Concept explainers
Show that
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
- The annual energy consumption in billions of Btu for both natural gas and coal is shown for a random selection of states. Gas 223 474 377 289 747 146 Coal 478 631 413 356 736 474 If 500 billion Btu of natural gas is used then what is the projected amount of coal that is usedThe projected amount of coal that is used is billion Btu. (Use your regression equation with the rounded values for m and b to find your answer to this question. Round your answer to THREE decimal places, add extra zeros at the end, if needed) (Round answer to 3 decimal places, for example, XXX.XXX)arrow_forwardThe lines of regression of y on x and x on y are, respectively, y = x + 5 and 16x - 9y = 94. Find the variance of x if the variance of y is 16. Also, find the covariance of x and yarrow_forwardAn owner of a home in the Midwest installed solar panels to reduce heating costs. After installing the solar panels, he measured the amount of natural gas used y (in cubic feet) to heat the home and outside temperature x (in degree-days, where a day's degree-days are the number of degrees its average temperature falls below 65° F) over a 23-month period. He then computed the least-squares regression line for predicting y from x and found it to be ŷ = 85 + 16x. The software used to compute the least-squares regression line for the equation above says that r2 = 0.98. This suggests which of the following? 1. Gas used increases by square root of 0.98 = 0.99 cubic feet for each additional degree-day? 2. Although degree-days and gas used are correlated, degree-days do not predict gas used very accurately. 3. Prediction of gas used from degree-days will be quite accurate.arrow_forward
- Consider the points (1, 1), (2, 3), (−3, 1). Find the least squares regression line.arrow_forwardIn the least-squares regression model, y, = B, X; + Bo +&, &, is a random error term with mean and standard deviation og In the least-squares regression model, y, = B4 X; + Bo + E, ɛ, is a random error term with mean V and standard deviation oarrow_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
- 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_forwardUse 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_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_forward
- 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).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_forwardox² = 3, oy² = 5, oxy = 2, Z = 2Y - 4X – 2 a. Determine the variance of Z.arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning