Consider the following observed values: (-5,-2) (-3,1) (0,4) (2,6) (1,3) (1) find the estimated regression line...
Consider the following observed values: (-5,-2) (-3,1) (0,4) (2,6) (1,3)
(1) find the estimated regression line based on the observed data
(2) for each xi, compute fitted value of yi
(3) compute the residuals ei
(4) calculate the coefficient of determination
Homework Answers
1)The regression line is given as y=3.4+x
It is obtained by the normal equations from the method of least squares
2)Predicted value of y is obtained by substituting the values of x in the equation given in 1.
3)Residual ,ei= actual y - predicted y
4) Coefficient of
determination,R2= MSS/TSS = 95.97/105 = 0.914
Add Answer to:
Consider the following observed values: (-5,-2) (-3,1) (0,4)
(2,6) (1,3)
(1) find the estimated regression line...
mealenk RI TAT Problem 2: ON Consider the tollowing observed values of (Ti, Yi): (2, –1)...
mealenk RI TAT Problem 2: ON Consider the tollowing observed values of (Ti, Yi): (2, –1) (0, 3), (1, 2), (-1,6), a. Find the estimated regression line ŷ = Bo + Bịa, based on the observed data. b. For each xi, compute the fitted value of 3Y; using c. Compute the residuals, e; = Yi-;- d. Find R-squared (the coefficient of determination). (a) (b) (c)
2. Consider a simple linear regression i ion model for a response variable Y, a single...
2. Consider a simple linear regression i ion model for a response variable Y, a single predictor variable ,i1.., n, and having Gaussian (i.e. normally distributed) errors: This model is often called "regression through the origin" since E(X) = 0 if xi = 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function Hint: The function g(x)log(x) +1-x...
5) Consider the simple linear regression model N(0, o2) i = 1,...,n Let g be the...
5) Consider the simple linear regression model N(0, o2) i = 1,...,n Let g be the mean of the yi, and let â and ß be the MLES of a and B, respectively. Let yi = â-+ Bxi be the fitted values, and let e; = yi -yi be the residuals a) What is Cov(j, B) b) What is Cov(â, ß) c) Show that 1 ei = 0 d) Show that _1 x;e; = 0 e) Show that 1iei =...
Consider the least-squares residuals ei-yi-yi, 1, 2, . . . , linear regression model. Find the...
Consider the least-squares residuals ei-yi-yi, 1, 2, . . . , linear regression model. Find the variance of the residuals Var(e). Is the vari- ance of the residuals a constant? Discuss. n,from the simple
4. We have n statistical units. For unit i, we have (xi; yi), for i-1,2,... ,n. We used the least squares line to obtain the estimated regression line у = bo +biz. (a) Show that the centroid (x, y) i...
4. We have n statistical units. For unit i, we have (xi; yi), for i-1,2,... ,n. We used the least squares line to obtain the estimated regression line у = bo +biz. (a) Show that the centroid (x, y) is a point on the least squares line, where x = (1/n) and у = (1/n) Σ¡ı yi. (Hint: E ) i-1 valuate the line at x = x. (b) In the suggested exercises, we showed that e,-0 and e-0, where...
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable...
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable ri, i-1,... , n, and having Gaussian (i.e. normally distributed) errors Ý,-BzitEj, Ejį.i.d. N(0, σ2) This model is often called "regression through the origin" since E(Yi) 0 if xi 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function. (Hint: The function...
3. Consider the multiple linear regression model iid where Xi, . . . ,Xp-1 ,i are observed covariate values for observation i, and Ei ~N(0,ơ2) (a) What is the interpretation of B1 in this model? (b)...
3. Consider the multiple linear regression model iid where Xi, . . . ,Xp-1 ,i are observed covariate values for observation i, and Ei ~N(0,ơ2) (a) What is the interpretation of B1 in this model? (b) Write the matrix form of the model. Label the response vector, design matrix, coefficient vector, and error vector, and specify the dimensions and elements for each. (c) Write the likelihood, log-likelihood, and in matrix form. aB (d) Solve : 0 for β, the MLE...
5. Consider the following data (You are allowed to use built-in function in R, but the...
5. Consider the following data (You are allowed to use built-in function in R, but the code must be attached) X 99.0 101.1 102.7 103.0 105.4 107.0 108.7 110.8 112.1 112.4 113.6 113.8 115.1 115.4 120.0 y 28.8 27.9 27.0 25.2 22.8 21.5 20.9 19.6 17.1 18.9 16.0 16.7 13.0 13.6 10.8 (a) Obtain sample statistics such as sample mean of y and x and swa = n. 2-1(;- T)?, Say = n-1 =1(xi – 7)(yi – ). (b) Obtain...
PROBLEMS 1. Find, by a hand computation, the estimated regression line for the following data: OWN-...
