8-17 Consider the following time series. t 1 2 3 4 5 yt 6 11 9 14 15 (a) Choose the correct time series plot. (i) (ii) (iii) (iv) What type of pattern exists in the data? (b) Use simple linear regression analysis to find the parameters for the line that minimizes MSE for this time series. If required, round your answers to two decimal places. y-intercept, b0 = Slope, b1 = MSE = (c) What is the forecast for t = 6? If required, round your answer to one decimal place. Problem 8-19 Because of high tuition costs at state and private universities, enrollments at community colleges have increased dramatically in recent years. The following data show the enrollment (in thousands) for Jefferson Community College for the nine most recent years. Click on the datafile logo to reference the data. Year Period (t) Enrollment (1,000s) 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 6.5 8.1 8.4 10.2 12.5 13.3 13.7 17.2 18.1 (a) Choose the correct time series plot. (i) (ii) (iii) (iv) What type of pattern exists in the data? (b) Use simple linear regression analysis to find the parameters for the line that minimizes MSE for this time series. If required, round your answers to two decimal places. y-intercept, b0 = Slope, b1 = MSE = (c) What is the forecast for year 10? Round your interim computations and final answer to two decimal places. Problem 8-23 Consider the following time series data. Quarter Year 1 Year 2 Year 3 1 4 6 7 2 2 3 6 3 3 5 6 4 5 7 8 (a Choose the correct time series plot. ) (i (i i iii ) i v What type of pattern exists in the data? b Use a multiple regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data. Qtr1 = 1 if Quarter 1, 0 otherwise; Qtr2 = 1 if Quarter 2, 0 otherwise; Qtr3 = 1 if Quarter 3, 0 otherwise. 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 = + Qtr1t + Qtr2t + Qtr3t c Compute the quarterly forecasts for next year based on the model you developed in part (b). If required, round your answers to three decimal places. Quarter 1 forecast Quarter 2 forecast Quarter 3 forecast Quarter 4 forecast d Use a multiple regression model to develop an equation to account for trend and seasonal effects in the data. Use the dummy variables you developed in part (b) to capture seasonal effects and create a variable t such that t = 1 for Quarter 1 in Year 1, t = 2 for Quarter 2 in Year 1,... t = 12 for Quarter 4 in Year 3. 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 = + Qtr1t + Qtr2t + Qtr3t + t e Compute the quarterly forecasts for next year based on the model you developed in part (d). Round your interim computations and final answers to three decimal places. Quarter 1 forecast Quarter 2 forecast Quarter 3 forecast Quarter 4 forecast (f Is the model you developed in part (b) or the model you developed in part (d) more effective? If required, round your intermediate calculations and final answer to three decimal places. Model developed in part (b) Model developed in part (d) MSE Justify your answer. Problem 8-25 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. Click on the datafile logo to reference the data. 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 a Choose the correct time series plot. ) (i ) (ii ) ii i) (i v) What type of pattern exists in the data? b 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 c Using the equation developed in part (b), compute estimates of the levels of nitrogen dioxide for July 18. ) If required, round your answers to three decimal places. 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 d 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 e Based on the seasonal effects in the data and linear trend estimated in part (d), compute estimates of the levels of ) nitrogen dioxide for July 18. If required, round your answers to three decimal places. 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 f 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 in part (b) Model developed in part (d) MSE Justify your answer. Problem 8-27 Hogs & Dawgs is an ice cream parlor on the border of north-central Louisiana and southern Arkansas that serves 43 flavors of ice creams, sherbets, frozen yoghurts, and sorbets. During the summer Hogs & Dawgs is open from 1:00 P.M. to 10:00 P.M. on Monday through Saturday, and the owner believes that sales change systematically from hour to hour throughout the day. She also believes her sales increase as the outdoor temperature increases. Hourly sales and the outside temperature at the start of each hour for the last week are provided in the DATAfile IceCreamSales. Click on the datafile logo to reference the data. a Choose the correct time series plot of hourly sales. ) i) ii) iii ) iv ) Choose the scatter plot of outdoor temperature and hourly sales. i) ii) iii ) iv ) What types of relationships exist in the data? b Use a simple regression model with outside temperature as the causal variable to develop an equation to account ) for the relationship between outside temperature and hourly sales in the data. 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) Sales = + Temperature Based on this model, compute an estimate of hourly sales for today from 2:00 P.M. to 3:00 P.M. if the temperature at 2:00 P.M. is 93oF. If required, round your answers to two decimal places. $ c Use a multiple linear regression model with the causal variable outside temperature and dummy variables as ) follows to develop an equation to account for both seasonal effects and the relationship between outside temperature and hourly sales in the data in the data: Hour1 = 1 if the sales were recorded between 1:00 P.M. and 2:00 P.M., 0 otherwise; Hour2 = 1 if the sales were recorded between 2:00 P.M. and 3:00 P.M., 0 otherwise; . . . Hour8 = 1 if the sales were recorded between 8:00 P.M. and 9:00 P.M., 0 otherwise. Note that when the values of the 8 dummy variables are equal to 0, the observation corresponds to the 9:00 P.M. to 10:00 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) Sales = + Temperature + Hour1 + Hour2 + Hour3 + Hour4 + Hour5 + Hour6 + Hour7 + Hour8 Based on this model, compute an estimate of hourly sales for today from 2:00 P.M. to 3:00 P.M. if the temperature at 2:00 P.M. is 93oF. If required, round your answers to two decimal places. $ Is the model you developed in part (b) or the model you developed in part (c) more effective? If required, round your answers to three decimal places. Model developed in part (b) MSE Justify your answer. Model developed in part (c)