Homework Answers
The general formula for method of least squares for linear regression is used after we take the equation to linear form.
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experiment and records the following observations: 2. An engineer conducts an 3 2 1 X 9.3...
experiment and records the following observations: 2. An engineer conducts an 3 2 1 X 9.3...
experiment and records the following observations: 2. An engineer conducts an 3 2 1 X 9.3 1.2 3.6 y model of the form, y = ax, find the optimal least squares Assuming that the recorded data obeys estimates of "a and b". Also, using the above estimated values of "a and b," find a predicted value of y for x 5. a [Note: You need not check the Hessian Matrix for sufficiency condition]. Hint: First convert the given model to...
2, Section 4 #3: In the context of the normal simple linear regression model Y | X =エ=エ~ N(9.3 ...
2, Section 4 #3: In the context of the normal simple linear regression model Y | X =エ=エ~ N(9.3 + 1.5, 16). Let Y, Y2 be independent observations corresponding to X-20 and X-25, respectively. (a) Find the 95 quantile of Yi (b) Find the probability that ½ > Y. (Hint: Use the fact that if Z1 ~ N(P1써) and
2, Section 4 #3: In the context of the normal simple linear regression model Y | X =エ=エ~ N(9.3 + 1.5,...
2.3*) Graph the following observations of x and y on graph paper. X 12 3 4...
2.3*) Graph the following observations of x and y on graph paper. X 12 3 4 5 6 la 10 8 55 23 bole (a) Using a ruler, draw a line that fits through the data. Measure the slope and intercept of the line you have drawn. (b) Use formulas (2.7) and (2.8) to compute, using only a hand calculator, the least squares estimates of the slope and the intercept. Plot this line on your graph. (c) Obtain the sample...
3. (20 pts) Suppose that we have 4 observations for 3 variables y , x\, X2...
3. (20 pts) Suppose that we have 4 observations for 3 variables y , x\, X2 and consider a problem of regressing y on two (qualitative) variables x\, xz. Data y (Income) x (Gender) X2 (Management Status) obs no. Female None 2 Male None 3 Female Yes 4 Male Yes Y4 To handle the qualitative variables x\, x2, we define dummy variables z1, 22 as Male for for 1, 1, T2= Yes Z1= Z2= -1 for for 1 1 =...
2. Suppose we are given data on n observations (x,Y), i 1,... , n, and we...
2. Suppose we are given data on n observations (x,Y), i 1,... , n, and we have a linear model, = SXY/SXX and A,-ㄚ-Ax be the least-square estimates so that E(X) = β0 +ATp Let given in lecture. (a) Show that E(5xx)-A5xx and E(Y)-Ao +A2. (b) Use (a) to show that E(A)-A and E(A)-A. În other words, these are unbiased estimators (c) The fitted values Yi = ArtAz; are used as estimates of E(K), and the residuals ei = Y-...
7. Suppose the data consist of repeated observations (y;it, X), t = 1, -.. ,T, for each in- dividual i = 1,... ,n. Here...
7. Suppose the data consist of repeated observations (y;it, X), t = 1, -.. ,T, for each in- dividual i = 1,... ,n. Here yit is the response and xt is a covariate vector. A linear mixed-effects model for analysing the population-averaged and subject-specific effects of Xit is of the following form Z;B + W;b; + €;, yi = where y (yi1;* . ,ViT)T; Z; is a T x p design matrix built from {xji} for the fixed effects B;...
Suppose the data consist of repeated observations (yt,X), t = 1,-.. ,T, for each in- dividual...
Suppose the data consist of repeated observations (yt,X), t = 1,-.. ,T, for each in- dividual i 1,.. , n. Here yit is the response and xjt is a covariate vector. A linear mixed-effects model for analysing the population-averaged and subject-specific effects of Xit is of the following form ZiBWibi Ei, Уi (yil ,yiT)T; Zi is a T x p design matrix built from {xjt} for the fixed are i.i.d. MVN(0, Q) > 0 where y effects B; Wi is...
2. Suppose we are given data on n observations (zi,Y), i = 1, , n, and...
2. Suppose we are given data on n observations (zi,Y), i = 1, , n, and we have a linear model, so that E(X) = β0+Axi. Let ßi-SXY/SXX and β0 = Y-Ax be the least-square estimates given in lecture. (a) Show that E(5xx) = ẢSXX and E(T) = β0+A2. (b) Use (a) to show that E(A) = and E(%) = A- In other words, these are unbiased estimators (c) The fitted values Y BotBr, are used as estimates of E(Y),...
3. (20 pts) Suppose that we have 4 observations for 3 variables y,I, 2 and consider...
3. (20 pts) Suppose that we have 4 observations for 3 variables y,I, 2 and consider a problem of regressing y on two (qualitative) variables r, 2. Data: 22 obs no. y (Income) 2 (Management Status) I (Gender) 1 None Female 2 None Male Yes Female Yes Male 4 To handle the qualitative variables r, 12, we define dummy variables 1, 22 as for 1, 22= Yes Male for 1, 219 22 -1. for 22= None for 1= Female -1,...
