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/ C,AF. 1.2 - Use the following for questions 1 — 4- Scenario 12-1 A high school guidance counselor wonders if it is possible to predict a student’s GPA in their senior year from their GPA in the first marking period of their freshman year. She selects a random sample of 15 seniors from the graduating class of 468 and records both full-year GPA in their senior year (‘Senior”) and first-marking-period GPA in their freshman year (“Fresh”). A computer regression analysis and a residual plot of these data are given below. Predictor Coef SE Coef T P Constant 1.6310 0.5328 3.06 0.009 Fresh 0.5304 0.1789 2.96 0.011 § = 0.3558 R-Sq = 40.3% R-Sq(adj) = 35.7% Residual plot for senior 0504 " W & -025+ . . MO.SO& ‘. - 2.00 2224 26 28 30 32 34 36 Fresh 1. Use Scenario 12-1. Enough information is provided above to confirm that four conditions for slope inference have been satisfied, but we need a further analysis of the data to confirm that the fifth condition for inference has been satisfied. Which condition requires further analysis of the data? " A) Mean Senior GPA is a linear function of F reshman GPA. B) Observations for each student are independent. C) For each value of Freshman GPA, the distribution of Senior GPA is roughly Normal. D) The variance of Senior GPA is roughly equal for each value of Freshman GPA. E) The data come from a random sample. ANS: @ TOP: Conditions for inference 2. Use Scenario 12-1Which of the following is the estimate from this sample for the standard deviation of the sampling distribution of slopes? A)0.1789 B) 0.3558 C) 0.5304 D) 0.5328 E)1.6310 . ANS: @ TOP: Identify SE of slope on computer output 206 Chapter 12: More About Regression

3. Use Scenario 12-1The equation of the least squares regression line from this sample is A) Fresh=1.63 10( Senior ) +0.5304 B) Fresh=1.6310+0.5304(Senior) C) Stnior =1.6310( Fresh)+0.5304 D) Stnior =1.6310+0.5304( Fresh) E) Sénior = 0.5304( Fresh)+0.1789 ANS: P TOP: Regression equation from computer output 4. Use Scenario 12-1The guidance counselor wants to test the hypotheses Hy:f=0versus H, : ##0 atthe ot = 0.01 level. Which of the following is the appropriate conclusion? i A) Since the P-value of 0.011 is less than o, we reject H,. There is convincing evidence of a linear ! relationship between freshman GPA and Senior GPA. B) Since the P-value of 0.011 is greater than o, we fail to reject H,. We do not have enough evidence to conclude that there is a linear relationship between freshman GPA and Senior GPA. C) Since the P-value of 0.011 is greater than o, we accept H,. We have convincing evidence that there is not a linear relationship between freshman GPA and Senior GPA. D) Since the P-value of 0.009 is less than a,, we reject H,. We have convincing evidence of a linear relationship between freshman GPA and Senior GPA. E) Since the P-value of 0.009 is less than o, we accept Hy. We have convincing evidence that there is not a linear relationship between freshman GPA and Senior GPA. ANS: ® TOP: Conclusion of significance test from computer output ¢ Chapter 12: More About Regression : 207

Use the following for questions 5 — 8: Scenario 12-2 Can we predict annual household electricity costs in a specific region from the number of rooms in the house? Below is a scatterplot of annual electricity costs (in dollars) versus number of rooms for 30 randomly-selected houses in Michigan, along with computer output for linear regression of electricity costs on number of rooms. Predictor Coef SE Coef T P Constant 406.9 164.8 2.47 0.020 [ Rooms 58.45 24.77 2.36 0.026 S = 246.735 R-Sq = 16.6% R-Sq(adj) = 13.6% Scatterplot of annual electricity cost vs. number of rooms z S 1200 | . » - - % 1000 . » . . . TR * . ] 8OO * . . . [ v . » [ k) : S 600+, e . . ° T E. 400 . N - . -« 200 T T T i T ] 'S i 3 4 5 6 7 ] 9 Number of rooms 5. Use Scenario 12-2. Both the scatterplot and the residual plot indicate that residuals for 3 rooms and 8 and 9 rooms tend to be lower than residuals for 4 through 7 rooms. Which condition for regression inference is not satisfied in this case? A) Mean Annual Electricity costs is a linear function of Number of rooms. B) Observations for each household are independent. C) For each number of rooms, the distribution of annual electricity costs is roughly Normal. D) The variance of annual electricity costs is roughly equal for each number of rooms. E) The data come from a random sample. ANS: @ TOP: Conditions for inference 6. Use Scenario 12-2. Which value in the computer output is the estimate from this sample for the standard deviation of the residuals? A)24.77 B) 58.45 C)164.8 D) 246.735 E) This quantity is not provided by the computer output. ANS: ® TOP: Interpret s 208 Chapter 12: More About Regression

