(25) 1. The cloud point (y) of a liquid is a measure of the degree of crystallization in a stock that can be measured by the refractive index. It has been suggested that the percentage of I-8 in the base stock (r) is an excellent predictor of cloud point using the second order method. The following data (sample size n = 19) were collected on stocks with known percentage of I-8. A scatterplot of the data is shown (next page) with the least squares line drawn. X 0 2 3 5 6 7 8 22. 24.5 26.0 26.8 28.2 28.9 30.0 30.4 31.4 21.0 2 6 8 10 3 6 9 26.1 28.5 30.3 31.5 33.1 22.8 27.3 20.8 31.8 The least squares line and quadratic for the above data are given below. Line : Residual Standard Error=0.7372 R-Square=0. 9548 Estimate Std. Err t-value Pr(>It[) Intercept 23.3464 0.2968 78.6730 X 1.0455 0.0552 18.9548 Quadratic: Residual Standard Error=0.3942 R-Square=0.9878 Estimate Std. Err t-value Pr(>|t[) Intercept 22.5612 0.1984 113.6984 X 1. 6680 0.0990 16.8568 x -2 -0.0680 0.0103 -6.5911 (a) Write the equation of the least squares line. (b) For r = 9, compute the value y on the least squares line. (e) For the last point (r, y) = (9,31.8), determine the residual from the least squares line (check the plot on next page to be sure you have the right sign). (d) Write the equation of the least squares quadratic. (c) Using to975, 16 = 2.12, obtained a 95% confidence interval for the quadratic coefficient / for the least squares quadratic. (f) Is the quadratic a much better fit to the data than the line? Give a reason for your answer. (g) The matrix form of the model is Y = XP + . For the data given above, write out the first five rows of the X matrix when fitting the least squares quadratic and A = (f, A , /)T