Deck 14: Time Series: Descriptive Analyses, Models, and Forecasting Available on CD

ملء الشاشة (f)
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سؤال
(Situation M) Fast food chains are closely watching what proposed legislation will do to consumption of "huge meals" in the United States. Researchers have accumulated statistics on the annual consumption of "huge for the past 25 years. The goal of the analysis is to use the past data to predict future consumption and then compare the predicted consumption to the actual consumption in those years.

-Propose a straight-line model that includes both a long-term trend and a seasonal component for the time series. Let t=t = the year in which the data was collected (t=1,2,,25)( t = 1,2 , \ldots , 25 ) and let Q1,Q2Q _ { 1 } , Q _ { 2 } , and Q3Q _ { 3 } be dummy variables used to model a seasonal effect.

A) E(Yt)=β0+β1Q1+β3Q2+β4Q3\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { Q } _ { 1 } + \beta _ { 3 } \mathrm { Q } _ { 2 } + \beta _ { 4 } \mathrm { Q } _ { 3 }
B) E(Yt)=β1t\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 1 \mathrm { t } }
C) E(Yt)=β0+β1t+β2Q1+β3Q2+β4Q3\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t } + \beta _ { 2 } \mathrm { Q } _ { 1 } + \beta _ { 3 } \mathrm { Q } _ { 2 } + \beta _ { 4 } \mathrm { Q } _ { 3 }
D) E(Yt)=β0+β1t\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t }
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سؤال
(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table.  Year  Commercial  Service  Construction  Manufacturing  and Mining  Retail  Trade  Wholesale  Trade 1985163722621645479910891986133117701360413910281987104114631122340688719887731204101328897401989930137811653183908199015942355159949101284199123663614222368821709199238404872368397302783199386275247443311,4293598199412,7876936575913,7874882199516,6477004566213,5014835199620,9117035564113,5094808\begin{array} { l c c c r c } \hline \text { Year } & \begin{array} { l } \text { Commercial } \\\text { Service }\end{array} & \text { Construction } & \begin{array} { c } \text { Manufacturing } \\\text { and Mining }\end{array} & \begin{array} { r } \text { Retail } \\\text { Trade }\end{array} & \begin{array} { l } \text { Wholesale } \\\text { Trade }\end{array} \\\hline 1985 & 1637 & 2262 & 1645 & 4799 & 1089 \\1986 & 1331 & 1770 & 1360 & 4139 & 1028 \\1987 & 1041 & 1463 & 1122 & 3406 & 887 \\1988 & 773 & 1204 & 1013 & 2889 & 740 \\1989 & 930 & 1378 & 1165 & 3183 & 908 \\1990 & 1594 & 2355 & 1599 & 4910 & 1284 \\1991 & 2366 & 3614 & 2223 & 6882 & 1709 \\1992 & 3840 & 4872 & 3683 & 9730 & 2783 \\1993 & 8627 & 5247 & 4433 & 11,429 & 3598 \\1994 & 12,787 & 6936 & 5759 & 13,787 & 4882 \\1995 & 16,647 & 7004 & 5662 & 13,501 & 4835 \\1996 & 20,911 & 7035 & 5641 & 13,509 & 4808 \\\hline\end{array}

-Using 1985 as the base period and using just construction failures, calculate the simple index for 1992.

A) 217.88
B) 46.43
C) 215.38
D) 487.20
سؤال
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.
 Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Using the exponential smoothing technique to the data from 1980 to 1985, forecast the number of Chevrolet passenger cars sold to U.S. and Canadian dealers in 1986 using w=0.3\mathrm { w } = 0.3

A) 2.448 million cars
B) 3.427 million cars
C) 4.882 million cars
D) 3.834 million cars
سؤال
(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991 The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is
shown in the table.  Wheat Harvested by Coop. Member  Year  Time (y, in thousands of bushels) 199117519922781993382199448219955841996685199778719988911999992200010922001119320021296200313101200414102\begin{array} { l c c } \hline & \text { Wheat Harvested by Coop. Member } \\\text { Year } & \text { Time } & ( \mathrm { y } , \text { in thousands of bushels) } \\\hline 1991 & 1 & 75 \\1992 & 2 & 78 \\1993 & 3 & 82 \\1994 & 4 & 82 \\1995 & 5 & 84 \\1996 & 6 & 85 \\1997 & 7 & 87 \\1998 & 8 & 91 \\1999 & 9 & 92 \\2000 & 10 & 92 \\2001 & 11 & 93 \\2002 & 12 & 96 \\2003 & 13 & 101 \\2004 & 14 & 102 \\\hline\end{array}

-A forecast was obtained for the year 2005 and the corresponding 95% prediction interval was found to be (103, 107). Interpret this interval.

A) We are 95% confident that the volume of wheat harvested in 2005 will be between 103,000 and 107,000 bushels.
B) We are 95% confident that the mean volume of wheat harvested in all years will be between 103,000 and 107,000 bushels.
C) We expect the volume of wheat harvested to increase between 103,000 and 107,000 bushels from one year to the next.
D) We are 95% confident that the 2005 harvest will be between 103,000 and 107,000 bushels larger than the harvest in 2004.
سؤال
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.  Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { c l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Using a smoothing constant of w=0.80\mathrm { w } = 0.80 ate the value of the exponentially smoothed series in 1983.

A) 1.289
B) 1.340
C) 1.228
D) 1.235
سؤال
The _______ generally describes fluctuations of the time series that are attributable to business and economic conditions.

A) seasonal variation
B) secular trend
C) cyclical effect
D) residual effect
سؤال
(Situation O) Using data from the post-Korean war period, an economist modeled annual consumption, yty _ { t } , as a function of total labor income, x1t\mathrm { x } _ { 1 \mathrm { t } } , and total property income, x2t\mathrm { x } _ { 2 \mathrm { t } } , with the following results. Assume data for n=40\mathrm { n } = 40 years were used in the analysis.
y^t=7.81+0.91x1t+0.57x2ts=1.29 Durbin-Watson d=2.09\hat { y } _ { t } = 7.81 + 0.91 x _ { 1 t } + 0.57 x _ { 2 t } \quad s = 1.29 \quad \text { Durbin-Watson } d = 2.09

-For the situation above, give the rejection region for the Durbin-Watson test for autocorrelation of residuals. Use ? = 0.10.

A) d<1.39\mathrm { d } < 1.39
B) d>1.60d > 1.60 or 4d>1.604 - d > 1.60
C) 1.39<d<1.601.39 < \mathrm { d } < 1.60
D) d<1.39\mathrm { d } < 1.39 or 4d<1.394 - \mathrm { d } < 1.39
سؤال
Using just the wool prices and a smoothing constant w = 0.8, find the exponentially smoothed value for May.

A) 272.0
B) 275.0
C) 287.0
D) 276.9
سؤال
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.  Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Using the exponential smoothing technique to the data from 1980 to 1985, forecast the number of Chevrolet passenger cars to be sold to U.S. and Canadian dealers in 1986 using w=0.7w = 0.7

A) 4.882 million cars
B) 3.427 million cars
C) 3.834 million cars
D) 2.448 million cars
سؤال
(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986 through 1992 are shown in the table below.
 Year  G.M.  Ford  Chrysler 1986899358101796198771014328122519886762431312831989624442551182199077694934149419918256558520341992930555512157\begin{array} { l r c r } \hline \text { Year } & \text { G.M. } & \text { Ford } & \text { Chrysler } \\\hline 1986 & 8993 & 5810 & 1796 \\1987 & 7101 & 4328 & 1225 \\1988 & 6762 & 4313 & 1283 \\1989 & 6244 & 4255 & 1182 \\1990 & 7769 & 4934 & 1494 \\1991 & 8256 & 5585 & 2034 \\1992 & 9305 & 5551 & 2157 \\\hline\end{array}

-Using 1986 as the base year, find the simple composite index for 1990.

A) 85.53
B) 116.92
C) 65.81
D) 151.95
سؤال
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.
 Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-The _______ is what remains of a time series value after the secular, cyclical, and seasonal components have been removed.

A) exponential effect
B) additive effect
C) residual effect
D) error effect
سؤال
The tendency of a series of values to increase or decrease over a long period of time is known as the _______ of a time series.

A) seasonal variation
B) cyclical fluctuation
C) secular trend
D) residual effect
سؤال
(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table.  Year  Commercial  Service  Construction  Manufacturing  and Mining  Retail  Trade  Wholesale  Trade 1985163722621645479910891986133117701360413910281987104114631122340688719887731204101328897401989930137811653183908199015942355159949101284199123663614222368821709199238404872368397302783199386275247443311,4293598199412,7876936575913,7874882199516,6477004566213,5014835199620,9117035564113,5094808\begin{array} { l r c c r c } \hline \text { Year } & \begin{array} { l } \text { Commercial } \\\text { Service }\end{array} & \text { Construction } & \begin{array} { c } \text { Manufacturing } \\\text { and Mining }\end{array} & \begin{array} { l } \text { Retail } \\\text { Trade }\end{array} & \begin{array} { l } \text { Wholesale } \\\text { Trade }\end{array} \\\hline 1985 & 1637 & 2262 & 1645 & 4799 & 1089 \\1986 & 1331 & 1770 & 1360 & 4139 & 1028 \\1987 & 1041 & 1463 & 1122 & 3406 & 887 \\1988 & 773 & 1204 & 1013 & 2889 & 740 \\1989 & 930 & 1378 & 1165 & 3183 & 908 \\1990 & 1594 & 2355 & 1599 & 4910 & 1284 \\1991 & 2366 & 3614 & 2223 & 6882 & 1709 \\1992 & 3840 & 4872 & 3683 & 9730 & 2783 \\1993 & 8627 & 5247 & 4433 & 11,429 & 3598 \\1994 & 12,787 & 6936 & 5759 & 13,787 & 4882 \\1995 & 16,647 & 7004 & 5662 & 13,501 & 4835 \\1996 & 20,911 & 7035 & 5641 & 13,509 & 4808 \\\hline\end{array}

-Using 1985 as the base year and using all five types of firms, calculate the simple composite index for 1995.

