)? What is the area to the LEFT of z-score(-3]? What is the area to the RIGHT of z-score(1)? What is the area to the RIGHT of z-score(-2)? What is the area in-between z-score (-2) and z-score(0)? What is the area in-between z-score(2) and z-score(3)? What is the area in-between z-score(-2) and z-score(1)?D Question 1 1 pts Suppose X1 has a Poisson distribution with mean /1 and X2 has a Poisson distribution with mean /42. Suppose X, and Xzare independent. True or False: X, + X2 also has a Poisson distribution. Note: the moment generating function of a Poisson random variable with mean /4 is Mx (t) = elite'-1) @ True False