Air pollution control specialists in southern California monitor the amount of ozone, carbon dioxide, and nitrogen dioxide in the air on an hourly basis. The hourly time series data exhibit seasonality, with the levels of pollutants showing patterns that vary over the hours in the day. On July 15, 16, and 17, the following levels of nitrogen dioxide were observed for the 12 hours from 6:00 A.M. to 6:00 P.M. July 15 25 28 35 50 60 60 40 35 30 25 25 20 July 16 28 30 35 48 60 65 50 40 35 25 20 20 July 17 35 42 45 70 72 75 60 45 40 25 25 25 Use a multiple linear regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data: Hour1 = 1 if the reading was made between 6:00 A.M. and 7:00A.M.; 0 otherwise Hour2 = 1 if the reading was made between 7:00 A.M. and 8:00 A.M.; 0 otherwise . . . Hour11 = 1 if the reading was made between 4:00 .P.M. and 5:00 P.M., 0 otherwise Note that when the values of the 11 dummy variables are equal to 0, the observation corresponds to the 5:00 P.M. to 6:00 P.M. hour. If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300) Value=___+____hour1+_____hour2+______hour3+_______hour4+______hour5+______ hour6+______hour7+______hour8+_______hour9+_____hour10+_______hour11 6:00 a.m. - 7:00 a.m. forecast 7:00 a.m. - 8:00 a.m. forecast 8:00 a.m. - 9:00 a.m. forecast 9:00 a.m. - 10:00 a.m. forecast 10:00 a.m. - 11:00 a.m. forecast 11:00 a.m. - noon forecast noon - 1:00 p.m. forecast 1:00 p.m. - 2:00 p.m. forecast 2:00 p.m. - 3:00 p.m. forecast 3:00 p.m. - 4:00 p.m. forecast 4:00 p.m. - 5:00 p.m. forecast 5:00 p.m. - 6:00 p.m. forecast __________ Let t = 1 to refer to the observation in hour 1 on July 15; t = 2 to refer to the observation in hour 2 of July 15; ...; and t = 36 to refer to the observation in hour 12 of July 17. Using the dummy variables defined in part (b) and ts, develop an equation to account for seasonal effects and any linear trend in the time series. If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300) Value=_______+______hour1+_______hour2+______hour3+________hour4+ _______hour5+_______hour6+_______hour7+_______hour8+_______hour9+ _______hour10+_______hour11+_______t Based upon the seasonal effects in the data and linear trend estimated in part (d), compute estimates of the levels of nitrogen dioxide for July 18. 6:00 a.m. - 7:00 a.m. forecast 7:00 a.m. - 8:00 a.m. forecast 8:00 a.m. - 9:00 a.m. forecast 9:00 a.m. - 10:00 a.m. forecast 10:00 a.m. - 11:00 a.m. forecast 11:00 a.m. - noon forecast noon - 1:00 p.m. forecast 1:00 p.m. - 2:00 p.m. forecast 2:00 p.m. - 3:00 p.m. forecast 3:00 p.m. - 4:00 p.m. forecast 4:00 p.m. - 5:00 p.m. forecast 5:00 p.m. - 6:00 p.m. forecast Is this model you developed in part (b) or the model you developed in part (d) more effective? Model developed in part (d) Model developed in part (b) MSE My solutions: Use a multiple linear regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data: Hour1 = 1 if the reading was made between 6:00 A.M. and 7:00A.M.; 0 otherwise Hour2 = 1 if the reading was made between 7:00 A.M. and 8:00 A.M.; 0 otherwise . . . Hour11 = 1 if the reading was made between 4:00 .P.M. and 5:00 P.M., 0 otherwise Note that when the values of the 11 dummy variables are equal to 0, the observation corresponds to the 5:00 P.M. to 6:00 P.M. hour. If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300) Value=21.7 +7.67 hour1+11.7 hour2+16.7 hour3+34.3 hour4+42.3 hour5+45.0 hour6+28.3 hour7+18.3 hour8+13.3 hour9+3.33 hour10+1.67 hour11 6:00 a.m. - 7:00 a.m. forecast 29.31 7:00 a.m. - 8:00 a.m. forecast 33.4 8:00 a.m. - 9:00 a.m. forecast 38.4 9:00 a.m. - 10:00 a.m. forecast 56 10:00 a.m. - 11:00 a.m. forecast 64 11:00 a.m. - noon forecast 66.7 noon - 1:00 p.m. forecast 50 1:00 p.m. - 2:00 p.m. forecast 40 2:00 p.m. - 3:00 p.m. forecast 35 3:00 p.m. - 4:00 p.m. forecast 25.03 4:00 p.m. - 5:00 p.m. forecast 23.37 5:00 p.m. - 6:00 p.m. forecast ___21.7_______ Let t = 1 to refer to the observation in hour 1 on July 15; t = 2 to refer to the observation in hour 2 of July 15; ...; and t = 36 to refer to the observation in hour 12 of July 17. Using the dummy variables defined in part (b) and ts, develop an equation to account for seasonal effects and any linear trend in the time series. If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300) Value=11.2 +12.5 hour1+16.0 hour2+20.06 hour3+37.8 hour4+45.4 hour5+47.6 hour6+30.5 hour7+20.1 hour8+14.6 hour9+4.21 hour10+2.10 hour11+0.437 t Based upon the seasonal effects in the data and linear trend estimated in part (d), compute estimates of the levels of nitrogen dioxide for July 18. 6:00 a.m. - 7:00 a.m. forecast 40 7:00 a.m. - 8:00 a.m. forecast 44 8:00 a.m. - 9:00 a.m. forecast 49 9:00 a.m. - 10:00 a.m. forecast 66 10:00 a.m. - 11:00 a.m. forecast 75 11:00 a.m. - noon forecast 77 noon - 1:00 p.m. forecast 60 1:00 p.m. - 2:00 p.m. forecast 51 2:00 p.m. - 3:00 p.m. forecast 45 3:00 p.m. - 4:00 p.m. forecast 36 4:00 p.m. - 5:00 p.m. forecast 34 5:00 p.m. - 6:00 p.m. forecast 32 Is the model you developed in part (b) or the model you developed in part (d) more effective? If required, round your answers to three decimal places. Model developed Model developed in part (b) in part (d) MSE Hour 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 Level 25 28 35 50 60 60 40 35 30 25 25 20 28 30 35 48 60 65 50 40 35 25 20 20 35 42 45 70 72 75 60 45 40 25 25 25