A) 217.88
B) 23.99
C) 476.49
D) 416.80
سؤال
(Situation O) Using data from the post-Korean war period, an economist modeled annual consumption, yty _ { t } as a function of total labor income, x1t\mathrm { x } _ { 1 } \mathrm { t } , and total property income, x2t\mathrm { x } _ { 2 } \mathrm { t } , with the following results. Assume data for n=40years\mathrm { n } = 40 \mathrm { years } were used in the analysis.
y^t=7.81+0.91x1t+0.57x2ts=1.29 Durbin-Watson d=2.09\hat { y } _ { t } = 7.81 + 0.91 x _ { 1 t } + 0.57 x _ { 2 t } \quad s = 1.29 \quad \text { Durbin-Watson } d = 2.09

-For the situation above, set up the null and alternative hypotheses for testing for the presence of autocorrelation of residuals.

A) H0:β1=β2=0\mathrm { H } _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = 0
Ha\mathrm { H } _ { \mathrm { a } } : At least one β0\beta \neq 0

B) H0\mathrm { H } _ { 0 } : No first-order autocorrelation
Ha\mathrm { H } _ { \mathrm { a } } : Negative first-order autocorrelation

C) H0\mathrm { H } _ { 0 } : No first-order autocorrelation
Ha\mathrm { H } _ { \mathrm { a } } : Positive or Negative first-order autocorrelation

D) H0\mathrm { H } _ { 0 } : No first-order autocorrelation
Ha\mathrm { H } _ { \mathrm { a } } : Positive first-order autocorrelation
سؤال
(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991-2004.
The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is shown in the table.
 Year  Time  Wheat Harvested by Coop. Member  (y, in thousands of bushels) 199117519922781993382199448219955841996685199778719988911999992200010922001119320021296200313101200414102\begin{array} { l c c } \hline \text { Year } & \text { Time } & \begin{array} { c } \text { Wheat Harvested by Coop. Member } \\\text { (y, in thousands of bushels) }\end{array} \\\hline 1991 & 1 & 75 \\1992 & 2 & 78 \\1993 & 3 & 82 \\1994 & 4 & 82 \\1995 & 5 & 84 \\1996 & 6 & 85 \\1997 & 7 & 87 \\1998 & 8 & 91 \\1999 & 9 & 92 \\2000 & 10 & 92 \\2001 & 11 & 93 \\2002 & 12 & 96 \\2003 & 13 & 101 \\2004 & 14 & 102 \\\hline\end{array}

-Suppose the least squares regression equation is y^t=75+2t\hat { y } _ { t } = 75 + 2 t Interpret the estimate of β1\beta _ { 1 } in terms of the problem.

A) We expect the mean volume of wheat harvested to increase 2000 bushels from one year to the next.
B) We expect the volume of wheat harvested to increase 2000 bushels for each additional corporate member.
C) We expect the volume of wheat harvested to be 2000 bushels in any given year.
D) We expect to harvest 2000 bushels of wheat in 2005.
سؤال
(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991 The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is
shown in the table.  Year  Time  Wheat Harvested by Coop. Member  (y, in thousands of bushels) 199117519922781993382199448219955841996685199778719988911999992200010922001119320021296200313101200414102\begin{array} { l c c } \hline \text { Year } & \text { Time } & \begin{array} { c } \text { Wheat Harvested by Coop. Member } \\\text { (y, in thousands of bushels) }\end{array} \\\hline 1991 & 1 & 75 \\1992 & 2 & 78 \\1993 & 3 & 82 \\1994 & 4 & 82 \\1995 & 5 & 84 \\1996 & 6 & 85 \\1997 & 7 & 87 \\1998 & 8 & 91 \\1999 & 9 & 92 \\2000 & 10 & 92 \\2001 & 11 & 93 \\2002 & 12 & 96 \\2003 & 13 & 101 \\2004 & 14 & 102\end{array}

-Suppose the least squares regression equation is y^t=75+2t\hat{ y } _ { \mathrm { t } } = 75 + 2 \mathrm { t } . Use the regression model to forecast the harvest in 2005.

A) 110,000 bushels
B) 102,000 bushels
C) 105,000 bushels
D) 103,000 bushels
سؤال
(Situation M) Fast food chains are closely watching what proposed legislation will do to consumption of "huge meals" in the United States. Researchers have accumulated statistics on the annual consumption of "huge for the past 25 years. The goal of the analysis is to use the past data to predict future consumption and then compare the predicted consumption to the actual consumption in those years.

-To test for first-order autocorrelation, we use the _______ test.

A) Paasche
B) Laspeyres
C) Durbin-Watson
D) Wilcoxon
سؤال
(Situation O) Using data from the post-Korean war period, an economist modeled annual consumption, yty _ { t } as a function of total labor income, x1t\mathrm { x } _ { 1 } \mathrm { t } , and total property income, x2t\mathrm { x } _ { 2 } \mathrm { t } , with the following results. Assume data for n=40years\mathrm { n } = 40 \mathrm { years } were used in the analysis.
y^t=7.81+0.91x1t+0.57x2ts=1.29 Durbin-Watson d=2.09\hat { y } _ { t } = 7.81 + 0.91 x _ { 1 t } + 0.57 x _ { 2 t } \quad s = 1.29 \quad \text { Durbin-Watson } d = 2.09

-Is there evidence of positive autocorrelation of residuals in the consumption model presented above? Test using α=0.10\alpha = 0.10 .

A) Yes, since the standard deviation s=1.29\mathrm { s } = 1.29 is small.
B) Yes, since the Durbin-Watson statistic d=2.09d = 2.09 falls in the rejection region.
C) No, since the standard deviation s=1.29\mathrm { s } = 1.29 is small.
D) No, since the Durbin-Watson statistic d=2.09\mathrm { d } = 2.09 falls in the nonrejection region.
سؤال
(Situation M) Fast food chains are closely watching what proposed legislation will do to consumption of "huge meals" in the United States. Researchers have accumulated statistics on the annual consumption of "huge for the past 25 years. The goal of the analysis is to use the past data to predict future consumption and then compare the predicted consumption to the actual consumption in those years.

-Propose a straight-line model for the long-term trend of the time series. Do not include a seasonal component. Let t=t = the year in which the data was collected (t=1,2,,25)( t = 1,2 , \ldots , 25 ) .

A) E(Yt)=β0+β1t+β2Q1+β3Q2+β4Q3E \left( Y _ { t } \right) = \beta _ { 0 } + \beta _ { 1 } t + \beta _ { 2 } Q _ { 1 } + \beta _ { 3 } Q _ { 2 } + \beta _ { 4 } Q _ { 3 }
B) E(Yt)=β1t\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 1 } \mathrm { t }
C) E(Yt)=β0+β1Q1+β3Q2+β4Q3\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { Q } _ { 1 } + \beta _ { 3 } \mathrm { Q } _ { 2 } + \beta _ { 4 } \mathrm { Q } _ { 3 }
D) E(Yt)=β0+β1t\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t }
سؤال
(Situation N) An economist wishes to study the monthly trend in the Dow Jones Industrial Average (DJIA). Data collected over the past 40 months were used to fit the model E(Yt)=β0+β1t, where E \left( Y _ { t } \right) = \beta _ { 0 } + \beta _ { 1 } t _ { \text {, where } } , monthly close of the DJIA and t=t = month (1,2,3,,40)( 1,2,3 , \ldots , 40 ) . The regression results appear below:
y^=88+0.25tR2=0.37 MSE =144F=4.25 Durbin-Watson d=0.96\hat { y } = 88 + 0.25 t \quad R ^ { 2 } = 0.37 \quad \text { MSE } = 144 \quad F = 4.25 \quad \text { Durbin-Watson } \mathrm { d } = 0.96

-Since the data are recorded over time (months), there is a strong possibility that the residuals are positively correlated. How could you check for residual correlation using a graphical technique?

A) Plot the residuals against y^\hat { y } and look for a funnel shape.
B) Plot the residuals against y^\hat { y } and look for outliers.
C) Plot the residuals against tt and look for long runs of positive and negative residuals.
D) Plot the residuals against y^\hat { y } and look for a linear trend.
سؤال
   <div style=padding-top: 35px>
   <div style=padding-top: 35px>
سؤال
(Situation K) Foreign Exchange rates, the values of foreign currency in U.S. dollars, are important to investors and international travelers. The table lists the monthly foreign exchange rates of the British pound (in U.S. dollars per pound) for a certain year.
 Month  Exchange Rate  January 1.13 February 1.10 March 1.13 April 1.23 May 1.25 June 1.28 July 1.38 August 1.39 September 1.36 October 1.42 November 1.44 December 1.44\begin{array} { l l } \underline {\text { Month }} & \underline { \text { Exchange Rate } } \\\text { January } & 1.13 \\\text { February } & 1.10 \\\text { March } & 1.13 \\\text { April } & 1.23 \\\text { May } & 1.25 \\\text { June } & 1.28 \\\text { July } & 1.38 \\\text { August } & 1.39 \\\text { September } & 1.36 \\\text { October } & 1.42 \\\text { November } & 1.44 \\\text { December } & 1.44\end{array}

-Calculate the value of the exponentially smoothed series for April using a smoothing constant of w=0.7w = 0.7
سؤال
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.  Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Using a smoothing constant of w=0.30w = 0.30 0.30, calculate the value of the exponentially smoothed series in 1982.

