ECO 391 Exam 3 - Spring 2020 (Past Paper)

***Last Name: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_*** **First Name: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_** **ECO 391 (Spring 2020)** **Exam 3 -- Form A (White)** **Directions:** - **You have two hours to complete the exam and it is out of 100 points.** - **Multiple choice 1-50 are worth 2 points per question.** **Good Luck!!!** 1. On the basis of sample information, we either \"accept the null hypothesis\" or \"reject the null hypothesis.\" a\. True b\. False 2. Consider the regression model *Y* = *β*0 + *β*1*X* + *β*2*D* + *β*3*XD* + *ε*. If the dummy variable *D* changes from 0 to 1, the estimated changes in the intercept and the slope are *β*0 + *β*2 and *β*2 respectively. a\. True b\. False 3. For the model ln(*y*) = *β*0 + *β*1*x* + *ε, β*1 is the approximate percentage change in *y* when *x* increases by 1%. a\. True b\. False 4. For the model *y* = *β*0 + *β*1ln(*x*) + *ε, β*1 is the approximate percentage change in *y* when *x* increases by 1%. a\. True b\. False 5. The term *multicollinearity* refers to the condition when the variance of the error term, of *x*1, *x*2,..., *xn*, is the same for all observations. a\. True b\. False 6. When some explanatory variables of a regression model are strongly correlated, this phenomenon is called autocorrelation. b\. False 7. A multiple regression model has the form Y = 7 + 2 X~1~ + 9 X~2~ As X~1~ increases by 1 unit (holding X~2~ constant), Y is expected to \_\_\_\_\_. a\. increase by 9 units b\. decrease by 9 units c\. increase by 2 units d\. decrease by 2 units 8. A machine that is programmed to package 1.202 pounds of cereal is being tested for its accuracy in a sample of 36 cereal boxes, the sample mean filling weight is calculated as 1.22 pounds. The population standard deviation is known to be 0.06 pounds. Which of the following conclusions is correct? Assume a CI of 95%. a\. The machine is operating improperly because the target is below the lower limit. b\. The machine is operating properly because the interval contains the target. c\. The machine is operating improperly because the target is above the upper limit. d\. There is not enough information to make a conclusion. Use the following scenario to answer questions 9 to 13 below: 9. The estimated regression equation is \_\_\_\_\_. a\. Y = β~0~ + β~1~X~1~ + β~2~X~2~ + ε b\. E(y) = β~0~ + β~1~X~1~ + β~2~X~2~ c\. [*Ŷ*]{.math.inline} = 12.924 - 3.682 X~1~+ 45.216 X~2~ d\. [*Ŷ*]{.math.inline} = 4.425 + 2.63 X~1~ + 12.56 X~2~ 10. The interpretation of the coefficient of X~1~ is that \_\_\_\_\_. a\. a one-unit change in X~1~ will lead to a 3.682-unit decrease in y b\. a one-unit increase in X~1~ will lead to a 3.682-unit decrease in y when all other variables are held constant c\. a one-unit increase in X~1~ will lead to a 3.682-unit decrease in X~2~ when all other variables are held constant d\. It is impossible to interpret the coefficient. 11. We want to test whether the parameter β~1~ is significant. The test statistic equals \_\_\_\_\_. 12. Carry out the test of significance for the parameter β~1~ at the 1% level. The null hypothesis should \_\_\_\_\_. 13. Refer to below regression results. ------------ ------ -------- ------- ----- ------------------ *df* *SS* *MS* *F* *Significance F* Regression 2 3,500 1,750 0.1000 Residual 20 13,500 675 Total 22 17,000 ------------ ------ -------- ------- ----- ------------------ When testing the overall significance of the regression model at the 5%, the decision is to \_\_\_\_\_\_\_\_. 14. The number of dummy variables representing a qualitative variable should be \_\_\_\_\_\_. a\. one less than the number of categories of the variable b\. two less than the number of categories of the variable c\. the same number as the number of categories of the variable d\. None of these choices is correct. 15\. When estimating = *β*0 + *β*1*x*1 + *β*2*x*2 , the following regression results were obtained. ------------ ---------------- ------------------ ---------- ----------- *df* *SS* *MS* *F* Regression 2 210.9 105.5 114.7 Residual 17 15.6 0.92 Total 19 226.5 *Coefficients* *Standard Error* *t-stat* *p-value* Intercept −1.6 0.57 −2.77 0.0132 \* x*1 −0.5 0.04 −15.11 2.77E-11 \* x*2 0.1 0.07 1.89 0.0753 ------------ ---------------- ------------------ ---------- ----------- Which of the following is the adjusted *R*2? a\. 