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7. Use Scenario 12-2. Which of the following statements is a correct interpretation of information in this computer output? A) A one-room house is predicted to have annual electricity costs of $58.45. B) For each additional room a house has, predicted annual electricity costs increase by $58.45. C) For each additional room a house has, predicted annual electricity costs increase by $24.77. D) The standard deviation in annual electrical costs for houses in this sample is $246.735. E) The standard deviation in annual electrical costs for houses in this sample is $24.77. ANS: ® TOP: Interpret Slope 8. Use Scenario 12-2. Assume the conditions for inference have been met. If we test the hypotheses Hy:f=0versus H,:f #0 atthe a. = 0.05 level. Which of the following is the appropriate conclusion? A) Since the P-value of 0.020 is less than o, we reject Hy. There is convincing evidence of a linear relationship between annual electricity costs and number of rooms. B) Since the P-value of 0.020 is greater than a., we fail to reject H,. We do not have enough evidence to conclude that there is a linear relafionship between annual electricity costs and number of rooms. C) Since the P-value of 0.020 is greater than a., we accept H,. We have convincing evidence that there is not a linear relationship between annual electricity costs and number of rooms. D) Since the P-value of 0.026 is less than o, we accept H,. We have convincing evidence that there is not a linear relationship between annual electricity costs and number of rooms. E) Since the P-value of 0.026 is less than ., we reject H,. We have convincing evidence of a linear relationship between annual electricity costs and number of rooms. ANS: @ TOP: Conclusion of significance test from computer output 9. A history teacher, who is concerned about how long his students spend writing papers for his class, asks all of his students how long (in minutes) they spent on a recent essay. He then performs a linear regression analysis, with writing time as the explanatory variable and essay score (out of 100 points) as the response variable. He is careful to make sure that the residual plot for his data shows a random scatter of points and that there is no evidence of changes in variance of essay score for different values of writing time. Nevertheless, his friend who teaches statistics tells him it would not be appropriate for him to test the null hypothesis that the slope of the regression line is 0. Why not? A) The variable “Essay score” is a discrete variable. B) He does not have evidence that the relationship between writing time and essay score is linear. C) He does not have evidence that the variable “writing time” is Normal ly distributed. D) There are too many lurking variables, such as student ability and motivation, which would surely bias the study. E) It is not appropriate for the teacher to use regression inference when he has information about the entire population, ¢ ANS: @ TOP: When is inference appropriate? Chapter 12: More About Regression 209

Use the following for questions 10— 14: Scenario 12-3 Are high school students who like their English class more likely to enjoy their history class as well? Here is a regression analysis and residual plot for 30 randomly-selected students who were asked to rate how much they liked both English and history on a 0 to 5 scale (a higher rating means the student liked the subject more). [Data from 2004-5 Census at Schools survey in Canada). 3 Predictor Coef SE Coef T P Constant 1.1867 0.5574 2.13 0.042 English 0.5254 0.1995 2.63 0.014 s = 1.37707 R-Sq = 19.8% R-Sqfladj) = 17.0% ‘Residinal plot for history rating vs. english rating 5 . yu:-‘:t “ » ".. 5 - - T+ » . B v‘.t .. = Iy o = » Te # W - 2 . L. .. &i 2 Tt 1Y s J * N L2 . & 3 2O 1 ¥ T T o 1 2 3 4 5 * English rating 10. Use Scenario 12-3. Which of the following conditions for inference does the residual plot suggest has not been satisfied? A) The data come from a random sample. B) Observations for cach student are independent. C) The variance of history rating is roughly equal for each value of English rating. D) For each value of English rating, the distribution of history rating is roughly Normal. E) Mean History rating is a linear function of English rating,. ANS: @ TOP: Conditions for inference 11. Use Scenario 12-3. Which condition for inference should be checked by examining a Normal Probability plot of the residuals? A) The data come from a random sample. B) Observations for each student are independent. C) The variance of history rating is roughly equal for cach value of English rating. D) For each value of English rating, the distribution of history rating is roughly Normal. E) Mean History rating is a linear function of English rating. ANS: B TOP: Conditions for inference 210 Chapter 12: More About Regression

