4. We are interested in the effect on test scores of the student-teacher ratio (STR). The...

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4. We are interested in the effect on test scores of the student-teacher ratio (STR). The following regression results have been obtained using the California data set. All the regressions used average test scores in the district as the dependent variable. Many factors potentially affect the average test scores in a district. Three variables are considered: (i) the fraction of students who are still learning English/HiEL (a dummy taking 1 if the fraction is larger than 10%, 0 otherwise), (ii) the percentage of students who are eligible for receiving subsidized free lunch at school, and (iii) average district income (in logarithm). Different regressions use different combinations of regressors, and quadratic and cubic terms of STR, and the interaction terms. The entries in the table include estimated coefficients, standard errors (in parenthesis), F-statistics and their p-values (in parenthesis below the values of F-statistic), and other summary statistics (SER, adjusted R). The significance of each variable is indicated by ** (1%) and (5%). Dependent variable: average test score in district; 420 observations. Regressor (1) (2) (3) (4) (5) (6) (7) Student-teacher ratio (STR) -1.00** -0.73** -0.97 -0.53 64.33** 83.70** 65.29** (0.27) (0.26) (0.59) (0.34) (24.86) (28.50) (25.26) STR -3.42** -4.38** -3.47** (1.25) (1.44) (1.27) STR 0.059** (0.021) 0.075** (0.024) 0.060** (0.021) % English learners -0.122 (0,033) -0.176** (0.034) **9910- (0.034) % English learners 10%? (Binary, HIEL) 5.64 (19.51) (9.80) 5.50 -5.47** 816.1* (1.03) (327.7) HIEL STR -1.28 -0.58 -123.3 (0.97) (0.50) (50.2) HIEL STR 6.12* (2.54) HIEL STR -0.101* (0.043) % Eligible for subsidized lunch -0.547** -0.398 (0.024) -0.411** -0.420** -0.418** -0.402** (0,033) (0.029) (0.029) (0.029) (0,033) Average district income (logarithm) 11.57** (1.81) 12.12** (1.80) 11.75** 11.80** (1.78) (1.78) 11.51** (1.81) Intercept 700.2** (5.6) 658.6** 682.2** (8.6) (11.9) 653.6** 252.0 (9.9) (163.6) 122.3 (185.5) 244.8 (165.7) F-Statistics and p-values on Joint Hypotheses (a) All STR variables and interactions -0 (b) STR, STR = 0 (c) HIEL X STR, HIEL STR, HIEL X STR=0 SER R 5.64 5.92 6.31 (0.004) (0.003) ( <0.001) ( <0.001) 4.96 5.91 (0.001) 6.17 ( <0.001) 5.81 (0.003) 5.96 (0.003) 2.69 (0.046) 9.08 8.64 15.88 8.63 8.56 8.55 8.57 0.773 0.794 0.305 0.795 0.798 0.799 0.798 These regressions were estimated using the data on K-8 school districts in California, described in Appendix 4.1. Standard errors are given in parentheses under coefficients, and p-values are given in parentheses under F-statistics. Individual coefficients are statistically significant at the *5% or **1% significance level. (d) Regression (5) examines whether the effect of changing the student-teacher ratio depends on student-teacher ratio by including nonlinear terms of STR, in addition to other control variables in regression (4). Based on the regression results, would you keep the non-linear terms (STR and STR)? Discuss both individual and joint significance tests. Note that p- values for the F-statistics are given in parenthesis. You do not have to calculate T-statistics nor F-statistics as the table present the individual significance and the realized values of the F-statistic. This question is asking you to use P-values to conduct the hypothesis testing. Present your statistical arguments for your choice. Would you choose to add the nonlinear terms? (e) Would you conclude that this effect in (d) depend on the value of the student-teacher ratio? Present statistical evidence (statistical arguments) for your answer. (f) Regression (6) further examines whether the effect of STR depends not just on the value of STR, but also on HiEL. What is your statis- tical conclusion on this, based on Regression (6)? Present your statistical arguments for your conclusion. 