#8 Please study Ch13 to finish this homework assignment. You can calculate the answers following steps in the textbook practice (Computing a and b for the prediction Equation on Page 424-427 and TESTING THE SIGNIFICANCE OF R2 USING ANOVA on Page 441-445) and weekly materials in BlackBoard. Also, please familiar with how to interpret the results using your own words following each definitions (See interpretation on Page 427-429). Table 13.4 (P.426) provides a clear, sequential example of the steps to follow to calculate the regression coefficients. By following the steps in this table, you not only learn about the calculation of the regression coefficients, but also about the logic of how the regression equation is constructed. Of course, one additional caveat: continue to remind yourself about the issue of rounding error and should develop a general classroom rule to round to a specific decimal point (i.e., 2 decimal points) for all calculations. Problem set 1: Use the following data to answer questions 1-4 (30 points): There is often thought to be a relationship between a person's educational attainment and the number of children he or she has. The hypothesis is that as one's educational level increases, he or she has fewer children. Investigate this conjecture with 25 cases drawn from the 2006 GSS file. The following table displays educational attainment, in years, and the number of children for each respondent. Education 16 12 12 6 14 14 16 12 Children 0 1 3 6 2 2 2 2 17 Education Children Education 12 14 13 12 12 12 11 3 4 0 1 2 3 1 Children 0 2 3 4 1 0 3 12 12 12 12 12 2 14 12 12 2 11 2 1. Calculate the Pearson correlation coefficient for these two variables. Does its value support the hypothesized relationship (10 pts)? EDUC 16 12 12 6 14 14 16 12 17 12 14 13 CHILDS (X - 0 1 3 6 2 2 2 2 2 3 4 0 ) X )2 (X X (Y - ) Y 1 )2 (Y Y (X - ) (Y X ) Y 12 12 12 11 12 11 12 12 12 12 12 14 12 X=314 1 2 3 1 2 2 0 2 3 4 1 0 3 Y=51 Mean X = X Mean Y = Y = X N Y N = = Variance (Y) = 2 SY (Y Y ) N1 2 Standard deviation (Y) = S Y ____ Variance (X) = S 2 X (X X )2 N1 Standard deviation (X) = S X ____ 2 = = Covariance (X,Y) = S YX r (X X )(Y Y ) N1 S YX * S X S Y (___)(___) *Answers may differ slightly due to rounding. 2. Calculate the least squares regression equation using education as a predictor variable. What is the value of the slope, b (5 pts)? What is the value of the intercept, a (5 pts)? 3. What is the predicted number of children for a person with a college degree (16 years of education) (5 pts)? 4. Does any respondent actually have this number of children? If so, what is his or her level of education? If not, is this a problem or an indication that the regression equation you calculated is incorrect? Why or why not (5 pts)? 3 Problem set 2: use the following information to answer questions 5-9 (40 points total): Let's examine the relationship between GNP per capita and the percentage of respondents willing to pay more in taxes. 5. In this chapter, we used Table 13.4 to illustrate how to calculate the slope and intercept in the regression table. Using table 13.4 as a model, create a similar table using the data below for GNP per capita and the percentage willing to pay higher taxes (10 pts). GNP per Capita (X) % Willing to Pay Higher Taxes (Y) Mean X = X Mean Y = Y X N Y N (X - ) X )2 (X - (Y - X ) Y Variance (Y) = 4 )2 (Y Y (X - ) (Y X ) Y Standard deviation (Y) = S Y _____ Variance (X) = S 2 X (X X )2 N1 Standard deviation (Y) = S Y _____ Covariance (X,Y) = S YX (X X )(Y Y ) N1 *Answers may differ due to rounding (2 decimal points); however the exact value of these column totals, properly calculated will always be equal to zero. Sources: The World Bank Group, Development Education Program Learning Model: Economics, GNP per Capita, 2004. International Social Survey Programme, 2000. 6. From the table that you created in #5, calculate a, b, and write out the regression equation (i.e., prediction equation) (10 pts). % Willing to Pay Higher Taxes (Y) Predicted % Willing to Pay Higher Taxes () (Y- ) (Y- )2 7. Calculate and interpret error type, E2 (5 pts). 5 = 8. Using your answer from #7, calculate the PRE measure, r2. Interpret (5 pts). 9. About what percentage of citizens are willing to pay higher taxes for a country with a GNP per capita of 3.0 (i.e., $3,000)? For a GNP per capita of 30.0 (i.e., $30,000) (10 pts)? Problem set 3: use the following information to answer questions 10-12 (30 points total): 10. Calculate the Y-intercept, a, of the regression equation using the following information: (10 pts) Mean number of years of schooling: 17 Mean monthly income: $2,750 Slope, b, of the regression equation: $650 11. Consider the following linear regression prediction equation: Y = 12 + -1x. If x = 7, what is the predicted value of, Y, for this observation (10 pts)? 12. Consider the following linear regression prediction equation: Y = 12 + -1x. If the predicted value of, Y, for an observation is 5, calculate the residual for an observation where the observed value, Y, is equal to 9 (10 pts)? 6