e A . 12. Use Scenario 12-3. Assume (despite the evidence above) that the conditions for inference have been met, and we wish to perform a z-test for regression slope. How many degrees of freedom does the t-statistic’s distribution have? A)2 B) 28 C)29 D) 30 E) 58 ANS: ® TOP: Degrees of freedom for critical t* for slope 13. Use Scenario 12-3. We wish to perform a /-test for regression slope. What is the test statistic for this test? A)0.014 B) 0.1995 C) 1.37707 D)2.13 E)2.63 ANS: @ TOP: Identify test statistic from computer output 14. Use Scenario 12-3. Assuming that the conditions for inference have been met, which of the following represents a 99% confidence interval for the rate of change in history rating for a one-unit change in English rating? A) 0.5254+2.763 (0. 1 995) 0.1995) J28 0.1995) V30 D) 0.5254+2.048(0.1995) 0.1995) B) 0.5254+ 2.763( C) 0.5254 2.763( J30 E) 0.5254 2.048( ANS: @ TOP: Confidence interval from computer output Use the following for questions 15— 17: Scenario 12-4 % A high school guidance counselor wonders if it is possible to predict a student’s GPA in their senior year from their GPA in the first marking period of their freshman year. She selects a random sample of 15 seniors from the graduating class of 468 and records both full-year GPA in their senior year (‘Senior”) and first-marking-period GPA in their freshman year (“Fresh”). A computer regression analysis of these data is given below. Assume the conditions for regression inference have been satisfied. Predictor Constant Fresh S = 0.355761 T P 3.06 0.009 2.96 0.011 R-Sqg(adj) = Chapter 12: More About Regression 211

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15. Use Scenario 12-4. Which of the following is a correct interpretation of s = 0.355761? A) The average distance between the Freshman GPA for each student and the mean Freshman GPA for all the students in this sample is about 0.355761. B) The average distance between each senior’s GPA and the mean Senior GPA for all the students in this sample is about 0.355761. C) Predictions of Senior GPA from Freshman GPA based on this regression model will be off by an average of about 0.355761. D) The sum of the squared deviations between each observed Freshman GPA and the Freshman GPA predicted by the regression equation is 0.355761. g E) The sum of the squared deviations between each observed Senior GPA and the Senior GPA predicted by the regression equation is 0.355761. ANS: @ TOP: Interpret s 16. Use Scenario 12-4. To construct the 90% confidence interval for the true slope of the population regression equation relating Senior GPA to Freshman GPA, how many degrees of freedom does the critical #* have? A)13 B) 14 A C)15 D) 28 E) 30 ANS: @ TOP: Degrees of freedom for critical t* for slope 17. Use Scenario 12-4. Which of the following expressions represents the 90% confidence interval for the true slope of the population regression equation relating Senior GPA to Freshman GPA? A) 0.5304+1.645(0.1789) B) 1.6310+1.645(0.5328) C) 0.5304 £1.761(0.1789) D) 1.6310+1.771(0.5328) E) 0.5304+1.771(0.1789) ANS: @ TOP: Confidence interval from computer output Use the following for questions 18 — 20: Scenario 12-5 Can we predict annual household electricity costs in a specific region from the number of rooms in the house? Below is computer output for a regression of annual electricity costs (in dollars) on number of rooms for 30 randomly-selected houses in Michigan. Assume the conditions for regression inference have been met. Predictor Coef SE Coef T P Constant 406.9 164.8 2.47 0.020 Rooms 58.45 24.77 2.36 0.0206 S = 246.735 R-Sq = 16.6% R-Sg{adj) = 13.6% 212 Chapter 12: More About Regression