4. We are interested in the effect on test scores of the student-teacher ratio (STR). The following regression results have been obtained using the California data set. All the regressions used average test scores in the district as the dependent variable. Many factors potentially affect the average test scores in a district. Three variables are considered: (i) the fraction of students who are still learning English/HiEL (a dummy taking 1 if the fraction is larger than 10%, 0 otherwise), (ii) the percentage of students who are eligible for receiving subsidized free lunch at school, and (iii) average district income (in logarithm). Different regressions use different combinations of regressors, and quadratic and cubic terms of STR, and the interaction terms. The entries in the table include estimated coefficients, standard errors (in parenthesis), F-statistics and their p-values (in parenthesis below the values of F-statistic), and other summary statistics (SER, adjusted R). The significance of each variable is indicated by ** (1%) and (5%). Dependent variable: average test score in district; 420 observations. Regressor (1) (2) (3) (4) (5) (6) (7) Student-teacher ratio (STR) -1.00** -0.73** -0.97 -0.53 64.33** 83.70** 65.29** (0.27) (0.26) (0.59) (0.34) (24.86) (28.50) (25.26) STR -3.42** -4.38** -3.47** (1.25) (1.44) (1.27) STR 0.059** (0.021) 0.075** (0.024) 0.060** (0.021) % English learners -0.122 (0,033) -0.176** (0.034) **9910- (0.034) % English learners 10%? (Binary, HIEL) 5.64 (19.51) (9.80) 5.50 -5.47** 816.1* (1.03) (327.7) HIEL STR -1.28 -0.58 -123.3 (0.97) (0.50) (50.2) HIEL STR 6.12* (2.54) HIEL STR -0.101* (0.043) % Eligible for subsidized lunch -0.547** -0.398 (0.024) -0.411** -0.420** -0.418** -0.402** (0,033) (0.029) (0.029) (0.029) (0,033) Average district income (logarithm) 11.57** (1.81) 12.12** (1.80) 11.75** 11.80** (1.78) (1.78) 11.51** (1.81) Intercept 700.2** (5.6) 658.6** 682.2** (8.6) (11.9) 653.6** 252.0 (9.9) (163.6) 122.3 (185.5) 244.8 (165.7) F-Statistics and p-values on Joint Hypotheses (a) All STR variables and interactions -0 (b) STR, STR = 0 (c) HIEL X STR, HIEL STR, HIEL X STR=0 SER R 5.64 5.92 6.31 (0.004) (0.003) ( <0.001) ( <0.001) 4.96 5.91 (0.001) 6.17 ( <0.001) 5.81 (0.003) 5.96 (0.003) 2.69 (0.046) 9.08 8.64 15.88 8.63 8.56 8.55 8.57 0.773 0.794 0.305 0.795 0.798 0.799 0.798 These regressions were estimated using the data on K-8 school districts in California, described in Appendix 4.1. Standard errors are given in parentheses under coefficients, and p-values are given in parentheses under F-statistics. Individual coefficients are statistically significant at the *5% or **1% significance level. (d) Regression (5) examines whether the effect of changing the student-teacher ratio depends on student-teacher ratio by including nonlinear terms of STR, in addition to other control variables in regression (4). Based on the regression results, would you keep the non-linear terms (STR and STR)? Discuss both individual and joint significance tests. Note that p- values for the F-statistics are given in parenthesis. You do not have to calculate T-statistics nor F-statistics as the table present the individual significance and the realized values of the F-statistic. This question is asking you to use P-values to conduct the hypothesis testing. Present your statistical arguments for your choice. Would you choose to add the nonlinear terms? (e) Would you conclude that this effect in (d) depend on the value of the student-teacher ratio? Present statistical evidence (statistical arguments) for your answer. (f) Regression (6) further examines whether the effect of STR depends not just on the value of STR, but also on HiEL. What is your statis- tical conclusion on this, based on Regression (6)? Present your statistical arguments for your conclusion.