0.82 b\. 0.86 c\. 0.92 d\. 0.96 16\. In an examination of purchasing patterns of shoppers, a sample of 16 shoppers revealed that they spent, on average, \$54 per hour of shopping. Based on previous years, the population standard deviation is thought to be \$21 per hour of shopping. Assuming that the amount spent per hour of shopping is normally distributed, find a 90% confidence interval for the mean amount. a\. \[\$45.36, \$62.64\] b\. \[\$47.27, \$60.73\] c\. \[\$51.84, \$56.16\] d\. \[\$52.32, \$55.68\] 17\. As the sample size increases, the margin of error \_\_\_\_\_. a\. increases b\. decreases c\. stays the same d\. None of the answers is correct. 18\. Professors at a local university earn an average salary of \$80,000 with a standard deviation of \$6,000. The salary distribution is approximately bell-shaped. What can be said about the percentage of salaries that are less than \$68,000 or more than \$92,000? a\. It is about 5%. b\. It is about 32%. c\. It is about 68%. d\. It is about 95%. Use the following scenario to answer questions 22 and 23 below: b\. E(y) = β~0~ + β~1~X~1~ d\. E(y) = β~0~ + β~1~X~1~ + β~2~X~2~ 19.Which equation describes the multiple regression model? a\. equation a b\. equation b 20\. Which equation gives the estimated regression line? a\. equation a b\. equation b c\. equation c d\. equation d 21\. Consider the following competing hypotheses: *H*0: *μ* = 0, *HA*: *μ* ≠ 0. The value of the test statistic is *Z* = −1.38. If we choose a 5% significance level, then we \_\_\_\_\_\_\_\_. 22\. What is the decision rule when using the *p*-value approach to hypothesis testing? a\. Reject *H*0 if the *p*-value \> *α.* b*.* Reject *H*0 if the *p*-value \< *α.* c*.* Do not reject *H*0 if the *p*-value \< 1 -- *α.* d*.* Do not reject *H*0 if the *p*-value \> 1 -- *α.* 23\. For the model ln(*y*) = *β*0 + *β*1*x* + *ε*, if *x* increases by one unit, then *Y* changes by approximately a\. *β*1 × 100% b\. *β*1 × 100 units c\. *β*1% d\. *β*1 units 24\. The adjusted multiple coefficient of determination (the adjusted R^2^ ([\${\\overline{R}}\^{2}\$]{.math.inline})) is adjusted for \_\_\_\_\_. a\. the number of dependent variables b\. the number of independent variables c\. the number of equations d\. detrimental situations 25\. A small stock brokerage firm wants to determine the average daily sales (in dollars) of stocks to their clients. A sample of the sales for 36 days revealed average daily sales of \$200,000. Assume that the standard deviation of the population is known to be \$18,000. Provide a 95% confidence interval estimate for the average daily sale. a\. It has a lower bound of \$194,120 and an upper bound of \$206,510 b\. It has a lower bound of \$193,490 and an upper bound of \$205,880 c\. It has a lower bound of \$194,120 and an upper bound of \$206,510 d\. It has a lower bound of \$194,120 and an upper bound of \$205,880 Use the following scenario to answer questions 29 to 31 below: The following estimated regression model was developed relating yearly income (y in \$1000s) of 30 individuals with their age (X~1~) and their gender (X~2~) (0 if male and 1 if female). [*Ŷ*]{.math.inline} = 30 + 0.7 X~1~ + 3 X~2~ 26\. From the above function, it can be said that the expected yearly income for \_\_\_\_\_. a\. male is \$3 more than females b\. female is \$3 more than males c\. male is \$3,000 more than females d\. female is \$3,000 more than males 27\. The yearly income of a 24-year-old female individual is \_\_\_\_\_. b\. \$19,800 d\. \$49,800 28\. The yearly income of a 24-year-old male individual is \_\_\_\_\_. a\. \$13.80 29\. Professors at a local university earn an average salary of \$80,000 with a standard deviation of \$6,000. With the beginning of the next academic year, all professors will get a 2% raise. What will be the average and the standard deviation of their new salaries? a\. \$80,000 and \$6,120. b\. \$81,600 and \$6,000. c\. \$81,600 and \$6,120. d\. \$82,000 and \$6,200. Use the following scenario to answer questions 30 and 31 below: The average monthly electric bill of a random sample of 256 residents of a city is \$90. The population standard deviation is assumed to be \$24. 