18. Use Scenario 12-5. Based on these data, the margin of error for a 95% confidence interval for the amount of increase in annual electricity costs for a one-room increase in house size is given by: A)24.77 B) 164.8 C) 1.96(164.8) 24.77 s ¢ E) 2.048(24.77) ANS: ® TOP: Margin of error from computer output 19. Use Scenario 12-5. Which of the following represents the proportion of variation in annual electricity costs that is accounted for by the regression of annual electricity costs on number of rooms? A) 0.020 B) 0.026 C) 0.166 ANS: @ TOP: Interpret r-squared 20. Use Scenario 12-5. If we test the hypotheses H,, : # =0 versus H_: ## 0 , what is the value of the test statistic? A) 0.026 B)2.36 C)2.47 D) 24.77 E) 58.45 ANS: ® TOP: Identify test statistic from computer output Use the following for questions 21 — 23: 4 Scenario 12-6 Are high school students who like their English class more likely to enjoy their history class as well? Here is a regression analysis and residual plot for 30 randomly-selected students who were asked to rate how much they liked both English and history on a 0 to 5 scale (a higher rating mean the student liked the subject more). [Data from 2004-5 Census at Schools survey in Canadal. Predictor Coef 8E Coef T P Constant 1.1867 0.5574 2.13 0.042 English 0.5254 0.1995 2.63 (0.014 9 S =1.37707 R-Sq = 19.8% R-Sqg(adj) = 17.0% Assume the conditions for regression inference have been satisfied. Chapter 12: More About Regression 213

21. Use Scenario 12-6. Which of the following is the estimate from this sample of the standard deviation of the sampling distribution of slopes? A)0.014 B) 0.042 C) 0.1995 D) 0.5574 E) 1.37707 ANS: @ TOP: 1dentify SE of slope on computer output 22. Use Scenario 12-6. What does the quantity R-Sq = 19.8% represent? A) The correlation of history rating and English rating—a measure of the strength of the 1¥ear relationship between the two variables. _ B) The average deviation of observed history ratings from the predicted history ratings, expressed as a percentage of the predicted history rating. C) The average deviation of observed English ratings from the predicted English ratings, expressed as a percentage of the predicted English rating. D) The percentage of variation in history rating that can be explained by the regression of history rating on English rating. E) The percentage of variation in English rating that can be explained by the regression of English rating on history rating. ANS: ® TOP: Interpret r-squared 23. Use Scenario 12-6. Which of the following is an appropriate conclusion for the test of the Rypotheses H,:pB=0versusH,: [ +#0 at the o. = 0.05 level in this context? A) Fail to reject I): The data provide convincing evidence that there is a linear relationship between English rating and history rating. B) Reject Hy. The data provide convincing evidence that there is a linear relationship between English rating and history rating. C) Accept H,. The data provide convincing evidence that there is a linear relationship between English rating and history rating. D) Reject Hy. The data do not provide convincing evidence that there is a linear relationship between English rating and history rating. ¥ E) Accept H,. The data do not provide convincing evidence that there is a linear relationship between English rating and history rating. ANS: ® TOP: Conclusion of significance test from computer output 24. Suppose we measure a response variable Y for several values of an explanatory variable X. A scatterplot of log Y versus log X looks approximately like a negatively-sloping straight line. We may conclude that A) the rate of growth of Y is positive, but slowing down over time. B) an exponential growth model would approximately describe the relationship between Y and X. C) a power model would approximately describe the relationship between Y and X. D) the relationship between Y and X is a positively-sloping straight line. E) the residual plot of the regression of log ¥ on log X would have a “U-shaped” pattern suggesting a non-linear relationship. ANS: @ TOP: Interpreting log-log scatterplot 214 Chapter 12: More About Regression

