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MA3518: Applied Statistics Page 1 Department of Mathematics Faculty of Science and Engineering City University of Hong Kong MA 3518: Applied Statistics Tutorial 4 Question 1: Consider the following simple linear regression model without intercept: Yi = b Xi + ei, i = 1, 2, …, n, where ei are i.i.d. with common distribution N(0, 2) (a) Find the least squares estimator Be for the parameter b. (b) E(Be) = b Question 2: Consider the following simple linear regression model: Yi = a + b Xi + ei, i = 1, 2, …, n, where the random errors {ei} are i.i.d. with common distribution N(0, 2) Let Ae and Be denote the least squares estimators of the unknown parameters a and b, respectively. Prove that Be = Sxy / Sxx and Ae = Y – Be X n where Sxy = i 1 n (Xi – X ) (Yi – Y ) and Sxx = i 1 (Xi – X )2 MA3518: Applied Statistics Page 2 Question 3: Consider again the simple linear regression model in Question 2. Prove that (a) E(Ae) = a (b) E(Be) = b Question 4: Suppose an experiment is conducted to examine the impact of the heating degree day on the daily gas consumption. Let X and Y represent the daily gas consumption and the heating degree day, respectively. The data collected for X and Y over different months are displayed as follows: Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec X 12.2 22.6 27.8 30.3 36.6 38.1 38.6 11.2 18.8 22.4 30.1 33.1 Y 3.0 3.2 3.8 4.0 5.1 6.2 6.8 3.2 3.6 3.1 2.6 2.8 (a) Create a scatter plot of the data by SAS and comment on the plot (b) Fit a simple linear regression model to the data (c) Let b denote the slope of the regression line. Perform a statistical test on the hypotheses H0: b = 0 against H0: b 0 at 5% significance level ~ End of Tutorial 4~