If we have a set of points that actually lie on a line, the process we've learned forlinear regression will certainly find the equation of that line! Moreover the linearmodel and the input data will match precisely (i.e. all of the residuals will be 0.)Consider the set of points {(3,4), (4, 5), (6,7)}, and the linear regression proce-dure.We would first construct the matrix314 11We would normally be solving the systemAA.x = vwhere v is the projection of the vector made from the y coordinates of the pointsonto the column space of A.Show that the projection onto Col(A) is unnecessary by verifying that314 16 1mbhas a unique solution. (This means that=4LO74was already in Col(A).)

Answered: Image /qna-images/answer/64682878-fa4e-4134-a084-15610a8a0915.jpg

Answered: If we have a set of points that…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 31EQ

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If the points (x1, y1), (x2, y2),..., (xn, yn) lie on a straight line, what can you say about the regression line associated with these points?
For a set of data: x = (0,1,2,3,4,5,6) and y=(36, 28, 25, 24, 23, 21, 19), is it wise to use a linear regression to extrapolate data for x = 50? Solution: Since the coefficient of determination is 0.8582, the linear model is a reasonably good fit for the data, so extrapolation for any x-value is acceptable. What is wrong with this solution?
Use the general equation for the least square regression line to show that this line always passes through the point (x,y) * bars above the x and y.That is, set x=x(with a bar above the x) and show that the line predicts that y=y (with a bar above the y).
Assuming that this simple linear regression model fulfills all the classical assumptions we discussed in class, please show that the OLS estimator for the intercept of the simple linear regression model is an unbiased estimator.
A fitted linear regression model is (y=10+2x ). If x = 0 and the corresponding observed value of y = 9, the residual at this observation is:
It is required to use the data given in the table to estimate the parameters of the simple linear regression equation by any of the estimation methods:
The best predicted crash fatality rate for a year in which there are 525 metric tons of lemon imports is ____________ fatalities per 100,000 population. Is the prediction worthwhile? A.Since common sense suggests there should not be much of a relationship between the two variables, the prediction does not make much sense. B. Since the sample size is small, the prediction is not appropriate. C. Since all of the requirements for finding the equation of the regression line are met, the prediction is worthwhile. D.Since there appears to be an outlier, the prediction is not appropriate.
Suppose there is 1 dependent variable (dissolved oxygen, Y) and 3 independent variables (water temp X1, depth X2, and hardness of water X3). Below is the result of the multiple linear regression.Which of the following is NOT true in the multiple linear regression outputs? In the F-test ANOVA result, if Ho is rejected, this means that the regression model overall predicts the dependent variable significantly well. If a predictor is having a significant impact on our ability to predict the outcome then the regression coefficient b should be significantly different from 1.0. The F-test ANOVA assesses all of the regression coefficients jointly whereas the t-test for each coefficient examines them individually. It is possible that a model is significant, but not enough to conclude that any individual variable is significant.
Consider a simple linear regression model with predictor variable x and response variable y, where the regression line is represented by the equation y = β0 + β1x. If β0 = -5 and β1 = 3, what is the predicted value of y for a given x = 4?
Suppose that researchers are interested in determining the bi-annual salary of statisticians of different levels using their years of experience and their education level (M = bachelors, P = doctorate). They fit the following model to a dataset that includes these variables and, after performing the proper steps of multiple linear regression, the following multiple linear regression model is obtained: yˆ = 42308 + 323x1 + 213x2 + 301(x1*x2) where the variables are as follows: yˆ = predicted bi−annual salary in dollars, x1 = number of years of experiencex2= {1 if the education level is a doctorate 0 if the education level is a bachelors What is the predicted bi-annual salary in dollars of an employee with 5 years of experience and a bachelor’s degree?
Suppose that researchers are interested in determining the bi-annual salary of statisticians of different levels using their years of experience and their education level (M = bachelors, P = doctorate). They fit the following model to a dataset that includes these variables and, after performing the proper steps of multiple linear regression, the following multiple linear regression model is obtained: yˆ = 42308 + 323x1 + 213x2 + 301(x1*x2) where the variables are as follows: yˆ = predicted bi−annual salary in dollars, x1 = number of years of experiencex2= {1 if the education level is a doctorate 0 if the education level is a bachelors What is the predicted bi-annual starting salary of an employee with a doctorate degree? (Someone with no work experience). $ What is the predicted bi-annual starting salary of an employee with a bachelor’s degree? (Someone with no work experience). $
If we collect monthly sales over two years for N=100 stores, we should not apply a simple linear regression model directly to the data, because the observations are not independent with each other. Is this statement True or False? A) True B) False

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