A) 1.150
B) 1.289
C) 1.301
D) 1.425
سؤال
The _______ of a time series can account for fluctuations that recur during specific time periods.

A) secular trend
B) cyclical fluctuation
C) seasonal effect
D) residual effect
سؤال
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.  Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Use the Holt forecasting model with trend to forecast the number of Chevrolet passenger cars sold to U.S. and Canadian dealers in 1990 using w=0.6 and v=0.5w = 0.6 \text { and } v = 0.5

A) 6.068 million cars
B) 8.72 million cars
C) 8.952 million cars
D) 6.39 million cars
سؤال
Which of the following statements about the Durbin-Watson d-statistic is true?

A) It can assume any value between 0 and 2 .
B) It can assume any value between 0 and 4.4 .
C) It can assume any value between 4- 4 and 0 .
D) It can assume any value between 4- 4 and 4 .
سؤال
(Situation H) The prices of coffee, gasoline, and sugar for each month of 1983 are shown below in the table. <strong>(Situation H) The prices of coffee, gasoline, and sugar for each month of 1983 are shown below in the table.   Using just the price of gasoline and a smoothing constant of   lculate the exponentially smoothed value for March.</strong> A) $1.102 B) $1.12 C) $1.076 D) $1.084 <div style=padding-top: 35px>
Using just the price of gasoline and a smoothing constant of <strong>(Situation H) The prices of coffee, gasoline, and sugar for each month of 1983 are shown below in the table.   Using just the price of gasoline and a smoothing constant of   lculate the exponentially smoothed value for March.</strong> A) $1.102 B) $1.12 C) $1.076 D) $1.084 <div style=padding-top: 35px> lculate the exponentially smoothed value for March.

A) $1.102
B) $1.12
C) $1.076
D) $1.084
سؤال
Consider the monthly time series shown in the table.  Month tY January 1185 February 2192 March 3189 April 4201 May 5195 June 6199 July 7206 August 8203 September 9208 October 10209 November 11218 December 12216\begin{array}{l|c|c}\hline \text { Month } & \mathrm{t} & \mathrm{Y} \\\hline \text { January } & 1 & 185 \\\hline \text { February } & 2 & 192 \\\hline \text { March } & 3 & 189 \\\hline \text { April } & 4 & 201 \\\hline \text { May } & 5 & 195 \\\hline \text { June } & 6 & 199 \\\hline \text { July } & 7 & 206 \\\hline \text { August } & 8 & 203 \\\hline \text { September } & 9 & 208 \\\hline \text { October } & 10 & 209 \\\hline \text { November } & 11 & 218 \\\hline \text { December } & 12 & 216 \\\hline\end{array}
a. Use the values of YY in the table to forecast the values of YY for the next two months, using simple exponential smoothing with w=0.7\mathrm { w } = 0.7 .
b. Use the Holt model with w=0.7w = 0.7 and v=0.7v = 0.7 to forecast the values of YY for the next two months.
سؤال
(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table.  Year  Commercial  Manufacturing  Retail  Trade  Wholesale  Trade 1985163722621645479910891986133117701360413910281987104114631122340688719887731204101328897401989930137811653183908199015942355159949101284199123663614222368821709199238404872368397302783199386275247443311,4293598199412,7876936575913,7874882199516,6477004566213,5014835199620,9117035564113,5094808\begin{array} { r r r r r r } \hline \text { Year } & \text { Commercial } & \text { Manufacturing } & \begin{array} { r } \text { Retail } \\\text { Trade }\end{array} & \begin{array} { l } \text { Wholesale } \\\text { Trade }\end{array} \\\hline 1985 & 1637 & 2262 & 1645 & 4799 & 1089 \\1986 & 1331 & 1770 & 1360 & 4139 & 1028 \\1987 & 1041 & 1463 & 1122 & 3406 & 887 \\1988 & 773 & 1204 & 1013 & 2889 & 740 \\1989 & 930 & 1378 & 1165 & 3183 & 908 \\1990 & 1594 & 2355 & 1599 & 4910 & 1284 \\1991 & 2366 & 3614 & 2223 & 6882 & 1709 \\1992 & 3840 & 4872 & 3683 & 9730 & 2783 \\1993 & 8627 & 5247 & 4433 & 11,429 & 3598 \\1994 & 12,787 & 6936 & 5759 & 13,787 & 4882 \\1995 & 16,647 & 7004 & 5662 & 13,501 & 4835 \\1996 & 20,911 & 7035 & 5641 & 13,509 & 4808 \\\hline\end{array}

-Using just the wholesale trade failures and a smoothing constant w=0.7\mathrm { w } = 0.7 , calculate the exponentially smoothed value for 1988.

A) 1015.6
B) 798.4
C) 932.9
D) 845
سؤال
Consider the table below which displays the price of a commodity for six consecutive
years.  Year  Price (dollars) 125022553253425552596261\begin{array}{c|c}\hline \text { Year } & \text { Price (dollars) } \\\hline 1 & 250 \\\hline 2 & 255 \\\hline 3 & 253 \\\hline 4 & 255 \\\hline 5 & 259 \\\hline 6 & 261 \\\hline\end{array}
a. Calculate the values in the exponentially smoothed series using w=0.6\mathrm { w } = 0.6 .
b. Calculate the forecast errors for Years 7-10 if the actual values in those years are 262, 264, 263, 266 respectively.
c. Calculate MAD, MAPE, and RMSE, using the forecast errors for Years 7-10.
سؤال
Consider the actual values Y and forecast values F given in the table below.  Time Period YF119.519.3221.520.9322.622.5\begin{array} { | c | c | c | } \hline \text { Time Period } & \mathrm { Y } & \mathrm { F } \\\hline 1 & 19.5 & 19.3 \\\hline 2 & 21.5 & 20.9 \\\hline 3 & 22.6 & 22.5 \\\hline\end{array} Calculate the root mean squared error (RMSE) of the forecasts.

A) 0.90
B) 0.30
C) 0.37
D) 1.42
سؤال
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.  Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Use the Holt forecasting model with trend to forecast the number of Chevrolet passenger cars sold to U.S. and Canadian dealers in 1990 using w=0.4 and v=0.5\mathrm { w } = 0.4 \text { and } \mathrm { v } = 0.5

A) 8.952 million cars
B) 8.72 million cars
C) 6.068 million cars
D) 6.39 million cars
سؤال
Consider the monthly time series shown in the table.  Month tY January 1185 February 2192 March 3189 April 4201 May 5195 June 6199 July 7206 August 8203 September 9208 October 10209 November 11218 December 12216\begin{array}{l|c|c}\hline {\text { Month }} & \mathrm{t} & \mathrm{Y} \\\hline \text { January } & 1 & 185 \\\hline \text { February } & 2 & 192 \\\hline \text { March } & 3 & 189 \\\hline \text { April } & 4 & 201 \\\hline \text { May } & 5 & 195 \\\hline \text { June } & 6 & 199 \\\hline \text { July } & 7 & 206 \\\hline \text { August } & 8 & 203 \\\hline \text { September } & 9 & 208 \\\hline \text { October } & 10 & 209 \\\hline \text { November } & 11 & 218 \\\hline \text { December } & 12 & 216 \\\hline\end{array}

a. Use the method of least squares to fit the model E(Yt)=β0+β1t\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t } to the data. Write the prediction equation.
b. Construct a residual plot for the model.
c. Is there evidence of a cyclical component? Explain.
سؤال
Consider the actual values Y and forecast values F given in the table below.  Time Period YF119.519.3221.520.9322.622.5\begin{array} { | c | c | c | } \hline \text { Time Period } & \mathrm { Y } & \mathrm { F } \\\hline 1 & 19.5 & 19.3 \\\hline 2 & 21.5 & 20.9 \\\hline 3 & 22.6 & 22.5 \\\hline\end{array} Calculate the mean absolute deviation (MAD) of the forecasts.

A) 1.42
B) 0.90
C) 0.30
D) 0.37
سؤال
A(n) _______ is a number that measures the change in a variable over time relative to the value of the variable during a base period.

A) index number
B) exponential smoothing constant
C) residual value
D) time series
سؤال
(Situation K) Foreign Exchange rates, the values of foreign currency in U.S. dollars, are important to investors and international travelers. The table lists the monthly foreign exchange rates of the British pound (in U.S. dollars per pound) for a certain year.
 Month  Exchange Rate  January 1.13 February 1.10 March 1.13 April 1.23 May 1.25 June 1.28 July 1.38 August 1.39 September 1.36 October 1.42 November 1.44 December 1.44\begin{array} { l l } \underline {\text { Month }} & \underline { \text { Exchange Rate } } \\\text { January } & 1.13 \\\text { February } & 1.10 \\\text { March } & 1.13 \\\text { April } & 1.23 \\\text { May } & 1.25 \\\text { June } & 1.28 \\\text { July } & 1.38 \\\text { August } & 1.39 \\\text { September } & 1.36 \\\text { October } & 1.42 \\\text { November } & 1.44 \\\text { December } & 1.44\end{array}

-Consider the table below which displays the price of a commodity for six consecutive
years.

