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- For the following table of data. x 1 2 3 4 5 6 7 8 9 10 y 0 0.5 1 2 2.5 3 3 4 4.5 5 a. draw a scatterplot. b. calculate the correlation coefficient. c. calculate the least squares line and graph it on the scatterplot. d. predict the y value when x is 11.Two random variables X andY are related by the expression Y = aX +b, where ta' and 'b' are any real numbers.Is there a linear correlation between x and y? Use α = 0.01. a.) Yes, because the correlation coefficient is not in the critical region. b.) Yes, because the correlation coefficient is in the critical region. c.) No, because the correlation coefficient is in the critical region. d.) No, because the correlation coefficient is not in the critical region. C. What is correlation coefficient when the point (2,9) is excluded? r=_________ (Round to three decimal places as needed.) Is there a linear correlation between x and y? Use α= 0.01 a.) No, because the correlation coefficient is in the critical region. b.) Yes, because the correlation coefficient is not in the critical region. c.) Yes, because the correlation coefficient is in the critical region. d.) No, because the correlation coefficient is not in the critical region.
- Let I be an indicator variable with p(I) = {p , I=1 } {1-p , I=0}. Show that the variance of I is p(1-p)Q2: (b) Let X₁, X2, X3 be uncorrelated random variables, having the same variance o². Consider the linear transformations Y₁ = X₁ + X₂, X2 + X3. Find the correlations of Y₁, Y; i, Y₂ = X₁ + X3 and Y3 for i + j. =Consider the model Y = B1 + B2X + e, which we estimate using a random sample with 12 observations. Let bj and b2 be the estimators for B1 and B2 and recall Eê, where ê; = Y; – bị – b2X;. Suppose the sample correlation between {X; }"_-1 and {Y;}"_, is 0.5 and E(Y; – Ỹ„)² = 100. What is ổ?? %3D n-2 Hint: (i) for simple regression, the regression R? is equal to the squared sample correlation between X and Y. (ii) R² = 1 – SSE where SSE = Eế and SST SST = O a. 7.5 O b. 8.5 О с. 9 O d. 10 O e. 8 O f. 7 Clear my choice
- Let f(x, y) = x + y for 0 < x < 1 and 0 < y < 1 The Conditional Variance of Y when X = ; is(d) Does a linear relation exist between the speed at which a ball is hit and the distance the ball travels? The variables speed at which the ball is hit and the distance the ball travels are positively associated because r is positive and the absolute value of the correlation coefficient is greater than the critical value ( ) (Round to three decimal places as needed.)3. Let X be the nxp data matrix, define the deviation vector as d; = (x1j-Tj, ..., Xnj T;)'. Let 0 be the angle between the jth and Ith deviation vectors. Calculate the lengths of ||d;|| and cos(0). Relate these quantities to the sample covariance and sample correlation.
- Suppose that the continuous two-dimensional random variable (X, Y ) is uniformly distributed over the square whose vertices are (1, 0), (0, 1), (−1, 0), and (0, −1). Find the Correlation Coefficient ρxya) The simple correlation coefficients between iemperature (X,), corn yield (X,) and and rainfall (X,) are r, = 0.59, r, = 0.46 and r, = 0.77. Calculate partial correlation coefficient multıple correlation coefficient R, .The correlation between X and Y Select one or more: a. is the covariance squared b. cannot be negative since variances are always positive. c. is given by corr(X, Y) = cov(X,Y)/var(X)var(Y)cov(X,Y)/var(X)var(Y) d. can be calculated by dividing the covariance between X and Y by the product of the two standard deviations