Suppose that the method of least squares is used to find the best-fit line through the following set of data points: (1,7), (3, 4), (5, 2), and (7, 1). In the notation of Section 6.3 in the workbook, let x = (c, d)t be the solution to the system A* Ax = A* y. Given that a1 A* A = az a4 find the value of the number a¡ . find the value of the number a2 . find the value of the number a4 .

4. Determine the equation of the least-squares approximating line that is the best fit for the data points (2, 6), (0, 2), and (1, 1).

Various doses of a poisonous substance were givento groups of 25 mice and the following results wereobserved: Dose (mg) Number of deathsx y4 16 38 610 812 1414 1616 20 (a) Find the equation of the least squares line fit tothese data.(b) Estimate the number of deaths in a group of 25 micethat receive a 7-milligram dose of this poison.

Ten rats were randomly assigned a solid diet and 10 in a liquid diet. At the end of the feeding period, the total protein intake x (in grams) and the weight gain y (in grams) were recorded for each of the rats. A straight line was fit to the 10 (x, y) data points for each diet by the method of least squares. The least square prediction equations follows: Liquid: y = 110 - 0.75x Solid: y = 84 +3.66x Interpret the estimated y-intercept and slope for each diet. For each diet, find the predicted weight gain of a rat with a protein instake of 10 grams.

Ten rats were randomly assigned a solid diet and 10 in a liquid diet. At the end of the feeding period, the total protein intake x (in grams) and the weight gain y (in grams) were recorded for each of the rats. A straight line was fit to the 10(x, y) data points for each diet by the method of least squares. The least square prediction equations follows: Liquid : y =110-0.75x Solid: y = 84 + 3.66x Interpret the estimated y-intercept and slope for each diet. For each diet, find the predicted weight gain of a rat with a protein instake of 10 grams.

Given the data points (1,4), (2,2), (3,1), and (5,-1): (a) Find the least squares line to best fit these points. (b) Find the least squares error of the line found in (a).

Find the equation y = Bo + B₁x of the least-squares line that best fits the given data points. Data points: (5, 1), (2, 2), (4, 3), (5, 1) Select one: O A. 37 1 y O B. 37 4 -31/2-1/2 x X O C. 47 X O D. 37 + 1x y= y = y = X

a. Rewrite the data points (2,1), (4,2), (8,6) and (9,6) with new x-coordinates in mean-deviation form. Let X be the associated design matrix. Why are the columns of X orthogonal? b. Write the normal equations for the data in part (a), and solve them to find the least-squares line, y =B₁ + B₁x², where x = x-5.75. The associated design matrix, X, is 1 - 1.75 2.25 3.25 1 the entries in one column are all 1 and the entries in the other column sum to 0. (Simplify your answer. Use ascending order.) B. Therefore, the columns of X are orthogonal because b. Write the normal equations for the data from part (a). Select the correct choice below and fill in the answer boxes within your choice. (Simplify your answer.) O A. [Bo B₁] = Bo 421-0

When the predicted overnight temperature is between 15°F and 32°F, roads in northern cities are salted to keep water from freezing on the roadways. Suppose that a small city was trying to determine the average amount of salt y (in tons) needed per night at temperature x. They found the following least squares prediction equation: y = 20,000 - 2,500x Interpet the slope. a) 2,500 tons is the decrease in the amount of salt needed for a 1 degree increase in temperature. b) 2,500 tons is the increase in the amount of salt needed for a 1 degree increase in temperature. c) 20,000 is the  increase in the amount of salt needed for a 1 degree increase in temperature. d) 2,500 tons is the expected amount of salt needed when the temperatures is 0° C.

Find the least-squares line y=Bo+B₂x that best fits the given data. Given: The data points (-2,2). (-1,5), (0.5), (1,4), (2,2). Suppose the errors in measuring the y-values of the last two data points are greater than for the other points. Weight these data points twice as much as the rest of the data. 1-2 1 -1 x= 1 1 1 2 PP 2 Next question y 5 CIT OA. y 7.1-0.73x OB. y 3.6-0.37x OC. y 3.3-0.53x OD. y 3.4-0.47x