PROBLEMS 1. Find, by a hand computation, the estimated regression line for the following data: OWN- * CONA (a) Plot the scatter diagram, and draw the estimated regression line. (b) Double all the data values and repeat part (a). Verify the value given in Example 12.2 for the estimated regression line either by a pencil-and-paper computation or by using a hand calculator The following pairs of data represent the amounts of damages in thou- sands of dollars) in fires at...
Step 1 of 6: Find the estimated slope. Round your answer to three decimal places Step...
Step 1 of 6: Find the estimated slope. Round your answer to
three decimal places
Step 2 of 6: Find the estimated y-intercept. Round your answer
to three decimal places.
Step 3 of 6: Determine if the statement "Not all points predicted
by the linear model fall on the same line" is true or false.
Step 4 of 6: Substitute the values you found in steps 1 and 2 into
the equation for the regression line to find the estimated...
mealenk RI TAT Problem 2: ON Consider the tollowing observed values of (Ti, Yi): (2, –1) (0, 3), (1, 2), (-1,6), a. Find the estimated regression line ŷ = Bo + Bịa, based on the observed data. b. For each xi, compute the fitted value of 3Y; using c. Compute the residuals, e; = Yi-;- d. Find R-squared (the coefficient of determination). (a) (b) (c)
2. Consider a simple linear regression i ion model for a response variable Y, a single predictor variable ,i1.., n, and having Gaussian (i.e. normally distributed) errors: This model is often called "regression through the origin" since E(X) = 0 if xi = 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function Hint: The function g(x)log(x) +1-x...
5) Consider the simple linear regression model N(0, o2) i = 1,...,n Let g be the mean of the yi, and let â and ß be the MLES of a and B, respectively. Let yi = â-+ Bxi be the fitted values, and let e; = yi -yi be the residuals a) What is Cov(j, B) b) What is Cov(â, ß) c) Show that 1 ei = 0 d) Show that _1 x;e; = 0 e) Show that 1iei =...
Consider the least-squares residuals ei-yi-yi, 1, 2, . . . , linear regression model. Find the variance of the residuals Var(e). Is the vari- ance of the residuals a constant? Discuss. n,from the simple
4. We have n statistical units. For unit i, we have (xi; yi), for i-1,2,... ,n. We used the least squares line to obtain the estimated regression line у = bo +biz. (a) Show that the centroid (x, y) is a point on the least squares line, where x = (1/n) and у = (1/n) Σ¡ı yi. (Hint: E ) i-1 valuate the line at x = x. (b) In the suggested exercises, we showed that e,-0 and e-0, where...
2. Consider a simple linear regression model for a response variable Yi, a single predictor variable ri, i-1,... , n, and having Gaussian (i.e. normally distributed) errors Ý,-BzitEj, Ejį.i.d. N(0, σ2) This model is often called "regression through the origin" since E(Yi) 0 if xi 0 (a) Write down the likelihood function for the parameters β and σ2 (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function. (Hint: The function...
3. Consider the multiple linear regression model iid where Xi, . . . ,Xp-1 ,i are observed covariate values for observation i, and Ei ~N(0,ơ2) (a) What is the interpretation of B1 in this model? (b) Write the matrix form of the model. Label the response vector, design matrix, coefficient vector, and error vector, and specify the dimensions and elements for each. (c) Write the likelihood, log-likelihood, and in matrix form. aB (d) Solve : 0 for β, the MLE...
5. Consider the following data (You are allowed to use built-in function in R, but the code must be attached) X 99.0 101.1 102.7 103.0 105.4 107.0 108.7 110.8 112.1 112.4 113.6 113.8 115.1 115.4 120.0 y 28.8 27.9 27.0 25.2 22.8 21.5 20.9 19.6 17.1 18.9 16.0 16.7 13.0 13.6 10.8 (a) Obtain sample statistics such as sample mean of y and x and swa = n. 2-1(;- T)?, Say = n-1 =1(xi – 7)(yi – ). (b) Obtain...
PROBLEMS 1. Find, by a hand computation, the estimated regression line for the following data: OWN- * CONA (a) Plot the scatter diagram, and draw the estimated regression line. (b) Double all the data values and repeat part (a). Verify the value given in Example 12.2 for the estimated regression line either by a pencil-and-paper computation or by using a hand calculator The following pairs of data represent the amounts of damages in thou- sands of dollars) in fires at...
Step 1 of 6: Find the estimated slope. Round your answer to
three decimal places
Step 2 of 6: Find the estimated y-intercept. Round your answer
to three decimal places.
Step 3 of 6: Determine if the statement "Not all points predicted
by the linear model fall on the same line" is true or false.
Step 4 of 6: Substitute the values you found in steps 1 and 2 into
the equation for the regression line to find the estimated...
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