 Year  Price 125022553253425552596261\begin{array}{c|c}\hline \text { Year } & \text { Price } \\\hline 1 & 250 \\\hline 2 & 255 \\\hline 3 & 253 \\\hline 4 & 255 \\\hline 5 & 259 \\\hline 6 & 261 \\\hline\end{array}

a.  Use the method of least squares to fit the model E(Yt)=β0+β1t to the data. Write the \text { Use the method of least squares to fit the model } E\left(Y_{t}\right)=\beta_{0}+\beta_{1} t \text { to the data. Write the } prediction equation.

b. Calculate the residuals and construct a residual plot.
c. Calculate the Durbin Watson d statistic.
سؤال
Consider the actual values Y and forecast values F given in the table below.  Time Period  Y  F 119.519.3221.520.9322.622.5\begin{array} { | c | c | c | } \hline \text { Time Period } & \text { Y } & \text { F } \\\hline 1 & 19.5 & 19.3 \\\hline 2 & 21.5 & 20.9 \\\hline 3 & 22.6 & 22.5 \\\hline\end{array} Calculate the mean absolute percentage error (MAPE) of the forecasts.

A) 0.37
B) 0.30
C) 1.42
D) 0.90
سؤال
(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991
The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is shown in the table.
 Year  Time  Wheat Harvested by Coop. Member  (y, in thousands of bushels) 199117519922781993382199448219955841996685199778719988911999992200010922001119320021296200313101200414102\begin{array} { l c c } \hline \text { Year } & \text { Time } & \begin{array} { c } \text { Wheat Harvested by Coop. Member } \\\text { (y, in thousands of bushels) }\end{array} \\\hline 1991 & 1 & 75 \\1992 & 2 & 78 \\1993 & 3 & 82 \\1994 & 4 & 82 \\1995 & 5 & 84 \\1996 & 6 & 85 \\1997 & 7 & 87 \\1998 & 8 & 91 \\1999 & 9 & 92 \\2000 & 10 & 92 \\2001 & 11 & 93 \\2002 & 12 & 96 \\2003 & 13 & 101 \\2004 & 14 & 102 \\\hline\end{array}

-Find the least squares prediction equation for the model yt=β0+β1t+εy _ { t } = \beta _ { 0 } + \beta _ { 1 } t + \varepsilon .

A) y^t=74.21.9165t\hat { y } _ { t } = - 74.2 - 1.9165 \mathrm { t }
B) y^t=1.916574.2t\hat { y } _ { \mathrm { t } } = 1.9165 - 74.2 \mathrm { t }
C) y^t=74.21.9165t\hat { y } _ { t } = 74.2 - 1.9165 \mathrm { t }
D) y^t=74.2+1.9165t\hat { y } _ { \mathrm { t } } = 74.2 + 1.9165 \mathrm { t }
سؤال
(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986
through 1992 are shown in the table below.  Year  G.M.  Ford  Chrysler 1986899358101796198771014328122519886762431312831989624442551182199077694934149419918256558520341992930555512157\begin{array} { l r r r } \hline \text { Year } & \text { G.M. } & \text { Ford } & \text { Chrysler } \\\hline 1986 & 8993 & 5810 & 1796 \\1987 & 7101 & 4328 & 1225 \\1988 & 6762 & 4313 & 1283 \\1989 & 6244 & 4255 & 1182 \\1990 & 7769 & 4934 & 1494 \\1991 & 8256 & 5585 & 2034 \\1992 & 9305 & 5551 & 2157\end{array}

-Using 1986 as the base year, and only using the Chrysler sales data, find the simple index for 1992.

A) 102.49
B) 83.26
C) 97.57
D) 120.10
سؤال
(Situation N) An economist wishes to study the monthly trend in the Dow Jones Industrial Average (DJIA). Data collected over the past 40 months were used to fit the model E(Yt)=β0+β1tE \left( Y _ { t } \right) = \beta _ { 0 } + \beta _ { 1 } t , where y=y = monthly close of the DJIA and t=t = month (1,2,3,,40)( 1,2,3 , \ldots , 40 ) . The regression results appear below:
y^=88+0.25tR2=0.37 MSE =144F=4.25 Durbin-Watson d=0.96\hat { y } = 88 + 0.25 t \quad R ^ { 2 } = 0.37 \quad \text { MSE } = 144 \quad F = 4.25 \quad \text { Durbin-Watson } \mathrm { d } = 0.96

-What is the value of the test statistic for testing whether autocorrelation exists in the data?

A) 4.25
B) 0.25
C) 0.37
D) 0.96
سؤال
It is common to use dummy variables to describe seasonal differences in a time series.
سؤال
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.  Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Using a smoothing constant of w=0.70,\mathrm { w } = 0.70 , 0.70, calculate the value of the exponentially smoothed series in 1985.
سؤال
The printout below shows a regression analysis for a time series that included 20 observations.

Regression Analysis: C2 versus C1

The regression equation is
C2=1.20+0.0362C1\mathrm { C } 2 = 1.20 + 0.0362 \mathrm { C } 1

 Predictor  Coef  SE Coef TP Constant 1.199470.0847114.160.000C10.0362410.0070725.120.000\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \mathrm { T } & \mathrm { P } \\ \text { Constant } & 1.19947 & 0.08471 & 14.16 & 0.000 \\ \mathrm { C } 1 & 0.036241 & 0.007072 & 5.12 & 0.000 \\\end{array}

 S=0.182361RSq=59.3%RSq( adj )=57.1%\begin{array} { lll}\mathrm {~S} = 0.182361 & \mathrm { R } - \mathrm { Sq } = 59.3 \% \quad \mathrm { R } - \mathrm { Sq } ( \text { adj } ) = 57.1 \% \end{array}

Analysis of Variance
 Source  DF  SS  MS  F  P  Regression 10.873400.8734026.260.000 Residual Error 180.598600.03326 Total 191.47200\begin{array}{lcrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\\text { Regression } & 1 & 0.87340 & 0.87340 & 26.26 & 0.000 \\\text { Residual Error } & 18 & 0.59860 & 0.03326 & & \\\text { Total } & 19 & 1.47200 & & &\end{array}

Locate the Durbin-Watson d-statistic and test the null hypothesis that there is no autocorrelation of residuals. Use α=0.10\alpha = 0.10 .
سؤال
Since the theoretical distributional properties of the forecast errors with smoothing methods are unknown, many analysts regard smoothing methods as descriptive procedures rather than inferential procedures.
سؤال
The Laspeyres index is a weighted index while the Paasche index is not weighted.
سؤال
A composite index number represents combinations of the prices or quantities of several commodities.
سؤال
Fourth-order autocorrelation in a quarterly time series may indicate seasonality.
سؤال
The table below shows the price of a commodity for each of ten consecutive years. The table below shows the price of a commodity for each of ten consecutive years.  <div style=padding-top: 35px>
سؤال
The d-test requires that the residuals be normally distributed.
سؤال
The table below shows the price of a commodity for each of ten consecutive years.  Year 12345678910 Price $1.19$1.22$1.23$1.45$1.39$1.42$1.47$1.55$1.62$1.65\begin{array}{l}\begin{array} { l l l l l l l l l l l } \hline \text { Year } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\hline\text { Price } & \$ 1.19 & \$ 1.22 & \$ 1.23 & \$ 1.45 & \$ 1.39 & \$ 1.42 & \$ 1.47 & \$ 1.55 & \$ 1.62 & \$ 1.65 \\\hline\end{array}\end{array}