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25. Using least-squares regression on data from 1990 through 2009, I determine that the (base 10) logarithm of the population of a country is approximately described by the equation @opulation =—-13.5+ 0.0l(Year) . Which of the following is the predicted population of the country in the year 2010? A)6.6 B) 735 C) 2,000,000 D) 3,981,072 E) 33,000,000 ANS: ® TOP: Prediction from log Y vs. X regression 26. Which of the following would provide evidence that a power law model describes the relationship between a response variable y and an explanatory variable x? A) A normal probability plot of the residuals of the regression of log y versus log x looks approximately linear. B) A normal probability plot of the residuals of the regression of log y versus x looks approximately linear. C) A scatterplot of log y versus x looks approximately linear. D) A scatterplot of y versus log x looks approximately linear. E) A scatterplot of log y versus log x looks approximately linear. ANS: @ TOP: Power functions and transformations 27. Suppose the relationship between a response variable y and an explanatory variable x is modeled well by the equation y = 3.6(0.32)x . Which of the following plots is most likely to be roughly linear? A) A plot of y against x. B) A plot of y against log x. C) A plot of log y against x. D) A plot of 10" against x. ? E) A plot of log y against log x. ANS: @ TOP: Exponential functions and transformations Use the following for questions 28 — 29 Scenario 12-7 Like most animals, small marine crustaceans are not able to digest all the food they eat. Moreover, the percentage of food eaten that is assimilated (that is, digested) decreases as the amount of food eaten increases. A scatterplot of this relationship for a certain species of crustacean (at right) indicates that it is non-linear. However, a scatterplot of In Assimilation versus In Food Intake is strongly linear. Below is a computer regression analysis of the transformed data (note that natural logarithms are used). & Predictor Coef SE Coef T P Constant 6.3324 0.5218 12.14 0.000 1n Food Intake -0.6513 0.1047 -6.22 0.000 S = 0.247460 R-Sq = 84.7% R-Sq(adj) = 82.5% Chapter 12: More About Regression 215

Scatterplot of assimilation vs, food intake 60 e =] 2504 ~ & ’ = 30 by ;g 20+ P 4 . 'z o] .o oy i 2 10— 0 30 100 150 200 250 360 350 Food intake: (micrograms per day) 28. Use Scenario 12-7. Which of the following best describe the model that is given by this computer printout? A) A power model with equation bn-Alssimilation = 6.3324 —0.6513 (ln Food Intake) c B) An exponential model with equation In—Assimilation = 6.3324 -0.6513 (ln Food Intake) C) A power model with equation brFood Intake = —0.6513 + 6.3324(ln Assimilation) D) An exponential model with equation InFbod Intake = —0.6513 +6.3324 (ln Assimilation) E) A power model with equation In-Alssimilation = —0.6513 +6.3324 (In Food Intake) ANS: @ TOP: Model/Equation for log-log transformation 29. Use Scenario 12-7. If, as described above, the scatterplot of In Assimilation versus In FoodIntake is strongly linear, which of the following best describes the residual plot for the regression of these two variables? A) A roughly straight line. B) A “U-shaped” pattern, with positive residuals for low and high values of food intake and negative residuals in between. C) A “U-shaped” pattern, with negative residuals for low and high values of food intake and positive residuals in between. D) A curved pattern similar to the scatterplot of the variables Food Intake and Assimilation before the logarithmic transformation. - - E) A random scattering of points on either side of the line whose equation is residuals = 0. ANS: @ TOP: Residuals for good linear fit 216 Chapter 12: More About Regression

Use the following for questions 30 -31 Scenario 12-8 Use of the Internet worldwide increased steadily from 1990 to 2002. A scatterplot of this growth (at right) shows a strongly non-linear pattern. However, a scatterplot of In Internet Users versus Year is much closer to linear. Below is a computer regression analysis of the transformed data (note that natural logarithms are used). Predictor Coef SE Coef T P Constant -951.10 43.45 -21.89 0.000 Year 0.4785 0.02176 21.99 0.000 5 = 0.2516 R-Sg = 98.2% R-Sg(adj) = 98.0% 30. Internet users (millions) Scatterplot of internet _ users vs, year 700 . 600+ ’.'v-‘ Sm ™~ * S 400 + . = 300+ » - 200 ~ . 100 ~ - . 0~ « ¢ * L A A ! 1990 19921994 1996 1998 2000 2002 Year Use Scenario 12-8. Which of the following best describe the model that is given by this computer printout? 31 A) A power model with equation In-frlternet Users = -951.10+0.4785(In Year) ¢ B) A power model with equation kntelnet Users = ~951.10+0.4785 (ln Year) C) A power model with equation bn-trternet Users = —951 .10+ 0.4785 (Year) D) An exponential model with equation In-tdternet Users = -951.10+0.4785 (Year) E) An exponential model with equation fn-tdternet Users = -951.10 +'O.4785( In Year) ANS: ® TOP: Model/Equation for semi-log transformation . Use Scenario 12-8. If| as described above, the scatterplot of In Internet Users versus Year is strongly linear, which of the following best describes the residual plot for the regression of these two variables? A) A “U-shaped” pattern, with positive residuals for early and late years and negative residuals in between. B) A “U-shaped” pattern, with negative residuals for for early and late years and positive residuals in between. C) A random scattering of points on either side of the line whose equation is residuals = 0. D) A curved pattern similar to the scatterplot of the variables Internet users and Year before the logarithmic transformation. E) A roughly straight line. ANS: @ TOP: Residuals for good linear fit Chapter 12: More About Regression 2]7

