Question

The regression model Yˆi = B_0 + B_1x_i has been adapted to a data set consisting of 23 observations (x_i, y_i) for i = 1, ..., 23. Using the least squares method, the estimates b_0 = 26.984 and b_1 = 0.748 have been found. Yˆ0 is the value of the custom model at point x_0. The following is stated            x = 1/23 * Σn = 23, i = 1, x_i = 17,            y = 1/23 * Σn = 23, i = 1, y_i = 39.7,            Sxx = Σn = 23, i = 1, (xi-X) * 2 = 523.4,            Syy = Σn = 23, i = 1, (y-y) ^ 2 = 393.2,            Sxy = Σn = 23, i = 1, (xi-X) (yi-Y) = 391.3. (a) First calculate an expectation estimate for the variance of Yˆ0 at the point x_0 = 8.7. You can use that Y¯ = 1/23 * ∑n = 23, i = 1, Y_i and B_1 are independent stochastic variables.     Number (b) Calculate a 95% confidence interval for the expectation value E (Yˆ0) at the point x_0 = 8.7. Lower limit: Number Upper limit: Number

Answer 1

$Estimate~of~Slope=\hat{\beta}=\frac{S_{xy}}{S_{xx}}=391.3/523.4=0.7476\\ Estimate~of~intercept=\hat{\alpha}=\bar{y}-\hat{\beta}\bar{x}=39.7-0.7476*17=26.9908\\ Regression~equation:~\hat{y}=26.9908+0.7476x\\ At~x_0=8.7,~y_0=26.9908+0.7476*8.7=33.4949\\ r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=0.8626\\ SSE=sum~of~square~due~to~error\\=(1-r^)S_{yy}=(1-0.8626^2)*393.2=100.6282\\ MSE=mean~sum~of~due~to~error\\ =SSE/21=100.6282/21=4.7918\\ t_{0.025,21}=2.0796,~n=23\\ 95\%~C.I.~for~the~y_0~at~the~point~x_0:\\ Lower~limit:~y_0-t_{0.025,21}\sqrt{MSE\left(\frac{1}{n}+\frac{(x_0-\bar{x})^2}{S_{xx}} \right )}\\ =33.4949-2.0796*sqrt(4.7918(1/23+(8.7-17)^2/523.4))=31.5900\\ Upper~limit:~y_0+t_{0.025,21}\sqrt{MSE\left(\frac{1}{n}+\frac{(x_0-\bar{x})^2}{S_{xx}} \right )}\\ =33.4949+2.0796*sqrt(4.7918(1/23+(8.7-17)^2/523.4))=35.3998\\$