a. Using Year 1 as the base period, calculate the simple index for the price of the commodity for each year.
b. Plot the simple indexes for years 1-10.
c. Use the simple index to interpret the trend in the price of the commodity.
سؤال
The table below shows the prices and quantities of three commodities for six consecutive
years.
\quad \quad \quad \quad Commodity A\text {Commodity A}\quad \quad Commodity B\text {Commodity B}\quad Commodity C\text {Commodity C}
 Year  Price  Quantity  Price  Quantity  Price  Quantity 125012001213200675180022551500115350070019003253270012824007142100425518001262800721250052592100129270072531006261200013525007343900\begin{array} { c c c c c c c } \hline \text { Year } & \text { Price } & \text { Quantity } & \text { Price } & \text { Quantity } & \text { Price } & \text { Quantity } \\\hline 1 & 250 & 1200 & 121 & 3200 & 675 & 1800 \\2 & 255 & 1500 & 115 & 3500 & 700 & 1900 \\3 & 253 & 2700 & 128 & 2400 & 714 & 2100 \\4 & 255 & 1800 & 126 & 2800 & 721 & 2500 \\5 & 259 & 2100 & 129 & 2700 & 725 & 3100 \\6 & 261 & 2000 & 135 & 2500 & 734 & 3900 \\\hline\end{array} a. Compute the Laspeyres price index for the six-year period, using Year 1 as the base period.
b. Compute the Paasche price index for the six-year period, using Year 1 as the base period.
c. Plot the Laspeyres and Paasche indexes on the same graph. Comment on the
differences.
سؤال
Consider the monthly time series shown in the table.  Month  t Y January 1185 February 2192 March 3189 April 4201 May 5195 June 6199 July 7206 August 8203 September 9208 October 10209 November 11218 December 12216\begin{array} { l | c | c } \hline \text { Month } & \text { t } & \mathrm { Y } \\\hline \text { January } & 1 & 185 \\\hline \text { February } & 2 & 192 \\\hline \text { March } & 3 & 189 \\\hline \text { April } & 4 & 201 \\\hline \text { May } & 5 & 195 \\\hline \text { June } & 6 & 199 \\\hline \text { July } & 7 & 206 \\\hline \text { August } & 8 & 203 \\\hline \text { September } & 9 & 208 \\\hline \text { October } & 10 & 209 \\\hline \text { November } & 11 & 218 \\\hline \text { December } & 12 & 216 \\\hline\end{array} a. Calculate the values in the exponentially smoothed series using w w=0.6w = 0.6 b. Graph the time series and the exponentially smoothed series on the same graph.
سؤال
Consider the monthly time series shown in the table.  Month tY January 1185 February 2192 March 3189 April 4201 May 5195 June 6199 July 7206 August 8203 September 9208 October 10209 November 11218 December 12216\begin{array} { l | c | c } \hline \text { Month } & \mathrm { t } & \mathrm { Y } \\\hline \text { January } & 1 & 185 \\\hline \text { February } & 2 & 192 \\\hline \text { March } & 3 & 189 \\\hline \text { April } & 4 & 201 \\\hline \text { May } & 5 & 195 \\\hline \text { June } & 6 & 199 \\\hline \text { July } & 7 & 206 \\\hline \text { August } & 8 & 203 \\\hline \text { September } & 9 & 208 \\\hline \text { October } & 10 & 209 \\\hline \text { November } & 11 & 218 \\\hline \text { December } & 12 & 216 \\\hline\end{array} a. Use the method of least squares to fit the mo E(Yt)=β0+β1\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } t to the data. Write the prediction equation.
b. Use the prediction equation to obtain forecasts for the next two months.
c. Find 95% forecast intervals for the next two months.
سؤال
The exponential smoothing constant can be any number between 0 and 100.
سؤال
We plot time series residuals against observed values of Y to determine whether a cyclical component is apparent.
سؤال
Consider the table below which displays the price of a commodity for six consecutive years.  Year  Price (dollars) 125022553253425552596261\begin{array} { c | c } \hline \text { Year } & \text { Price (dollars) } \\\hline 1 & 250 \\\hline 2 & 255 \\\hline 3 & 253 \\\hline 4 & 255 \\\hline 5 & 259 \\\hline 6 & 261 \\\hline\end{array} a. Use the Holt model to forecast values for Years 7-10 using w=0.6\mathrm { w } = 0.6 and v=0.5\mathrm { v } = 0.5 .
b. Calculate the forecast errors for Years 7-10 if the actual values in those years are 263, 267, 269, 268 respectively.
c. Calculate MAD, MAPE, and RMSE, using the forecast errors for Years 7-10.
سؤال
(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986 through 1992 are shown in the table below.  Year  G.M.  Ford  Chrysler 1986899358101796198771014328122519886762431312831989624442551182199077694934149419918256558520341992930555512157\begin{array} { l r c r } \hline \text { Year } & \text { G.M. } & \text { Ford } & \text { Chrysler } \\\hline 1986 & 8993 & 5810 & 1796 \\1987 & 7101 & 4328 & 1225 \\1988 & 6762 & 4313 & 1283 \\1989 & 6244 & 4255 & 1182 \\1990 & 7769 & 4934 & 1494 \\1991 & 8256 & 5585 & 2034 \\1992 & 9305 & 5551 & 2157 \\\hline\end{array}

-Using 1986 as the base year, find the simple composite index for 1992.
سؤال
Price indexes measure changes in the price of a commodity or group of commodities over time.
سؤال
Consider the table below which displays the price of a commodity for six consecutive years.
 Year  Price (dollars) 125022553253425552596261\begin{array}{c|c}\hline \text { Year } & \text { Price (dollars) } \\\hline 1 & 250 \\\hline 2 & 255 \\\hline 3 & 253 \\\hline 4 & 255 \\\hline 5 & 259 \\\hline 6 & 261 \\\hline\end{array}

a. Use the method of least squares to fit the model E(Yt)=β0+β1tE \left( Y _ { t } \right) = \beta _ { 0 } + \beta _ { 1 } t to the data. Write the prediction equation.
b. Use the prediction equation to obtain forecasts of the prices in years 7 and 8 .
c. Find 95%95 \% prediction intervals for years 7 and 8 .
سؤال
Retail sales for a home improvement store in quarters 1-4 over a five-year period are shown (in millions of dollars) in the table below.  Quarter  Year 123411.21.41.51.121.31.61.51.231.41.81.81.641.41.71.91.651.62.02.11.9\begin{array} { c | c | c | c | c } \hline { \text { Quarter } } \\\hline \text { Year } & 1 & 2 & 3 & 4 \\\hline 1 & 1.2 & 1.4 & 1.5 & 1.1 \\\hline 2 & 1.3 & 1.6 & 1.5 & 1.2 \\\hline 3 & 1.4 & 1.8 & 1.8 & 1.6 \\\hline 4 & 1.4 & 1.7 & 1.9 & 1.6 \\\hline 5 & 1.6 & 2.0 & 2.1 & 1.9 \\\hline\end{array} a. Write a regression model that contains trend and seasonal components to describe the sales data.
b. Use least squares regression to fit the model.
c. Use the regression model to forecast the quarterly sales during Year 6. Give 95% prediction intervals for the forecasts.
سؤال
The least squares model is an excellent choice for forecasting time series since it works particularly well outside the region of known observations.
سؤال
Smaller values of the trend smoothing constant v assign more weight to the most recent trend of the series and less to past trends.
سؤال
One of the major weaknesses of exponential smoothing is that it is not easily adapted to forecasting.
سؤال
Smoothing techniques are used to remove rapid fluctuations in a time series so the general trend can be seen.
سؤال
With N time periods in your data, a good rule of thumb is to forecast ahead no more than 2N time periods.
سؤال
A major advantage of forecasting with smoothing techniques is that the standard deviation of the forecast errors is known prior to observing the future values.
سؤال
The Holt forecasting model consists of both an exponentially smoothed component and a trend component.
سؤال
The straight-line regression model accounts for both the secular trend and the cyclical effect in a time series.
سؤال
The exponentially smoothed forecast takes into account both changes in trend and seasonality.
سؤال
The value of the Durbin-Watson d-statistic always falls in the interval from 0 to 1.
سؤال
The choice of exponential smoothing constant w has little or no effect on forecast values found using exponential smoothing.
سؤال
The Laspeyres index uses the purchase quantities of the period as weights.
سؤال
Smaller choices of the exponential smoothing constant w assign more weight to the current value of the series and yield a smoother series.
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Deck 14: Time Series: Descriptive Analyses, Models, and Forecasting Available on CD
1
(Situation M) Fast food chains are closely watching what proposed legislation will do to consumption of "huge meals" in the United States. Researchers have accumulated statistics on the annual consumption of "huge for the past 25 years. The goal of the analysis is to use the past data to predict future consumption and then compare the predicted consumption to the actual consumption in those years.

-Propose a straight-line model that includes both a long-term trend and a seasonal component for the time series. Let t=t = the year in which the data was collected (t=1,2,,25)( t = 1,2 , \ldots , 25 ) and let Q1,Q2Q _ { 1 } , Q _ { 2 } , and Q3Q _ { 3 } be dummy variables used to model a seasonal effect.

A) E(Yt)=β0+β1Q1+β3Q2+β4Q3\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { Q } _ { 1 } + \beta _ { 3 } \mathrm { Q } _ { 2 } + \beta _ { 4 } \mathrm { Q } _ { 3 }
B) E(Yt)=β1t\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 1 \mathrm { t } }
C) E(Yt)=β0+β1t+β2Q1+β3Q2+β4Q3\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t } + \beta _ { 2 } \mathrm { Q } _ { 1 } + \beta _ { 3 } \mathrm { Q } _ { 2 } + \beta _ { 4 } \mathrm { Q } _ { 3 }
D) E(Yt)=β0+β1t\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t }
E(Yt)=β0+β1t+β2Q1+β3Q2+β4Q3\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t } + \beta _ { 2 } \mathrm { Q } _ { 1 } + \beta _ { 3 } \mathrm { Q } _ { 2 } + \beta _ { 4 } \mathrm { Q } _ { 3 }
2
(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table.  Year  Commercial  Service  Construction  Manufacturing  and Mining  Retail  Trade  Wholesale  Trade 1985163722621645479910891986133117701360413910281987104114631122340688719887731204101328897401989930137811653183908199015942355159949101284199123663614222368821709199238404872368397302783199386275247443311,4293598199412,7876936575913,7874882199516,6477004566213,5014835199620,9117035564113,5094808\begin{array} { l c c c r c } \hline \text { Year } & \begin{array} { l } \text { Commercial } \\\text { Service }\end{array} & \text { Construction } & \begin{array} { c } \text { Manufacturing } \\\text { and Mining }\end{array} & \begin{array} { r } \text { Retail } \\\text { Trade }\end{array} & \begin{array} { l } \text { Wholesale } \\\text { Trade }\end{array} \\\hline 1985 & 1637 & 2262 & 1645 & 4799 & 1089 \\1986 & 1331 & 1770 & 1360 & 4139 & 1028 \\1987 & 1041 & 1463 & 1122 & 3406 & 887 \\1988 & 773 & 1204 & 1013 & 2889 & 740 \\1989 & 930 & 1378 & 1165 & 3183 & 908 \\1990 & 1594 & 2355 & 1599 & 4910 & 1284 \\1991 & 2366 & 3614 & 2223 & 6882 & 1709 \\1992 & 3840 & 4872 & 3683 & 9730 & 2783 \\1993 & 8627 & 5247 & 4433 & 11,429 & 3598 \\1994 & 12,787 & 6936 & 5759 & 13,787 & 4882 \\1995 & 16,647 & 7004 & 5662 & 13,501 & 4835 \\1996 & 20,911 & 7035 & 5641 & 13,509 & 4808 \\\hline\end{array}

-Using 1985 as the base period and using just construction failures, calculate the simple index for 1992.

