0. Exam 2 - ECO 391 - Fall 2022 (Q&A) 42 questions UPLOADED-1

Last Name: ____________________ First Name: ____________________ ECO 391 (Fall 2022) Exam 2 Directions: ● You have 60 minutes to complete the exam and it is out of 100 points. ● Multiple choice 1 to 42 are worth 2.38 points per question. Good Luck!!! 1. The table below shows a portion of regression results. 1

SS MS F Significance F Regression 17248773 17248773 926.170433 9.80086E-68 Residual 2923930 18623.76 Total 20172703 What is the R 2 value? a. 0.56 b. 0.66 c. 0.76 d. 0.86 2. Which of the following corresponds to the explained part of the variation in the dependent variable? a. The total sum of squares b. The regression sum of squares c. The residual sum of squares d. The degrees of freedom 3. A model has unexplained variation of 153.47 and total variation of 1234.50. What is the R 2 value of this model? a. 0.1243 b. 0.8756 c. 0.95 d. 1.04 4. A researcher is interested in the determinants of monthly consumption of natural gas in North Carolina. She estimates a simple model with only mean monthly temperature as the independent variable. What does the R 2 of this model represent? a. The variance of the coefficient on mean temperature b. The percent of variation in gas consumption explained by variations in mean temperature c. The expected increase in gas consumption associated with a one unit increase in mean temperature d. The mean difference between the predicted level of gas consumption and the actual value 2

b. 22 c. 26 d. 31 10. Consider the sample regression equation ^ y = 15 + 2 x 1 + 7 x 2 − x 3 . What is the predicted change in Yhat when x 1 increases by 1 unit, x 2 does not change, and x 3 increases by 1 unit? a. Decrease 2 units b. Decrease 1 unit c. Increase 1 unit d. Increase 2 units 11. Consider the sample regression equation ^ y = 15 + 2 x 1 + 7 x 2 − x 3 . What is the value of the residual when x 1 is 0, x 2 is 0, x 3 is 5, and Y is 20? a. 10 b. 15 c. 20 d. 25 12. What is one advantage of using dummy variables? a. They can take any possible value, making it easier to work with b. They are simpler to add to your model than other variables. c. They allow you to capture qualitative measures. d. They have no advantage over other variables. 13. A researcher estimates the model crime = β 0 + β 1 ∗ education + β 2 ∗ wage + β 3 ∗ youth + β 4 Expenditure + ε . She obtains the results below. Coefficients Intercept -111.688723 Education 1.66861011 Wage 0.00332358 Youth -0.89949622 Expenditure 0.79326242 Increases in which variables are associated with increases in crime? a. Education b. Education, wage 4

c. Education, wage, youth d. Education, wage, expenditure 14. In a multiple regression analysis involving 12 independent variables and 166 observations, SSR = 878 and SSE = 122. Thus, the coefficient of determination is __________. a. 0.139 b. 0.122 c. 0.878 d. 0.731 15. A researcher estimates the model crime = β 0 + β 1 ∗ education + β 2 ∗ wage + β 3 ∗ youth + β 4 ∗ Expenditure + ε . She obtains the results below. Coefficients Standard Error t Stat P-value Intercept -111.688723 72.60820393 -1.53824 0.131491 Education 1.66861011 3.305592294 0.504784 0.616349 Wage 0.00332358 0.065738859 0.050557 0.959918 Youth 0.89949622 0.332354675 2.706435 0.009789 Expenditure 0.79326242 0.172211478 4.606327 0.000037 What is the predicted increase in the crime rate associated with a one unit increase in education, holding constant wage, youth, and expenditure? a. -111.69 b. 1.67 c. 72.61 d. 3.31 16. A researcher estimates the model crime = β 0 + β 1 ∗ education + β 2 ∗ wage + β 3 ∗ youth + β 4 ∗ Expenditure + ε . She obtains the results below. Coefficients Standard Error t Stat P-value Intercept -111.688723 72.60820393 -1.53824 0.131491 5

