Question 2: (28 points) The folowing table gives a sample of mathematics achievement test scores and final calculus grades for ten independently selected college freshmen. Let X be the variable representing mathematics achievement test score and Y be the variable representing final calculus grade. Student 2 3 5 6 8 9 10 Mathematics Achievement Test Score (X) 39 43 21 64 57 28 75 34 52 Final Calculus Grade (Y) 65 78 52 82 92 89 73 98 56 75 a. (5) Use R to draw the scatter plot between X and Y. The axes should be labeled properly. The points should have filled circle shape and colored darkgreen. Add a black line of best fit into the plot. b. (1) Find s', the sample variance of X. c. (1) Find sy, the sample variance of Y. d. (4) Find and interpret sxy, the sample covariance between X and Y. e. (4) Find and interpret rxy, the sample correlation coefficient between X and Y. f. (4) Assume X and Y each follows a normal distribution. Test if the mathematics achievement test core and the final calculus grade are independent at 5% level of significance. Make sure to include the four steps. g. (2) If X and Y does not follow a normal distribution, is the test in part f valid? Why? h. (2) Calculate the slope , of the line of best fit between X and Y. Save it in an object called beta1. Print out the value of beta1. i. (3) Let Z = Y - B,X. Use R to draw a scatter plot between X and Z. The axes should be labeled properly. The points should have filled circle shape and colored darkgreen. Add a black line of best fit of into the plot. j. (2) Calculate the sample correlation coefficient between X and Z. What relationship you find among X, Y and Z