A) 217.88
B) 46.43
C) 215.38
D) 487.20
215.38
3
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.
 Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Using the exponential smoothing technique to the data from 1980 to 1985, forecast the number of Chevrolet passenger cars sold to U.S. and Canadian dealers in 1986 using w=0.3\mathrm { w } = 0.3

A) 2.448 million cars
B) 3.427 million cars
C) 4.882 million cars
D) 3.834 million cars
2.448 million cars
4
(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991 The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is
shown in the table.  Wheat Harvested by Coop. Member  Year  Time (y, in thousands of bushels) 199117519922781993382199448219955841996685199778719988911999992200010922001119320021296200313101200414102\begin{array} { l c c } \hline & \text { Wheat Harvested by Coop. Member } \\\text { Year } & \text { Time } & ( \mathrm { y } , \text { in thousands of bushels) } \\\hline 1991 & 1 & 75 \\1992 & 2 & 78 \\1993 & 3 & 82 \\1994 & 4 & 82 \\1995 & 5 & 84 \\1996 & 6 & 85 \\1997 & 7 & 87 \\1998 & 8 & 91 \\1999 & 9 & 92 \\2000 & 10 & 92 \\2001 & 11 & 93 \\2002 & 12 & 96 \\2003 & 13 & 101 \\2004 & 14 & 102 \\\hline\end{array}

-A forecast was obtained for the year 2005 and the corresponding 95% prediction interval was found to be (103, 107). Interpret this interval.

A) We are 95% confident that the volume of wheat harvested in 2005 will be between 103,000 and 107,000 bushels.
B) We are 95% confident that the mean volume of wheat harvested in all years will be between 103,000 and 107,000 bushels.
C) We expect the volume of wheat harvested to increase between 103,000 and 107,000 bushels from one year to the next.
D) We are 95% confident that the 2005 harvest will be between 103,000 and 107,000 bushels larger than the harvest in 2004.
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(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.  Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { c l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Using a smoothing constant of w=0.80\mathrm { w } = 0.80 ate the value of the exponentially smoothed series in 1983.

A) 1.289
B) 1.340
C) 1.228
D) 1.235
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6
The _______ generally describes fluctuations of the time series that are attributable to business and economic conditions.

A) seasonal variation
B) secular trend
C) cyclical effect
D) residual effect
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7
(Situation O) Using data from the post-Korean war period, an economist modeled annual consumption, yty _ { t } , as a function of total labor income, x1t\mathrm { x } _ { 1 \mathrm { t } } , and total property income, x2t\mathrm { x } _ { 2 \mathrm { t } } , with the following results. Assume data for n=40\mathrm { n } = 40 years were used in the analysis.
y^t=7.81+0.91x1t+0.57x2ts=1.29 Durbin-Watson d=2.09\hat { y } _ { t } = 7.81 + 0.91 x _ { 1 t } + 0.57 x _ { 2 t } \quad s = 1.29 \quad \text { Durbin-Watson } d = 2.09

-For the situation above, give the rejection region for the Durbin-Watson test for autocorrelation of residuals. Use ? = 0.10.

A) d<1.39\mathrm { d } < 1.39
B) d>1.60d > 1.60 or 4d>1.604 - d > 1.60
C) 1.39<d<1.601.39 < \mathrm { d } < 1.60
D) d<1.39\mathrm { d } < 1.39 or 4d<1.394 - \mathrm { d } < 1.39
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8
Using just the wool prices and a smoothing constant w = 0.8, find the exponentially smoothed value for May.

A) 272.0
B) 275.0
C) 287.0
D) 276.9
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(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.  Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Using the exponential smoothing technique to the data from 1980 to 1985, forecast the number of Chevrolet passenger cars to be sold to U.S. and Canadian dealers in 1986 using w=0.7w = 0.7

A) 4.882 million cars
B) 3.427 million cars
C) 3.834 million cars
D) 2.448 million cars
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10
(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986 through 1992 are shown in the table below.
 Year  G.M.  Ford  Chrysler 1986899358101796198771014328122519886762431312831989624442551182199077694934149419918256558520341992930555512157\begin{array} { l r c r } \hline \text { Year } & \text { G.M. } & \text { Ford } & \text { Chrysler } \\\hline 1986 & 8993 & 5810 & 1796 \\1987 & 7101 & 4328 & 1225 \\1988 & 6762 & 4313 & 1283 \\1989 & 6244 & 4255 & 1182 \\1990 & 7769 & 4934 & 1494 \\1991 & 8256 & 5585 & 2034 \\1992 & 9305 & 5551 & 2157 \\\hline\end{array}

-Using 1986 as the base year, find the simple composite index for 1990.

A) 85.53
B) 116.92
C) 65.81
D) 151.95
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11
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.
 Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-The _______ is what remains of a time series value after the secular, cyclical, and seasonal components have been removed.

A) exponential effect
B) additive effect
C) residual effect
D) error effect
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12
The tendency of a series of values to increase or decrease over a long period of time is known as the _______ of a time series.

A) seasonal variation
B) cyclical fluctuation
C) secular trend
D) residual effect
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13
(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table.  Year  Commercial  Service  Construction  Manufacturing  and Mining  Retail  Trade  Wholesale  Trade 1985163722621645479910891986133117701360413910281987104114631122340688719887731204101328897401989930137811653183908199015942355159949101284199123663614222368821709199238404872368397302783199386275247443311,4293598199412,7876936575913,7874882199516,6477004566213,5014835199620,9117035564113,5094808\begin{array} { l r c c r c } \hline \text { Year } & \begin{array} { l } \text { Commercial } \\\text { Service }\end{array} & \text { Construction } & \begin{array} { c } \text { Manufacturing } \\\text { and Mining }\end{array} & \begin{array} { l } \text { Retail } \\\text { Trade }\end{array} & \begin{array} { l } \text { Wholesale } \\\text { Trade }\end{array} \\\hline 1985 & 1637 & 2262 & 1645 & 4799 & 1089 \\1986 & 1331 & 1770 & 1360 & 4139 & 1028 \\1987 & 1041 & 1463 & 1122 & 3406 & 887 \\1988 & 773 & 1204 & 1013 & 2889 & 740 \\1989 & 930 & 1378 & 1165 & 3183 & 908 \\1990 & 1594 & 2355 & 1599 & 4910 & 1284 \\1991 & 2366 & 3614 & 2223 & 6882 & 1709 \\1992 & 3840 & 4872 & 3683 & 9730 & 2783 \\1993 & 8627 & 5247 & 4433 & 11,429 & 3598 \\1994 & 12,787 & 6936 & 5759 & 13,787 & 4882 \\1995 & 16,647 & 7004 & 5662 & 13,501 & 4835 \\1996 & 20,911 & 7035 & 5641 & 13,509 & 4808 \\\hline\end{array}

-Using 1985 as the base year and using all five types of firms, calculate the simple composite index for 1995.

A) 217.88
B) 23.99
C) 476.49
D) 416.80
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14
(Situation O) Using data from the post-Korean war period, an economist modeled annual consumption, yty _ { t } as a function of total labor income, x1t\mathrm { x } _ { 1 } \mathrm { t } , and total property income, x2t\mathrm { x } _ { 2 } \mathrm { t } , with the following results. Assume data for n=40years\mathrm { n } = 40 \mathrm { years } were used in the analysis.
y^t=7.81+0.91x1t+0.57x2ts=1.29 Durbin-Watson d=2.09\hat { y } _ { t } = 7.81 + 0.91 x _ { 1 t } + 0.57 x _ { 2 t } \quad s = 1.29 \quad \text { Durbin-Watson } d = 2.09

-For the situation above, set up the null and alternative hypotheses for testing for the presence of autocorrelation of residuals.

A) H0:β1=β2=0\mathrm { H } _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = 0
Ha\mathrm { H } _ { \mathrm { a } } : At least one β0\beta \neq 0

B) H0\mathrm { H } _ { 0 } : No first-order autocorrelation
Ha\mathrm { H } _ { \mathrm { a } } : Negative first-order autocorrelation

C) H0\mathrm { H } _ { 0 } : No first-order autocorrelation
Ha\mathrm { H } _ { \mathrm { a } } : Positive or Negative first-order autocorrelation

D) H0\mathrm { H } _ { 0 } : No first-order autocorrelation
Ha\mathrm { H } _ { \mathrm { a } } : Positive first-order autocorrelation
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15
(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991-2004.
The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is shown in the table.
 Year  Time  Wheat Harvested by Coop. Member  (y, in thousands of bushels) 199117519922781993382199448219955841996685199778719988911999992200010922001119320021296200313101200414102\begin{array} { l c c } \hline \text { Year } & \text { Time } & \begin{array} { c } \text { Wheat Harvested by Coop. Member } \\\text { (y, in thousands of bushels) }\end{array} \\\hline 1991 & 1 & 75 \\1992 & 2 & 78 \\1993 & 3 & 82 \\1994 & 4 & 82 \\1995 & 5 & 84 \\1996 & 6 & 85 \\1997 & 7 & 87 \\1998 & 8 & 91 \\1999 & 9 & 92 \\2000 & 10 & 92 \\2001 & 11 & 93 \\2002 & 12 & 96 \\2003 & 13 & 101 \\2004 & 14 & 102 \\\hline\end{array}

-Suppose the least squares regression equation is y^t=75+2t\hat { y } _ { t } = 75 + 2 t Interpret the estimate of β1\beta _ { 1 } in terms of the problem.