Observations 47 What percent of the variation in the crime rate is explained by the variation in wage? a. 43 b. 18 c. 16 d. 47 20. A researcher is interested in the determinants of crime across states for a particular year. She estimates the model crime = β 0 + β 1 ∗ southern + β 2 ∗ education + β 3 ∗ wage + ε where crime is the number of offenses per million population, education is the average years of schooling, and wage is the median weekly wage. Southern is a dummy variable that is equal to one if a state is in the south. True or false: according to the coefficients below, southern states are predicted to have a lower crime rate, holding education and wage constant. Coefficients Intercept -7.005004455 Southern 21.96978944 Education -0.108248777 Wage 0.197333765 a. True b. False 21. A researcher obtains the results below from estimating crime = β 0 + β 1 ∗ southern + β 2 ∗ education + β 3 ∗ wage + ε . The estimate for which of the following is significant at the 1% level? Coefficients Standard Error t Stat P-value Intercept -7.005004455 52.14699 -0.13433 0.893767 Southern 21.96978944 10.60892 2.070878 0.044407 Education -0.108248777 4.091165 -0.02646 0.979014 Wage 0.197333765 0.053475 3.690235 0.000626 7

a. β 1 b. β 1 + β 3 c. β 2 d. β 3 22. A researcher estimates the model y = β 0 + β 1 ∗ D + ε , where y is ACT score and D is equal to one for females and 0 for males. Which of the following is a correct interpretation for β 0 ? a. The average ACT score across the entire sample b. The average ACT score for females c. The average ACT score for males d. The lowest ACT score in the sample 23. A researcher is interested in the relationship between consumption of natural gas in North Carolina and month of the year. She estimates the model using dummy variables to represent the months, with Month_EQ_2 equaling 1 for February, Month_EQ_3 equaling 1 for March, and so on. She obtains the results below. Which month does she use as the base case? Coefficients Standard Error t Stat P-value Intercept 45382.42857 3357.579455 13.51641 1.1E-20 Month_EQ__02 -7144.71429 4748.334402 -1.50468 0.137178 Month_EQ__03 -9920.14286 4748.334402 -2.08918 0.04055 Month_EQ__04 -21631.1429 4748.334402 -4.55552 2.31E-05 Month_EQ__05 -21464.7143 4748.334402 -4.52047 2.63E-05 Month_EQ__06 -19708.8571 4748.334402 -4.15069 9.7E-05 Month_EQ__07 -19469.2619 4942.223139 -3.93937 0.0002 Month_EQ__08 -19966.5952 4942.223139 -4.04 0.000142 Month_EQ__09 -22168.5952 4942.223139 -4.48555 2.98E-05 Month_EQ__10 -22725.4286 4942.223139 -4.59822 1.98E-05 Month_EQ__11 -13780.2619 4942.223139 -2.78827 0.006916 8

25. A researcher estimates the following model: y = β 0 + β 1 ∗ D + β 2 ∗ X + β 3 ∗ D ∗ X + ε . If the coefficient on the dummy variable and the coefficient on the interaction term are both significant, what does the estimated regression represent? a. One regression line b. Two regression lines with the same intercept but different slopes c. Two regression lines with the same slope but different intercepts d. Two regression lines with different intercepts and different slopes 26. Suppose R 2 = 0.60 and SSR = 90 . Compute SSE . a. 36 b. 54 c. 60 d. 150 27. Consider the model y = β 0 + β 1 ∗ x + β 2 ∗ D ∗ x + ε . When D is equal to one, what is the increase in Y associated with a one unit increase in x? a. β 0 b. β 0 + β 1 c. β 1 + β 2 d. β 2 28. Consider the model y = β 0 + β 1 ∗ x + β 2 ∗ D ∗ x + ε . When D moves from zero to one, which of the following is affected in the graph of the line? a. Intercept b. Slope 29. True or False: all independent variables in a multiple regression must be in the same units. a. True b. False 30. What is the formula for the standard error of the mean? a. σ b. σ / √ n c. z ∗ σ / √ n d. z / σ 10

31. Which of the following is equal to half the width of a confidence interval? a. Standard deviation b. Standard error c. Margin of error d. Variance 32. True or False: sample size is one of the factors that determines the width of confidence intervals. a. True b. False 33. True or False: the distance of the sample mean from the true population mean is one of the factors that determines the width of confidence intervals. a. True b. False 34. A restaurant collects a survey from 50 respondents that ranks their service on a scale of 1 to 10. The average rating is 8.9. Assume a population standard deviation of 1 unit. What is the standard error? a. 0.14 b. 0.24 c. 0.34 d. 0.44 35. A restaurant collects a survey from 50 respondents that ranks their service on a scale of 1 to 10. The average rating is 8.9. Assume a population standard deviation of 1 unit. What is the 95% confidence interval? a. [8.62, 9.18] b. [8.55, 9.25] c. [7.51, 10.09] d. [6.91, 11.26] 36. True or False: a higher confidence level indicates a higher likelihood that the sample mean is within a given distance of the population mean. a. True b. False 37. If the sample mean is 10, the standard deviation is 2 and the sample size is 100, what is the standard error? a. 0.1 b. 0.2 11

