Multiple Choice 1) When the estimated slope coefficient in the simple regression model, ο’Μ 1 , is zero, then a. There is a strong relationship between X and Y b. There is a moderate relationship between X and Y c. Y cannot be explained by X. d. The intercept is zero 2) In the simple linear regression model, the regression slope a. indicates by how many percent Y increases, given a one percent increase in X. b. when multiplied with the explanatory variable will give you the predicted Y. c. indicates by how many units Y increases, given a one unit increase in X. d. represents the elasticity of Y on X. 3) The OLS estimator is derived by a. connecting the Yi corresponding to the lowest Xi observation with the Yi corresponding to the highest Xi observation. b. making sure that the standard error of the regression equals the standard error of the slope estimator. c. minimizing the sum of absolute residuals. d. minimizing the sum of squared residuals. Interpreting the intercept in a sample regression function is a. not reasonable because you never observe values of the explanatory variables around the origin. (x=0) b. reasonable because under certain conditions the estimator is BLUE. c. reasonable if your sample contains values of Xi around the origin. d. not reasonable because economists are interested in the effect of a change in X on the change in Y. 5) The OLS residuals a. can be calculated using the coefficient of determination from the regression function. b. can be calculated by subtracting the fitted values from the actual values. c. are unknown since we do not know the population regression function. d. should not be used in practice since they indicate that your regression does not run through all your observations 6. When the estimated slope coefficient in the simple regression model, ο’Μ 1 , is zero, a. b. c. d. R2 = Y . 0 < R2 < 1. R2 = 0. R2 > (SSR/TSS). 7. The regression R 2 is defined as follows: πππ a. πππ b. πΈππ⁄πππ c. 1 - πππ⁄πππ d. two of the above 8. 9. Which of the following statements is correct? a. TSS = ESS + USS b. TSS = ESS + SST c. ESS > TSS d. R2 = 1 – (ESS/USS) The regression R2 is a measure of a. whether or not X causes Y. b. the goodness of fit of your regression line. c. whether or not ESS > TSS. d. the square of the determinant of R. Analytical questions 1. Consider the regression model : Yi = ο’0 + ο’1 Xi + ο₯i , where Yi denotes average test scores from fifth-grade classes (TS, measured in percentages) and Xi denotes the data on fifth grade class size (CS) . You collected data on 25 classes with class sizes ranging from 25 to 36 and obtained the following measures ο₯X = 75 ο₯Y = 50, ο₯X2 = 625 ο₯Y2 = 228, ο₯XY = 30 a. What does the term ο₯I represent? b. Find the estimated regression equation and interpret it fully. c. Find the R2 and interpret the value. What are the units of measurement for the R2? d. A classroom has 22 students. What is the regression prediction for the classroom’s average test score? Is this prediction reliable? Why or why not? c. The sample average class size across the 25 classrooms is 5. What is the average test scores across the 25 classrooms? 2. Sir Francis Galton, a cousin of James Darwin, examined the relationship between the height of children and their parents towards the end of the 19th century. It is from this study that the name “regression” originated. You decide to update his findings by collecting data from 110 college students, and estimate the following relationship: Studenth = 19.6 + 0.73×Midparh, R2 = 0.45, Se = 2.0 where Studenth is the height of students in inches, and Midparh is the average of the parental heights. (Following Galton’s methodology, both variables were adjusted so that the average female height was equal to the average male height.). SER is the standard error of regression (a) (b) (c) Interpret the estimated equation. Is the estimated intercept meaningful? Why or why not. What is the meaning of the R-squared value in this problem? Given the positive intercept and the fact that the slope lies between zero and one, what can you say about the height of students who have quite tall parents? Who have quite short parents? Q.3 In a simple regression and correlation analysis based on 72 observations, we find r = 0.8 and Se=10 (a) Find the amount of unexplained variation (Residual Sum of Squares) (b) Find the proportion of unexplained variation to the total variation. (c) Find the total variation of the dependent variable (TSS) Q.4. From the text book Question 4.3 (a) and (d) Question 4.4 Q.5 SCENARIO: Suppose you are interested in learning about the determinants of college GPA for VIU students. Being aware of the common factors, you wish to determine the effect of skipping classes on GPA. You have collected a random sample of 100 VIU students and have recorded their ages (Agei in years), College GPA (colgpai), high school GPA (hsgpa), provincial achievement exam score (Prov), average lectures missed per week (Skipped) and the average number of days per week of alcohol consumption (Alcohol) The statistical analyses you perform are given in the Exhibits file below Based on the Regression results, a. what is the verbal interpretation of the estimated regression equation ? b. what college GPA of a randomly selected student who skipped 6 classes? Is this prediction reliable? Why or why not? c. Estimate R2 for the regression and interpret your result. d. Based on the computer output, find TSS, ESS and USS. DESCRIPTIVE STATISTICS Mean Median Maximum Minimum Std. Dev. Skewness Sum Sum Sq. Deviation Observations AGE 20.88652 21.00000 30.00000 19.00000 1.271064 3.229636 2945.000 226.1844 ALCOHOL 1.901064 2.000000 7.000000 0.000000 1.374701 0.981049 268.0500 264.5723 COLGPA 3.056738 3.000000 4.000000 2.200000 0.372310 0.324620 431.0000 19.40610 HSGPA 3.402128 3.400000 4.000000 2.400000 0.319926 -0.310957 479.7000 14.32936 PROV 24.15603 24.00000 33.00000 16.00000 2.844252 0.062098 3406.000 1132.567 SKIPPED 1.076241 1.000000 5.000000 0.000000 1.088882 1.234972 151.7500 165.9929 100 100 100 100 100 100 CORRELATION MATRIX AGE ALCOHOL COLGPA HSGPA PROV SKIPPED AGE 1.000000 -0.022005 -0.019504 -0.259368 -0.082002 -0.077569 ALCOHOL COLGPA HSGPA PROV SKIPPED 1.000000 0.017187 -0.045805 0.169029 0.337670 1.000000 0.414555 0.206754 -0.261820 1.000000 0.345806 -0.089662 1.000000 0.115485 1.000000 REGRESSION OUTPUT Dependent Variable: COLGPA Method: Least Squares Date: 01/18/06 Time: 11:51 Sample: 1: 100 Included observations: 100 Variable Coefficient Std. Error t-Statistic Prob. C SKIPPED 3.153084 -0.089521 0.042775 73.71302 0.027990 -3.198383 0.0000 0.0017 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.061849 18.07582 -55.25029 1.983121 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) 3.056738 0.372310 0.812061 0.853887 10.22965 0.001712