Quiz 5

The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app.

Show all work unless instructed otherwise.

The joint and marginal distributions of X and Y are given below. Find \Cov(X, Y) and \rho_{X, Y}.

Joint Distribution Table Y = 1 Y = 2 Y = 3 X = 0 0.1 0.15 0.1 X = 1 0.15 0.2 0.3
Distribution Table for <m>X</m> x 0 1 \Pr(X = x) 0.35 0.65
Distribution Table for <m>Y</m> y 1 2 2 \Pr(Y = y) 0.25 0.35 0.4

\Cov(X, Y) = \E(XY) - \E(X)\E(Y). We have: \E(X) \amp = 0(0.35) + 1(0.65) = 0.65 \E(Y) \amp = 1(0.25) + 2(0.35) + 3(0.4) = 2.15 \E(XY) \amp = (0)(1)(0.1) + (0)(2)(0.15) + (0)(3)(0.1) + \amp \quad\, (1)(1)(0.15) + (1)(2)(0.2) + (1)(3)(0.3) = 1.45 \\ \Cov(X, Y) \amp = 1.45 - (0.65)(2.15) = \boxed{0.0525} Since X is an indicator random variable, we have \Var(X) = (0.65)(0.35) = 0.2275. For \Var(Y): \E(Y^2) \amp = 1^2(0.25) + 2^2(0.35) + 3^2(0.4) = 5.25 \Var(Y) \amp = \E(Y^2) - (\E(Y))^2 = 5.25 - (2.15)^2 = 0.6275 So the correlation is: \rho_{X, Y} \amp = \frac{\Cov(X, Y)}{\sqrt{\Var(X)\Var(Y)}} = \frac{0.0525}{\sqrt{(0.2275)(0.6275)}} \approx \boxed{0.139}.

A sample of 30 measurements are taken and a best fit line is calculated, resulting in the data below (the final row of the table shows the sums for each column). Find the RSS, SST, and coefficient of determination.

Sample Data x_i y_i best fit predicted y_i res^2 (y - \textrm{avg }y)^2 1.94 3.31 10.53 52.13 886.55 2.63 10.38 17.37 48.76 515.34 \vdots \vdots \vdots \vdots \vdots 3.94 19.47 30.31 117.35 185.24 6.35 53.42 54.22 0.64 413.51 sum: 992.55 992.55 953.03 38397.35

The RSS is the sum of res^2, 953.03, and the SST is the sum of (y - \textrm{avg})^2, 38397.35. Then the coefficient of determination is: r^2 = 1 - \frac{\text{RSS}}{\text{SST}} = 1 - \frac{953.03}{38397.35} \approx \boxed{0.975}