2. (20 pts) The following price demand for three goods x, y and z are given:...
2. (20 pts) The following price demand for three goods x, y and z are given: Px = 130 - 3x-y-2z Py = 200 - x +4y P, = 150 - y-3z The total cost is given by the following function: TC = x2 + xy + y2 + yz + z2 (2 pts) Establish the profit function (3 pts) Use the F.O.C. in the profit function with respect to each of the three variables (12 pts) Now that you...
experiment and records the following observations: 2. An engineer conducts an 3 2 1 X 9.3 1.2 3.6 y model of the form, y = ax, find the optimal least squares Assuming that the recorded data obeys estimates of "a and b". Also, using the above estimated values of "a and b," find a predicted value of y for x 5. a [Note: You need not check the Hessian Matrix for sufficiency condition]. Hint: First convert the given model to...
2, Section 4 #3: In the context of the normal simple linear regression model Y | X =エ=エ~ N(9.3 + 1.5, 16). Let Y, Y2 be independent observations corresponding to X-20 and X-25, respectively. (a) Find the 95 quantile of Yi (b) Find the probability that ½ > Y. (Hint: Use the fact that if Z1 ~ N(P1써) and
2, Section 4 #3: In the context of the normal simple linear regression model Y | X =エ=エ~ N(9.3 + 1.5,...
2.3*) Graph the following observations of x and y on graph paper. X 12 3 4 5 6 la 10 8 55 23 bole (a) Using a ruler, draw a line that fits through the data. Measure the slope and intercept of the line you have drawn. (b) Use formulas (2.7) and (2.8) to compute, using only a hand calculator, the least squares estimates of the slope and the intercept. Plot this line on your graph. (c) Obtain the sample...
3. (20 pts) Suppose that we have 4 observations for 3 variables y , x\, X2 and consider a problem of regressing y on two (qualitative) variables x\, xz. Data y (Income) x (Gender) X2 (Management Status) obs no. Female None 2 Male None 3 Female Yes 4 Male Yes Y4 To handle the qualitative variables x\, x2, we define dummy variables z1, 22 as Male for for 1, 1, T2= Yes Z1= Z2= -1 for for 1 1 =...
2. Suppose we are given data on n observations (x,Y), i 1,... , n, and we have a linear model, = SXY/SXX and A,-ㄚ-Ax be the least-square estimates so that E(X) = β0 +ATp Let given in lecture. (a) Show that E(5xx)-A5xx and E(Y)-Ao +A2. (b) Use (a) to show that E(A)-A and E(A)-A. În other words, these are unbiased estimators (c) The fitted values Yi = ArtAz; are used as estimates of E(K), and the residuals ei = Y-...
7. Suppose the data consist of repeated observations (y;it, X), t = 1, -.. ,T, for each in- dividual i = 1,... ,n. Here yit is the response and xt is a covariate vector. A linear mixed-effects model for analysing the population-averaged and subject-specific effects of Xit is of the following form Z;B + W;b; + €;, yi = where y (yi1;* . ,ViT)T; Z; is a T x p design matrix built from {xji} for the fixed effects B;...
Suppose the data consist of repeated observations (yt,X), t = 1,-.. ,T, for each in- dividual i 1,.. , n. Here yit is the response and xjt is a covariate vector. A linear mixed-effects model for analysing the population-averaged and subject-specific effects of Xit is of the following form ZiBWibi Ei, Уi (yil ,yiT)T; Zi is a T x p design matrix built from {xjt} for the fixed are i.i.d. MVN(0, Q) > 0 where y effects B; Wi is...
2. Suppose we are given data on n observations (zi,Y), i = 1, , n, and we have a linear model, so that E(X) = β0+Axi. Let ßi-SXY/SXX and β0 = Y-Ax be the least-square estimates given in lecture. (a) Show that E(5xx) = ẢSXX and E(T) = β0+A2. (b) Use (a) to show that E(A) = and E(%) = A- In other words, these are unbiased estimators (c) The fitted values Y BotBr, are used as estimates of E(Y),...
3. (20 pts) Suppose that we have 4 observations for 3 variables y,I, 2 and consider a problem of regressing y on two (qualitative) variables r, 2. Data: 22 obs no. y (Income) 2 (Management Status) I (Gender) 1 None Female 2 None Male Yes Female Yes Male 4 To handle the qualitative variables r, 12, we define dummy variables 1, 22 as for 1, 22= Yes Male for 1, 219 22 -1. for 22= None for 1= Female -1,...
2. (20 pts) The following price demand for three goods x, y and z are given: Px = 130 - 3x-y-2z Py = 200 - x +4y P, = 150 - y-3z The total cost is given by the following function: TC = x2 + xy + y2 + yz + z2 (2 pts) Establish the profit function (3 pts) Use the F.O.C. in the profit function with respect to each of the three variables (12 pts) Now that you...
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