A) We expect the mean volume of wheat harvested to increase 2000 bushels from one year to the next.
B) We expect the volume of wheat harvested to increase 2000 bushels for each additional corporate member.
C) We expect the volume of wheat harvested to be 2000 bushels in any given year.
D) We expect to harvest 2000 bushels of wheat in 2005.
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16
(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991 The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is
shown in the table.  Year  Time  Wheat Harvested by Coop. Member  (y, in thousands of bushels) 199117519922781993382199448219955841996685199778719988911999992200010922001119320021296200313101200414102\begin{array} { l c c } \hline \text { Year } & \text { Time } & \begin{array} { c } \text { Wheat Harvested by Coop. Member } \\\text { (y, in thousands of bushels) }\end{array} \\\hline 1991 & 1 & 75 \\1992 & 2 & 78 \\1993 & 3 & 82 \\1994 & 4 & 82 \\1995 & 5 & 84 \\1996 & 6 & 85 \\1997 & 7 & 87 \\1998 & 8 & 91 \\1999 & 9 & 92 \\2000 & 10 & 92 \\2001 & 11 & 93 \\2002 & 12 & 96 \\2003 & 13 & 101 \\2004 & 14 & 102\end{array}

-Suppose the least squares regression equation is y^t=75+2t\hat{ y } _ { \mathrm { t } } = 75 + 2 \mathrm { t } . Use the regression model to forecast the harvest in 2005.

A) 110,000 bushels
B) 102,000 bushels
C) 105,000 bushels
D) 103,000 bushels
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17
(Situation M) Fast food chains are closely watching what proposed legislation will do to consumption of "huge meals" in the United States. Researchers have accumulated statistics on the annual consumption of "huge for the past 25 years. The goal of the analysis is to use the past data to predict future consumption and then compare the predicted consumption to the actual consumption in those years.

-To test for first-order autocorrelation, we use the _______ test.

A) Paasche
B) Laspeyres
C) Durbin-Watson
D) Wilcoxon
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18
(Situation O) Using data from the post-Korean war period, an economist modeled annual consumption, yty _ { t } as a function of total labor income, x1t\mathrm { x } _ { 1 } \mathrm { t } , and total property income, x2t\mathrm { x } _ { 2 } \mathrm { t } , with the following results. Assume data for n=40years\mathrm { n } = 40 \mathrm { years } were used in the analysis.
y^t=7.81+0.91x1t+0.57x2ts=1.29 Durbin-Watson d=2.09\hat { y } _ { t } = 7.81 + 0.91 x _ { 1 t } + 0.57 x _ { 2 t } \quad s = 1.29 \quad \text { Durbin-Watson } d = 2.09

-Is there evidence of positive autocorrelation of residuals in the consumption model presented above? Test using α=0.10\alpha = 0.10 .

A) Yes, since the standard deviation s=1.29\mathrm { s } = 1.29 is small.
B) Yes, since the Durbin-Watson statistic d=2.09d = 2.09 falls in the rejection region.
C) No, since the standard deviation s=1.29\mathrm { s } = 1.29 is small.
D) No, since the Durbin-Watson statistic d=2.09\mathrm { d } = 2.09 falls in the nonrejection region.
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19
(Situation M) Fast food chains are closely watching what proposed legislation will do to consumption of "huge meals" in the United States. Researchers have accumulated statistics on the annual consumption of "huge for the past 25 years. The goal of the analysis is to use the past data to predict future consumption and then compare the predicted consumption to the actual consumption in those years.

-Propose a straight-line model for the long-term trend of the time series. Do not include a seasonal component. Let t=t = the year in which the data was collected (t=1,2,,25)( t = 1,2 , \ldots , 25 ) .

A) E(Yt)=β0+β1t+β2Q1+β3Q2+β4Q3E \left( Y _ { t } \right) = \beta _ { 0 } + \beta _ { 1 } t + \beta _ { 2 } Q _ { 1 } + \beta _ { 3 } Q _ { 2 } + \beta _ { 4 } Q _ { 3 }
B) E(Yt)=β1t\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 1 } \mathrm { t }
C) E(Yt)=β0+β1Q1+β3Q2+β4Q3\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { Q } _ { 1 } + \beta _ { 3 } \mathrm { Q } _ { 2 } + \beta _ { 4 } \mathrm { Q } _ { 3 }
D) E(Yt)=β0+β1t\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t }
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20
(Situation N) An economist wishes to study the monthly trend in the Dow Jones Industrial Average (DJIA). Data collected over the past 40 months were used to fit the model E(Yt)=β0+β1t, where E \left( Y _ { t } \right) = \beta _ { 0 } + \beta _ { 1 } t _ { \text {, where } } , monthly close of the DJIA and t=t = month (1,2,3,,40)( 1,2,3 , \ldots , 40 ) . The regression results appear below:
y^=88+0.25tR2=0.37 MSE =144F=4.25 Durbin-Watson d=0.96\hat { y } = 88 + 0.25 t \quad R ^ { 2 } = 0.37 \quad \text { MSE } = 144 \quad F = 4.25 \quad \text { Durbin-Watson } \mathrm { d } = 0.96

-Since the data are recorded over time (months), there is a strong possibility that the residuals are positively correlated. How could you check for residual correlation using a graphical technique?

A) Plot the residuals against y^\hat { y } and look for a funnel shape.
B) Plot the residuals against y^\hat { y } and look for outliers.
C) Plot the residuals against tt and look for long runs of positive and negative residuals.
D) Plot the residuals against y^\hat { y } and look for a linear trend.
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21

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22
(Situation K) Foreign Exchange rates, the values of foreign currency in U.S. dollars, are important to investors and international travelers. The table lists the monthly foreign exchange rates of the British pound (in U.S. dollars per pound) for a certain year.
 Month  Exchange Rate  January 1.13 February 1.10 March 1.13 April 1.23 May 1.25 June 1.28 July 1.38 August 1.39 September 1.36 October 1.42 November 1.44 December 1.44\begin{array} { l l } \underline {\text { Month }} & \underline { \text { Exchange Rate } } \\\text { January } & 1.13 \\\text { February } & 1.10 \\\text { March } & 1.13 \\\text { April } & 1.23 \\\text { May } & 1.25 \\\text { June } & 1.28 \\\text { July } & 1.38 \\\text { August } & 1.39 \\\text { September } & 1.36 \\\text { October } & 1.42 \\\text { November } & 1.44 \\\text { December } & 1.44\end{array}

-Calculate the value of the exponentially smoothed series for April using a smoothing constant of w=0.7w = 0.7
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23
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.  Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Using a smoothing constant of w=0.30w = 0.30 0.30, calculate the value of the exponentially smoothed series in 1982.

A) 1.150
B) 1.289
C) 1.301
D) 1.425
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24
The _______ of a time series can account for fluctuations that recur during specific time periods.

A) secular trend
B) cyclical fluctuation
C) seasonal effect
D) residual effect
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25
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.  Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Use the Holt forecasting model with trend to forecast the number of Chevrolet passenger cars sold to U.S. and Canadian dealers in 1990 using w=0.6 and v=0.5w = 0.6 \text { and } v = 0.5

A) 6.068 million cars
B) 8.72 million cars
C) 8.952 million cars
D) 6.39 million cars
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Which of the following statements about the Durbin-Watson d-statistic is true?

A) It can assume any value between 0 and 2 .
B) It can assume any value between 0 and 4.4 .
C) It can assume any value between 4- 4 and 0 .
D) It can assume any value between 4- 4 and 4 .
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(Situation H) The prices of coffee, gasoline, and sugar for each month of 1983 are shown below in the table. <strong>(Situation H) The prices of coffee, gasoline, and sugar for each month of 1983 are shown below in the table.   Using just the price of gasoline and a smoothing constant of   lculate the exponentially smoothed value for March.</strong> A) $1.102 B) $1.12 C) $1.076 D) $1.084
Using just the price of gasoline and a smoothing constant of <strong>(Situation H) The prices of coffee, gasoline, and sugar for each month of 1983 are shown below in the table.   Using just the price of gasoline and a smoothing constant of   lculate the exponentially smoothed value for March.</strong> A) $1.102 B) $1.12 C) $1.076 D) $1.084 lculate the exponentially smoothed value for March.

