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The U.S. Department of Energy's Fuel Economy Guide provides fuel efficiency data for cars and trucks. A portion of the data for 311 compact, midsize, and large cars follows. The Class column identifies the size of the car: Compact, Midsize, or Large. The Displacement column shows the engine's displacement in liters. The Fuel Type column shows whether the car uses premium (P) or regular (R) fuel, and the Hwy MPG column shows the fuel efficiency rating for highway driving in terms of miles per gallon. The complete data set is contained in the file FuelData. (Let x1 represent the engine's displacement, x2 represent ClassMidsize, x3 represent ClassLarge, x4 represent FuelPremium, and y represent HwyMPG.) a) Develop an estimated regression equation that can be used to predict the fuel efficiency for highway driving given the engine's displacement. (Round your numerical values to four decimal places.) y = ? Test for significance using the 0.05 level of significance. (Use the t test.) Find the p-value. (Round your answer to four decimal places.) p-value = ? b) How much of the variation in the sample values of HwyMPG (in %) does this estimated regression equation explain? (Round your answers to two decimal places.) ? % c) Now consider the addition of the dummy variables ClassMidsize and ClassLarge to the simple linear regression model in part (a). The value of ClassMidsize is 1 if the car is a midsize car and 0 otherwise; the value of ClassLarge is 1 if the car is a large car and 0 otherwise. Thus, for a compact car, the value of ClassMidsize and the value of ClassLarge are both 0. Develop the estimated regression equation that can be used to predict the fuel efficiency for highway driving, given the engine's displacement and the dummy variables ClassMidsize and ClassLarge. (Round your numerical values to four decimal places.) y = ? How much of the variation in the sample values of HwyMPG (in %) is explained by this estimated regression equation? (Round your answer to two decimal places.) ? % d) Use a significance level of 0.05 to determine whether the dummy variables added to the model in part (c) are significant. Test whether the regression parameter β2 is equal to zero. Find the p-value. (Round your answer to four decimal places.) p-value = ? Test whether the regression parameter β3 is equal to zero. Find the p-value. (Round your answer to four decimal places.) p-value = ? e) Consider the addition of the dummy variable FuelPremium, where the value of FuelPremium is 1 if the car uses premium fuel and 0 if the car uses regular fuel. Develop the estimated regression equation that can be used to predict the fuel efficiency for highway driving given the engine's displacement, the dummy variables ClassMidsize and ClassLarge, and the dummy variable FuelPremium. (Round your numerical values to four decimal places.) y = ? How much of the variation in the sample values of HwyMPG (in %) does this estimated regression equation explain? (Round your answer to two decimal places.) ? % f) For the estimated regression equation developed in part (e), test for the significance of the relationship between each of the independent variables and the dependent variable using the 0.05 level of significance for each test. Test whether the regression parameter β1 is equal to zero. Find the p-value. (Round your answer to four decimal places.) p-value = ? Test whether the regression parameter β2 is equal to zero. Find the p-value. (Round your answer to four decimal places.) p-value = ? Test whether the regression parameter β3 is equal to zero. Find the p-value. (Round your answer to four decimal places.) p-value = ? Test whether the regression parameter β4 is equal to zero. Find the p-value. (Round your answer to four decimal places.) p-value = ? g) An automobile manufacturer is designing a new compact model with a displacement of 3.0 liters with the objective of creating a model that will achieve at least 24 estimated highway MPG. The manufacturer must now decide if the car can be designed to use premium fuel and still achieve the objective of 24 MPG on the highway. Use the model developed in part (e) to recommend a decision to this manufacturer. (Round your answers to one decimal place.) Using the estimated multiple linear regression equation developed in part (e), a new model that is designed to use regular fuel would have ??? MPG on the highway and a new model that is designed to use premium fuel would have ??? MPG on the highway. Therefore, the new model should be designed to use regular fuel.
Sri K.
