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Explanation of Solution
The formulas for sum of squares are as follows:
The total sum of squares is calculated is as follows:
Consider
Thus, the required TSS is as follows:
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Chapter 13 Solutions
Mathematical Statistics with Applications
- Based on a sample on n observations, (x1, y1 ), (x2, y2 ), c, (xn, yn), the sample regression of y on x is calculated. Show that the sample regression line passes through the point (x = x̄, y = ȳ), where x̄ and ȳ are the sample means.arrow_forwardIn a typical multiple linear regression model where x1 and x2 are non-random regressors, the expected value of the response variable y given x1 and x2 is denoted by E(y | 2,, X2). Build a multiple linear regression model for E (y | *,, *2) such that the value of E(y | x1, X2) may change as the value of x2 changes but the change in the value of E(y | X1, X2) may differ in the value of x1 . How can such a potential difference be tested and estimated statistically?arrow_forwardFind the least-squares regression line ŷ = bo + bjx through the points (-2, 0), (3, 7), (5, 14), (9, 19), (12, 26), and then use it to find point estimates ŷ corresponding to x = 4 and x = 10. For x = 4, y = For x = 10, y =arrow_forward
- ox² = 3, oy² = 5, oxy = 2, Z = 2Y - 4X – 2 a. Determine the variance of Z.arrow_forwardFind the least-squares regression line ŷ = bo + b1r through the points %3D (-1,2), (2, 6), (5, 13), (7, 20), (10, 23), and then use it to find point estimates y corresponding to x = 3 and x = 6. For = 3, y = %3D For I = 6, y = %3Darrow_forwardN(0, o?) is For the simple linear model Y = a + BX + €, where the error variable e ~ independent of X, use the law of total variance to show that Var(Y) = 3² Var(X)+o².arrow_forward
- Use a table to obtain the formula for the best least-squares fit to the data following data points: (1,2) (2, 3) (3,7) (4,9) (5, 12) Results from your Table ● Σα ● X = Συ · Σxy Σα2 Regression Line •y= - -arrow_forwardDo fast i will give like soonarrow_forwardConsider the linear model y=B,+B,x,+B,x,,+B,x+B,+u, You estimate the model y =B,+B,x,+B,x, observations and obtain the OLS residuals . You then estimate the auxiliary regression +u based on 123 31 3" 3i 41 The LM statistic +1 4/ you obtain to test the null hypothesis that H:B,=B,=0 is 20.91. What is the R2 of the auxiliary regression? It is not possible to say O 0.17 O 0.175714 0.177203arrow_forward
- Consider a simple linear regression model Yi = Bo + B1xi + Ei, i= 1,2, 3 with x; = i/3 for i = 1, 2, 3. Assume that %3| E1 1 -1 0 E = E2 ~ N -1 E3. 3 What is the smallest variance for an unbiased estimate of B1?arrow_forwardThe least-squares regression line relating two statistical variables is given as = 24 + 5x. Compute the residual if the actual (observed) value for y is 38 when x is 2. 4 38 2arrow_forwardCompute Var(ys) for the following model, where e, - wn(0, 0.01), i.e., a white noise process with mean zero and variance 0.01. Yt = 1+0.5y-1 + et, yo = 1. Please give the exact answer.arrow_forward
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