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Confidence Interval Formula R-squared Adjusted R-squared X ± z α 2 ∗ σ √ n R 2 = SSR SST Adjusted R 2 = 1 – (1 – R 2 ) * ( n − 1 n − 1 − p ) Z α 2 confidence level 90% confidence level  Z α 2 = 1.64 14

18. Solution A: Salary is probably determined in part by years of education. Especially for middle- aged individuals, it is less likely that years of education is determined by salary. 19. Solution: B . The variation in the dependent variable that is explained by variation in the independent variable is the R 2 . 20. Solution: B. The coefficient on Southern is positive, indicating that being in a southern state is associated with an increase in crime, holding constant other variables. 21. Solution: D. The coefficient on wage, β 3 , is the only one with a p-value below 0.01. 22. Solution: C. For males, D = 0, so the predicted score for males is β 0 . In a simple regression where the explanatory variable is a dummy, the predicted score when D = 0 is also the average score when D = 0. 23. Solution: A. There is no dummy for January, so we can conclude that January is the base case. The model’s prediction for January can be found by plugging 0 in for the other months. 24. Solution: A. The p-value for February is higher than 0.05, so it is not significant at the 5% level. 25. Solution: D. When D = 0, the intercept is β 0 but when D = 1, the intercept is β 0 + β 1 . Similarly, when D = 0 the slope is β 2 but when D = 1 the slope is β 2 + β 3 26. Solution: C. Note that R 2 = RSS TSS . Also note that TSS = RSS − ESS . We are given R 2 and RSS . Thus, we can first use basic algebra to obtain TSS : 0.60 = 90 TSS →TSS = 90 0.60 = 150 Now we can use basic algebra to obtain ESS: TSS = RSS + ESS→ESS = TSS − RSS = 150 − 90 = 60 27. Solution: C. Due to the interaction term, when D is equal to one, the regression can be simplified to y = β 0 +( β 1 + β 2 )∗ x + ε . Notice there is a new slope coefficient. 28. Solution: B. The D is in an interaction term, meaning it affects the slope of the regression line. The slope is the relationship between x and y. 29. Solution: B. The variables in a regression do not need to be in the same units for the coefficients to have an interpretation. 30. Solution: B. Answer choice b is the formula for the standard error. 31. Solution: C. The formula for confidence interval is X bar ± z ∗ σ / √ n or X bar ±marginof error . So, the width of a confidence interval is two times the margin of error. 32. Solution: A. X bar ± z ∗ σ / √ n is the formula for confidence intervals. “n” is for sample size. 33. Solution: B. The true population mean is usually not known and is not one of the factors that determines the width of a confidence interval. 34. Solution: A. The formula for standard error is σ / √ n or 1 / √ 50 ≈ 0.14 . 35. Solution: A. The formula for a confidence interval is X bar ± z ∗ σ / √ n Plug in to get 8.9 ± 1.96 ∗ 1 / √ 50 36. Solution: B . The confidence level determines the width of the confidence interval, but it does not indicate how close the true population mean is to the sample mean. 37. Solution: B. The formula for standard error is σ / √ n or 2 / √ 100 ≈ 0.2 . 38. Solution: B. The formula for margin of error is z ∗ σ / √ n or 1.96 ∗ 1 / √ 50 ≈ 0.28 , where 1.96 is the z-score corresponding to the 5% significance level. 39. Solution: B. The margin of error accounts for variation due to random chance but does not account for any bias in the sampling process. 16

40. Solution: D. The formula for margin of error is z ∗ σ / √ n or 1.96 ∗ 2 / √ 100 ≈ 0.4 , where 1.96 is the z-score corresponding to the 5% significance level. 41. Solution: A. The coefficient of determination, or R 2 , is equal to the SSR divided by the SST or total sum of squares: R 2 ¿ SSR / SST . Also note that SSE or sum of squared errors is equal to SST – SSR. In other words, we know that R 2 ¿ ( 1 − SSE )/ SST . Therefore, R 2 ¿ ( 1 − 9698 )/ 18007 ≈ 0.46 42. Solution: B. False. Adding a variable may increase, decrease, or may not affect the adj-R 2 . 17