Consider a multinomial experiment with n = 307 and k = 4. The null hypothesis to be tested is H0: p1 = p2 = p3 = p4 = 0.25. The observed frequencies resulting from the experiment are: Category 1 2 3 4 Frequency 85 58 89 75 a. Choose the appropriate alternative hypothesis. Not all population proportions are equal to 0.25. All population proportions differ from 0.25. b-1. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.) b-2. Find the p-value. p-value < 0.01 0.01 < p-value < 0.025 0.025 < p-value < 0.05 0.05 < p-value < 0.10 c. At the 10% significance level, what is the conclusion to the hypothesis test? Reject H0 since the p-value is less than the significance level. Reject H0 since the p-value is greater than the significance level. Do not reject H0 since the p-value is greater than the significance level. Do not reject H0 since the p-value is less than the significance level. 2. An analyst is trying to determine whether the prices of certain stocks on the NASDAQ are independent of the industry to which they belong. She examines four industries and classifies the stock prices in these industries into one of three categories (high-priced, average-priced, low-priced). Industry Stock Price I II III IV High 18 12 24 24 Average 19 18 20 25 Low 8 8 9 12 a. Choose the competing hypotheses to determine whether stock price depends on the industry. H0: Stock price is independent of the industry.; HA: Stock price is dependent on the industry. H0: Stock price is dependent on the industry.; HA: Stock price is independent of the industry. b-1. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.) b-2. Find the p-value. p-value < 0.01 0.01 < p-value < 0.025 0.025 < p-value < 0.05 0.05 < p-value < 0.10 p-value = 0.10 c. At a 1% significance level, what can the analyst conclude? Do not reject H0; there is not enough evidence to support the claim that the stock price is dependent on the industry. Reject H0; there is enough evidence to support the claim that the stock price is dependent on the industry. Reject H0; there is not enough evidence to support the claim that the stock price is dependent on the industry. Do not reject H0; there is enough evidence to support the claim that the stock price is dependent on the industry. 3. Using 20 observations, the multiple regression model y = β0 + β1x1 + β2x2 + ε was estimated. A portion of the regression results is shown in the accompanying table: df SS MS F Significance F Regression 2 2.10E+12 1.12E+12 63.503 1.30E-08 Residual 17 3.10E+11 1.77E+10 Total 19 2.43E+12 Coefficients Standard Error t Stat p-value Lower 95% Upper 95% Intercept -988,484 130,933 -7.550 0.000 -1,264,728 -712,240 x1 28,503 32,372 0.880 0.391 -39,796 96,802 x2 29,494 33,046 0.893 0.385 -40,227 99,215 a. At the 5% significance level, are the explanatory variables jointly significant? No, since the p-value of the appropriate test is less than 0.05. Yes, since the p-value of the appropriate test is more than 0.05. Yes, since the p-value of the appropriate test is less than 0.05. No, since the p-value of the appropriate test is more than 0.05. b. At the 5% significance level, is each explanatory variable individually significant? Yes, since both p-values of the appropriate test are less than 0.05. Yes, since both p-values of the appropriate test are more than 0.05. No, since both p-values of the appropriate test are not less than 0.05. No, since both p-values of the appropriate test are not more than 0.05. c. What is the likely problem with this model? Multicollinearity since the standard errors are biased. Multicollinearity since the explanatory variables are individually and jointly significant. Multicollinearity since the explanatory variables are individually significant but jointly insignificant. Multicollinearity since the explanatory variables are individually insignificant but jointly significant. 4. The following table lists a portion of Major League Baseball's (MLB's) leading pitchers, each pitcher's salary (In $ millions), and earned run average (ERA) for 2008. Salary ERA J. Santana 17.0 2.31 C. Lee 3.0 2.39 ⋮ ⋮ C. Hamels 0.2 3.00 a-1. Estimate the model: Salary = β0 + β1ERA + ε. (Negative values should be indicated by a minus sign. Enter your answers, in millions, rounded to 2 decimal places.) a-2. Interpret the coefficient of ERA. A one-unit increase in ERA, predicted salary decreases by $2.89 million. A one-unit increase in ERA, predicted salary increases by $2.89 million. A one-unit increase in ERA, predicted salary decreases by $11.48 million. A one-unit increase in ERA, predicted salary increases by $11.48 million. b. Use the estimated model to predict salary for each player, given his ERA. For example, use the sample regression equation to predict the salary for J. Santana with ERA = 2.31. (Round coefficient estimates to at least 4 decimal places and final answers, in millions, to 2 decimal places.) c. Derive the corresponding residuals. (Negative values should be indicated by a minus sign. Round coefficient estimates to at least 4 decimal places and final answers, in millions, to 2 decimal places.)
Sheryl E.