30\. Construct a 90% confidence interval for the mean monthly electric bills of all residents. 31\. Construct a 95% confidence interval for the mean monthly electric bills of all residents. 32\. If the *p*-value for a hypothesis test is 0.027 and the chosen level of significance is *α* = 0.05, then the correct conclusion is to \_\_\_\_\_\_\_\_. b\. not reject the null hypothesis c\. reject the null hypothesis if σ = 10 33\. In general, higher confidence levels provide \_\_\_\_\_. a\. wider confidence intervals b\. narrower confidence intervals c\. a smaller standard error d\. unbiased estimates 34\. Multicollinearity is suspected when \_\_\_\_\_\_\_\_. b\. there is a high *R*2 coupled with significant explanatory variables c\. there is a low *R*2 coupled with insignificant explanatory variables d\. there is a high *R*2 coupled with insignificant explanatory variables 35\. Given the following portion of regression results, which of the following conclusions is true with regard to the *F* test at the 5% significance level? ------------ ------ -------- ------- ------ ------------------ *df* *SS* *MS* *F* *Significance F* Regression 2 7,562 3,781 8.04 0.0028 Residual 20 9,395 470 Total 22 16,957 ------------ ------ -------- ------- ------ ------------------ a\. Neither of the explanatory variables is related to the response variable. b\. Both of the explanatory variables are related to the response variable. c\. Exactly one of the explanatory variables is related to the response variable. 36\. The following data show the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup. The brewing pot temperature is approximately 180º F. ------------ ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- Time (min) 0 5 8 11 15 18 22 25 30 34 38 42 45 50 Temp (ºF) 179.5 168.7 158.1 149.2 141.7 134.6 125.4 123.5 116.3 113.2 109.1 105.7 102.2 100.5 ------------ ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- The output for an exponential model, ln(*Temp*) = *β*0 + *β*1*Time* + *ε*, is below. ------------ ---------------- ------------------ ---------- *df* *SS* *MS* Regression 1 0.45476 0.45476 Residual 12 0.01400 0.00117 Total 13 0.46876 *Coefficients* *Standard Error* *t-stat* Intercept 5.1444 0.0173 297.74 *Time* −0.0118 0.0006 −19.74 ------------ ---------------- ------------------ ---------- During one minute, the predicted temperature decreases by approximately \_\_\_\_\_\_\_\_. a\. 0.0118° F b\. 1.18° F c\. 1.18% d\. 11.8% 37\. In a multiple regression model, the values of the error term, ε, are assumed to be \_\_\_\_\_. c\. independent of each other 38\. For a multiple regression model, SSE = 600 and SSR = 200. The coefficient of determination (R^2^) is \_\_\_\_\_. 39\. In an accounting class of 200 students, the mean and standard deviation of scores was 70 and 5, respectively. Use the empirical rule to determine the number of students who scored less than 65 or more than 75. a\. It is about 32. b\. It is about 64. c\. It is about 68. d\. It is about 136. 40\. Given the following portion of regression results, which of the following is the value of the *F*(2,20) test statistic? ------------ ------ -------- ------- ----- ------------------ *df* *SS* *MS* *F* *Significance F* Regression 2 3,500 1,750 0.1000 Residual 20 13,500 675 Total 22 17,000 ------------ ------ -------- ------- ----- ------------------ a\. 1.96 b\. 2.59 c\. 3.49 d\. 10 41\. If the variance of the error term is not the same for all observations, we \_\_\_\_\_\_\_\_. a\. get coefficient estimates that are biased b\. can perform tests of significance (t-tests) c\. cannot conduct tests of significance (t-tests) 42\. A real estate analyst believes that the three main factors that influence an apartment\'s rent in a college town are the number of bedrooms, the number of bathrooms, and the apartment\'s square footage. For 40 apartments, she collects data on the rent (*y*, in \$), the number of bedrooms (*x*1), the number of bathrooms (*x*2), and its square footage (*x*3). She estimates the following model: *Rent* = *β*0 + *β*1*Bedroom* + *β*2*Bath* + *β*3*Sqft* + *ε*. The following table shows a portion of the regression results. ------------ ---------------- ------------------ ----------- ----------- ------------------ ------------- *df* *SS* *MS* *F* *Significance F* Regression 3 5,694,717 1,898,239 50.88 0.000 Residual 36 1,343,176 37,310 Total 39 7,037,893 *Coefficients* *Standard Error* *t-stat* *p-value* *Lower 95%* *Upper 95%* Intercept 300 84.0 3.57 0.0010 130.03 470.79 *Bedroom* 226 60.3 3.75 0.0006 103.45 348.17 *Bath* 89 55.9 1.59 0.1195 −24.24 202.77 *Sq ft* 0.2 0.09 2.22 0.0276 0.024 0.39 ------------ ---------------- ------------------ ----------- ----------- ------------------ ------------- When testing whether the explanatory variables are jointly significant at the 5% level, she \_\_\_\_\_\_\_\_. 