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Use the following for questions 32 — 33: Scenario 12-9 Like most animals, small marine crustaceans are not able to digest all the food they eat. Moreover, the percentage of food eaten that is assimilated (that is, digested) decreases as the amount of food eaten increases. A residual plot for the regression of Assimilation rate (as a percentage of food intake) on Food Intake (in wg/day) is shown below. Residual plot for assimilation -vs, food intake Residual frame T T T T 7 0 50 100 150 200 250 300 350 Food intake 32. Use Scenario 12-9. Suppose we use the regression whose residuals are shown here to predict assimilation when food intake is 50 ug/day. Which of the following best describes the accuracy of that prediction. A) The prediction would probably be accurate. B) The prediction would probably overestimate the true assimilation rate. C) The prediction would probably underestimate the true assimilation rate. D) We do not have enough information determine the accuracy of the estimate. E) Whether the prediction underestimated or overestimated would depend on sampling variability, so that each type of error is equally likely. ANS: @ TOP: Accuracy of estimate from residuals 33. Use Scenario 12-9. A scatterplot of In Assimilation versus In Food Intake is strongly linear, suggesting that a linear regression of these transformed variables may be more appropriate. Below is a computer regression analysis of the transformed data (note that natural logarithms are used). Predictor Coef SE Coef T P Constant 6.33214 0.5218 12.14 0.000 1n Food Intake -0.6513 0.1047 -6.22 0.000 S = 0.247460 R-Sq = 84.7% R-Sq(adj) = 82.5% When food intake is 250 ug/day , what is the predicted assimilation rate from this model? A)2.7% B) 15.4% C)27.4% D) 34.3% E) 54.4% ANS: ® TOP: Prediction from log Y vs. log X regression 218 Chapter 12: More About Regression

S A e i, Use the following for 34 — 35: Scenario 12-10 Use of the internet worldwide increased steadily from 1990 to 2002. A residual plot for the regression of worldwide Internet Users (in millions) on Year is shown below. Residual plot for internet HISCTS Y8, year 200 4 150 100 — 50 - . 0 = . -50 -4 . . * w1 EK) . * e T 1 | 1 T 1990 1992 1994 1996 1998 2000 2002 Year Residual 34. Use Scenario 12-10. Suppose we use the regression whose residuals are shown here to predict the number of Internet users in 1991. Which of the followin g best describes the accuracy of that prediction? A) The prediction would probably underestimate the true number of Internet users in 1991. B) The prediction would probably overestimate the true number of Internet users in 1991 C) The prediction would probably be accurate. D) We do not have enough information determine the accuracy of the estimate. E) Since the sample is subject to random variable, the estimate will underestimate and overestimate with about the same frequency. ANS: @ TOP: Accuracy of estimate from residuals 35. Use Scenario 12-10. A scatterplot of In Internet Users (in millions) versus Year is strongly linear, suggesting that a linear regression of this transformation may be more appropriate. Below is a computer regression analysis of the transformed data (note that natural logarithms are used). Predictor Coef SE Coef T P Constant -951.10 43.45 -21.89 0.000 Year 0.4785 0.02176 21.99 0.000 S = 0.2516 R-Sg = 98.2% R-Sg{adj) = 98.0% What is the predicted number of Internet users (in millions) in 1991, based on this model? A) 1.59 B) 4.46 C)4.92 D) 38.90 E) 86.77 ANS: @ TOP: Prediction from log Y vs. X regression Chapter 12: More About Regression 219

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