Can the cost of flying a commercial airliner be predicted using regression analysis? If so, what variables are related to this cost? A few of the many variables that can potentially contribute are the type of plane, distance, number of passengers, amount of luggage/freight, weather condition, direction of destination, or even pilot skill. Suppose a study is conducted using only Boeing 737s traveling 800 km on comparable routes during the same season of the year. Can the number of passengers predict the cost of flying such routes? It seems logical that more passengers result in more mass and more baggage, which could, in turn, result in increased fuel consumption and other costs. Suppose the data displayed below are the cost and associated number of passengers for thirty-six 800-km commercial airline flights using Boeing 737s during the same season of the year. We will use these data to develop a regression model to predict cost by the number of passengers. The data contains the data on the cost and number of passengers of 36 observations. Cost Passengers 4.24 88 3.39 95 2.6 88 2.27 66 3.28 87 3.67 88 3.09 81 1.71 60 3.48 86 4.22 93 3.24 80 4.9 96 0.77 62 1.49 61 2.36 69 3.21 76 2.59 74 3.06 86 2.71 80 4.8 98 3.42 91 2.08 59 1.62 71 3.33 84 3.63 89 3.67 92 2.43 75 4.88 92 3.07 85 2.35 74 1.72 73 4.12 90 3.67 73 2.94 73 2.3 77 1.67 69 (a) Use software to estimate this model. Use three decimals in each of your least-squares estimates your answer. Costi = ___ + ____numberofpassengersi (b) Find the coefficient of determination. Express as a percentage, and use two decimal places in your answer. r^2 = (c) Find the standard deviation of the prediction/regression, using three decimals in your answer. Se = (d, i) You wish to test if the data collected supports the statistical model listed above. That is, can the cost of flying an 800-km commercial flight using Boeing 737s be expressed as a linear function of the number of passengers? Select the correct statistical hypotheses which you are to test. A. H0: β0 = 0, HA: β0 < 0 B. H0: β0 = 0, HA: β0 ≠0 C. H0: β1 = 0, HA: β1 > 0 D. H0: β1 = 0, HA: β1 < 0 E. H0: β0 = 0, HA: β0 > 0 F. H0: β1 = 0, HA: β1 ≠0 G. H0: β0 ≠0, HA: β0 ≠0 H. H0: β1 ≠0, HA: β1 ≠0 (d, ii) Use the F-test to test the statistical hypotheses determined in (d, i). Find the value of the test statistic, using two decimals in your answer. Fcalc = (e, iii) Find the P-value of your result in (d, ii). Use three decimals in your answer. P-value = (e, iv) Use the t-test to test the statistical hypotheses determined in (d, i). Find the value of the test statistic, using three decimals in your answer. Tcalc = (e, v) Find the P-value of your t-test in (e, iv). Use three decimals in your answer. P-value = (f) Using an α of 5%, this data indicates that the cost of flying a commercial flight using Boeing 737s cannot be expressed as a linear function of the number of passengers. (g) Find a 95% confidence interval for the slope term of the model, β1. Lower Bound = (use three decimals in your answer) Upper Bound = (use three decimals in your answer) (h) Choose the correct interpretation of the meaning of your confidence interval for β1, in the context of the data. A. As the cost of flying a commercial flight using Boeing 737s increases by 1 unit, the number of passengers increases by an amount that is somewhere between the lower and upper bounds found in (g). B. There is a statistical relationship between the cost of flying a commercial flight using Boeing 737s and the number of passengers. C. As the number of passengers increases by 1, the cost of flying a commercial flight using Boeing 737s increases by an amount that is somewhere between the lower and upper bounds found in (g). D. There is no statistical relationship between the cost of flying a commercial flight using Boeing 737s and the number of passengers. E. As the number of passengers increases by 1, the cost of flying a commercial flight using Boeing 737s will increase, on average, by an amount that is between the lower and upper bounds found in (g). F. As the cost of flying a commercial flight using Boeing 737s increases by 1 unit, the number of passengers will increase, on average, by an amount that is somewhere between the lower and upper bounds found in (g). (i) With 95% confidence, find the average cost of flying a commercial flight using Boeing 737s when the number of passengers is 70. Lower Bound = (use three decimals in your answer) Upper Bound = (use three decimals in your answer)
Dominador T.
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