43\. The Department of Education would like to test the hypothesis that the average debt load of graduating students with a bachelor\'s degree is equal to \$17,000. A random sample of 34 students had an average debt load of \$18,200. It is believed that the population standard deviation for student debt load is \$4,200. The α is set to 0.05. The *p*-value for this hypothesis test would be \_\_\_\_\_\_\_\_. a\. 0.1310 b\. 0.0957 c\. 0.0475 d\. 0.0219 44\. A variable that cannot be measured in terms of how much or how many but instead is assigned values to represent categories is called a(n) \_\_\_\_\_. b\. constant variable 45\. A manager at a local bank analyzed the relationship between monthly salary (*y*, in \$) and length of service (*x*, measured in months) for 30 employees. She estimates the model: *Salary* = *β*0 + *β*1 *Service* + *ε*. The following table summarizes a portion of the regression results. ------------ ---------------- ------------------ ---------- ----------- *df* *SS* *MS* *F* Regression 1 555,420 555,420 7.64 Residual 27 1,962,873 72,699 Total 28 2,518,293 *Coefficients* *Standard Error* *t-stat* *p-value* Intercept 784.92 322.25 2.44 0.02 *Service* 9.19 3.20 2.87 0.01 ------------ ---------------- ------------------ ---------- ----------- How much of the variation in *Salary* is unexplained by the sample regression equation? a\. 1% b\. 22.06% c\. 18.39% d\. 77.94% 46\. When confronted with multicollinearity, a good remedy is to \_\_\_\_\_\_\_\_ if we can justify its redundancy. a\. add one more collinear variable b\. drop one of the collinear variables c\. remove both the collinear variables d\. add as many collinear variables as possible 47\. As the goodness of fit for the estimated multiple regression equation increases, \_\_\_\_\_. a\. the value of the adjusted multiple coefficient of determination decreases 48. In a multiple regression analysis involving 15 independent variables and 200 observations, SST = 800 and SSE = 240 (Sum of Squares of the Errors). The coefficient of determination is \_\_\_\_\_\_. a\. 0.300 b\. 0.192 c\. 0.500 d\. 0.700 Use the following scenario to answer the questions below: A regression model between sales (y in \$1000s), unit price (X~1~ in dollars) and television advertisement (X~2~ in dollars) resulted in the following function: Y= 7 -- 3X~1~ + 5X~2~ 49\. The coefficient of X~2~ indicates that if television advertising is increased by \$1 (holding the unit price constant), sales are expected to \_\_\_\_\_. 50\. How can autocorrelation occur? The Z-table (Right tail test) ============================= **Answer Key:** 1. **False --** Based on sample information, we either \"reject the null hypothesis\" or \"fail to reject the null hypothesis.\" Rejecting the null hypothesis does not mean that we accept the alternative hypothesis as true. 2. **False \--** The estimated regression for this model is ![](media/image4.jpeg) = *b*0 + *b*1*x* + *b*2*d* + *b*3*xd*. That estimated regression reduces to = *b*0 + *b*1*x* for *d* = 0, and it reduces to ![](media/image4.jpeg) = (*b*0 + *b*2) + (*b*1 + *b*3)*x* for *d* = 1. Thus, if *d* changes from 0 to 1, the intercept changes by *b*2 (from *b*0 to *b*0 + *b*2), while the slope changes by *b*3 (from *b*1 to *b*1 + *b*3). 3. **False** -- The correct interpretation of [*β*~1~]{.math.inline}is: If we increase x~1~ by 1 unit, we would expect our y variable to change by [100 \* *β*~1~]{.math.inline} percent. 4. **False --** The correct interpretation of [*β*~1~]{.math.inline}is: If we increase x~1~ by one percent, we would expect y to increase by [\$\\frac{\\beta\_{1}}{100}\$]{.math.inline} percent. 5. 6. 7. **C --** The coefficient of X~1~ is 2. Holding all else constant, increasing X~1~ by 1 will correlate with a 2-unit increase in Y. 8. **B --** The formula to compute a confidence interval is [\$\\overline{x} \\pm \\ z\_{\\alpha/2}\\frac{\\sigma}{\\sqrt{n}}\$]{.math.inline}. First, find the *z-*score for α/2. For a 95% confidence interval, α=0.05, so look for 0.025 on the negative *z-*table (or 0.975 on the positive side); that gives a *z-*score of -1.96), so [*z*~*α*/2~]{.math.inline} = -1.645. The lower bound for the 95% confidence interval will be [\$1.22 - \\left( 1.96 \\right)\\frac{0.06}{6} = 1.2004\$]{.math.inline} and the lower bound will be [\$1.22 + \\left( 1.96 \\right)\\frac{0.06}{6} = 1.2396\$]{.math.inline}. Because the target weight of 1.202 is between the upper and lower bound, we can say that the machine is operating properly. 9. **C --** The estimated regression line is represented by [*Ŷ*]{.math.inline}. Answer C correctly uses the model's coefficients in the estimate, while answer D uses the standard errors. 10. **B --** Holding all else constant, a one-unit increase in X~1~ will correspond to an increase in Y equal to the coefficient on X~1~. 11. **A --** The significance of a regression coefficient is determined by dividing the coefficient by the standard error. -3.682/2.63 = -1.4 12. **B --** Using the *z-*table, the probability of *z* ≤ -1.4 is 0.0808. That's higher than the critical value of 0.01, so we cannot reject the null hypothesis that **β~1\ =~** 0. 13. 14. **A --** To avoid the dummy variable trap, the number of dummy variables must be one less than the number of categories of the variable. For example, a model with dummy variables representing the four seasons would include *Spring*, *Summer*, and *Fall* as dummy variables; for winter observations, all three of the dummy variables would equal zero. 15. **C --** The R^2^ for this model is SSR/SST = 210.9/226.5 = 0.9311. The adjusted R^2^ is found using the formula: Adjusted R^2^ = 1 - (1 - *R*^2^)[\$\\lbrack\\frac{n - 1}{n - k - 1}\\rbrack\$]{.math.inline}. (1-R^2^) = 0.0689, *n* = 17, and *k* = 2 (the number of independent variables), so Adjusted R^2^ is 0.92. 16. **A --** The formula to compute a confidence interval is [\$\\overline{x} \\pm \\ z\_{\\alpha/2}\\frac{\\sigma}{\\sqrt{n}}\$]{.math.inline}. First, find the *z-*score for α/2. For a 90% confidence interval, α=0.10, so look for 0.05 on the negative *z-*table (or 0.95 on the positive side); that gives a *z-*score of between -1.64 (0.0505) and -1.65 (0.0495), so [*z*~*α*/2~]{.math.inline} = -1.645. The lower bound for the 90% confidence interval will be [\$\\\$ 54 - \\left( 1.645 \\right)\\frac{21}{4} = \\\$ 45.36\$]{.math.inline} and the lower bound will be [\$\\\$ 54 + \\left( 1.645 \\right)\\frac{21}{4} = \\\$ 62.64\$]{.math.inline}. 17. **B** -- A confidence interval is found using the equation [\$\\overline{x} \\pm \\ z\_{\\alpha/2}\\frac{\\sigma}{\\sqrt{n}}\$]{.math.inline}. The margin of error refers to how far the confidence interval extends on either side of the mean. Because the range of the confidence interval becomes smaller as the *n* (the sample size) increases, the margin of error also decreases as the sample size increases. 18. **A --** A salary less than \$68,000 is more than two standard deviations below the mean, while a salary more than \$92,000 is more than two standard deviations above the mean. Using the empirical rule, 95% of observations are expected to fall within two standard deviations of the mean (68% within one standard deviation, plus 27% between one and two). That leaves 5% of observations at least two standard deviations away. 19. **A --** A multiple regression model includes more than one X variable; the model will also include an error term. 20. **C --** The estimated outcome of the model, given values of X~1~ and X~2~, is represented by [*Ŷ*]{.math.inline}. 21. **D** -- Using the *z-*table, we see that the probability of *Z* ≤ -1.38 is 0.0838. That's higher than the critical value of 0.025 (half of 5%, since *H~A~*: *μ* ≠ 0 means we're testing against both tails of the distribution). Therefore, we cannot reject the null hypothesis that the mean equals zero. 22. **B --** The decision rule is to reject the null hypothesis if the *p*-value \

ECO 391 Exam 3 - Spring 2020 (Past Paper)

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