A) $1.102
B) $1.12
C) $1.076
D) $1.084
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Consider the monthly time series shown in the table.  Month tY January 1185 February 2192 March 3189 April 4201 May 5195 June 6199 July 7206 August 8203 September 9208 October 10209 November 11218 December 12216\begin{array}{l|c|c}\hline \text { Month } & \mathrm{t} & \mathrm{Y} \\\hline \text { January } & 1 & 185 \\\hline \text { February } & 2 & 192 \\\hline \text { March } & 3 & 189 \\\hline \text { April } & 4 & 201 \\\hline \text { May } & 5 & 195 \\\hline \text { June } & 6 & 199 \\\hline \text { July } & 7 & 206 \\\hline \text { August } & 8 & 203 \\\hline \text { September } & 9 & 208 \\\hline \text { October } & 10 & 209 \\\hline \text { November } & 11 & 218 \\\hline \text { December } & 12 & 216 \\\hline\end{array}
a. Use the values of YY in the table to forecast the values of YY for the next two months, using simple exponential smoothing with w=0.7\mathrm { w } = 0.7 .
b. Use the Holt model with w=0.7w = 0.7 and v=0.7v = 0.7 to forecast the values of YY for the next two months.
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29
(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table.  Year  Commercial  Manufacturing  Retail  Trade  Wholesale  Trade 1985163722621645479910891986133117701360413910281987104114631122340688719887731204101328897401989930137811653183908199015942355159949101284199123663614222368821709199238404872368397302783199386275247443311,4293598199412,7876936575913,7874882199516,6477004566213,5014835199620,9117035564113,5094808\begin{array} { r r r r r r } \hline \text { Year } & \text { Commercial } & \text { Manufacturing } & \begin{array} { r } \text { Retail } \\\text { Trade }\end{array} & \begin{array} { l } \text { Wholesale } \\\text { Trade }\end{array} \\\hline 1985 & 1637 & 2262 & 1645 & 4799 & 1089 \\1986 & 1331 & 1770 & 1360 & 4139 & 1028 \\1987 & 1041 & 1463 & 1122 & 3406 & 887 \\1988 & 773 & 1204 & 1013 & 2889 & 740 \\1989 & 930 & 1378 & 1165 & 3183 & 908 \\1990 & 1594 & 2355 & 1599 & 4910 & 1284 \\1991 & 2366 & 3614 & 2223 & 6882 & 1709 \\1992 & 3840 & 4872 & 3683 & 9730 & 2783 \\1993 & 8627 & 5247 & 4433 & 11,429 & 3598 \\1994 & 12,787 & 6936 & 5759 & 13,787 & 4882 \\1995 & 16,647 & 7004 & 5662 & 13,501 & 4835 \\1996 & 20,911 & 7035 & 5641 & 13,509 & 4808 \\\hline\end{array}

-Using just the wholesale trade failures and a smoothing constant w=0.7\mathrm { w } = 0.7 , calculate the exponentially smoothed value for 1988.

A) 1015.6
B) 798.4
C) 932.9
D) 845
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30
Consider the table below which displays the price of a commodity for six consecutive
years.  Year  Price (dollars) 125022553253425552596261\begin{array}{c|c}\hline \text { Year } & \text { Price (dollars) } \\\hline 1 & 250 \\\hline 2 & 255 \\\hline 3 & 253 \\\hline 4 & 255 \\\hline 5 & 259 \\\hline 6 & 261 \\\hline\end{array}
a. Calculate the values in the exponentially smoothed series using w=0.6\mathrm { w } = 0.6 .
b. Calculate the forecast errors for Years 7-10 if the actual values in those years are 262, 264, 263, 266 respectively.
c. Calculate MAD, MAPE, and RMSE, using the forecast errors for Years 7-10.
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31
Consider the actual values Y and forecast values F given in the table below.  Time Period YF119.519.3221.520.9322.622.5\begin{array} { | c | c | c | } \hline \text { Time Period } & \mathrm { Y } & \mathrm { F } \\\hline 1 & 19.5 & 19.3 \\\hline 2 & 21.5 & 20.9 \\\hline 3 & 22.6 & 22.5 \\\hline\end{array} Calculate the root mean squared error (RMSE) of the forecasts.

A) 0.90
B) 0.30
C) 0.37
D) 1.42
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32
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.  Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Use the Holt forecasting model with trend to forecast the number of Chevrolet passenger cars sold to U.S. and Canadian dealers in 1990 using w=0.4 and v=0.5\mathrm { w } = 0.4 \text { and } \mathrm { v } = 0.5

A) 8.952 million cars
B) 8.72 million cars
C) 6.068 million cars
D) 6.39 million cars
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33
Consider the monthly time series shown in the table.  Month tY January 1185 February 2192 March 3189 April 4201 May 5195 June 6199 July 7206 August 8203 September 9208 October 10209 November 11218 December 12216\begin{array}{l|c|c}\hline {\text { Month }} & \mathrm{t} & \mathrm{Y} \\\hline \text { January } & 1 & 185 \\\hline \text { February } & 2 & 192 \\\hline \text { March } & 3 & 189 \\\hline \text { April } & 4 & 201 \\\hline \text { May } & 5 & 195 \\\hline \text { June } & 6 & 199 \\\hline \text { July } & 7 & 206 \\\hline \text { August } & 8 & 203 \\\hline \text { September } & 9 & 208 \\\hline \text { October } & 10 & 209 \\\hline \text { November } & 11 & 218 \\\hline \text { December } & 12 & 216 \\\hline\end{array}

a. Use the method of least squares to fit the model E(Yt)=β0+β1t\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } \mathrm { t } to the data. Write the prediction equation.
b. Construct a residual plot for the model.
c. Is there evidence of a cyclical component? Explain.
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34
Consider the actual values Y and forecast values F given in the table below.  Time Period YF119.519.3221.520.9322.622.5\begin{array} { | c | c | c | } \hline \text { Time Period } & \mathrm { Y } & \mathrm { F } \\\hline 1 & 19.5 & 19.3 \\\hline 2 & 21.5 & 20.9 \\\hline 3 & 22.6 & 22.5 \\\hline\end{array} Calculate the mean absolute deviation (MAD) of the forecasts.

A) 1.42
B) 0.90
C) 0.30
D) 0.37
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35
A(n) _______ is a number that measures the change in a variable over time relative to the value of the variable during a base period.

A) index number
B) exponential smoothing constant
C) residual value
D) time series
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36
(Situation K) Foreign Exchange rates, the values of foreign currency in U.S. dollars, are important to investors and international travelers. The table lists the monthly foreign exchange rates of the British pound (in U.S. dollars per pound) for a certain year.
 Month  Exchange Rate  January 1.13 February 1.10 March 1.13 April 1.23 May 1.25 June 1.28 July 1.38 August 1.39 September 1.36 October 1.42 November 1.44 December 1.44\begin{array} { l l } \underline {\text { Month }} & \underline { \text { Exchange Rate } } \\\text { January } & 1.13 \\\text { February } & 1.10 \\\text { March } & 1.13 \\\text { April } & 1.23 \\\text { May } & 1.25 \\\text { June } & 1.28 \\\text { July } & 1.38 \\\text { August } & 1.39 \\\text { September } & 1.36 \\\text { October } & 1.42 \\\text { November } & 1.44 \\\text { December } & 1.44\end{array}

-Consider the table below which displays the price of a commodity for six consecutive
years.

 Year  Price 125022553253425552596261\begin{array}{c|c}\hline \text { Year } & \text { Price } \\\hline 1 & 250 \\\hline 2 & 255 \\\hline 3 & 253 \\\hline 4 & 255 \\\hline 5 & 259 \\\hline 6 & 261 \\\hline\end{array}

a.  Use the method of least squares to fit the model E(Yt)=β0+β1t to the data. Write the \text { Use the method of least squares to fit the model } E\left(Y_{t}\right)=\beta_{0}+\beta_{1} t \text { to the data. Write the } prediction equation.

b. Calculate the residuals and construct a residual plot.
c. Calculate the Durbin Watson d statistic.
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37
Consider the actual values Y and forecast values F given in the table below.  Time Period  Y  F 119.519.3221.520.9322.622.5\begin{array} { | c | c | c | } \hline \text { Time Period } & \text { Y } & \text { F } \\\hline 1 & 19.5 & 19.3 \\\hline 2 & 21.5 & 20.9 \\\hline 3 & 22.6 & 22.5 \\\hline\end{array} Calculate the mean absolute percentage error (MAPE) of the forecasts.

A) 0.37
B) 0.30
C) 1.42
D) 0.90
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38
(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991
The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is shown in the table.
 Year  Time  Wheat Harvested by Coop. Member  (y, in thousands of bushels) 199117519922781993382199448219955841996685199778719988911999992200010922001119320021296200313101200414102\begin{array} { l c c } \hline \text { Year } & \text { Time } & \begin{array} { c } \text { Wheat Harvested by Coop. Member } \\\text { (y, in thousands of bushels) }\end{array} \\\hline 1991 & 1 & 75 \\1992 & 2 & 78 \\1993 & 3 & 82 \\1994 & 4 & 82 \\1995 & 5 & 84 \\1996 & 6 & 85 \\1997 & 7 & 87 \\1998 & 8 & 91 \\1999 & 9 & 92 \\2000 & 10 & 92 \\2001 & 11 & 93 \\2002 & 12 & 96 \\2003 & 13 & 101 \\2004 & 14 & 102 \\\hline\end{array}

-Find the least squares prediction equation for the model yt=β0+β1t+εy _ { t } = \beta _ { 0 } + \beta _ { 1 } t + \varepsilon .

A) y^t=74.21.9165t\hat { y } _ { t } = - 74.2 - 1.9165 \mathrm { t }
B) y^t=1.916574.2t\hat { y } _ { \mathrm { t } } = 1.9165 - 74.2 \mathrm { t }
C) y^t=74.21.9165t\hat { y } _ { t } = 74.2 - 1.9165 \mathrm { t }
D) y^t=74.2+1.9165t\hat { y } _ { \mathrm { t } } = 74.2 + 1.9165 \mathrm { t }
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39
(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986
through 1992 are shown in the table below.  Year  G.M.  Ford  Chrysler 1986899358101796198771014328122519886762431312831989624442551182199077694934149419918256558520341992930555512157\begin{array} { l r r r } \hline \text { Year } & \text { G.M. } & \text { Ford } & \text { Chrysler } \\\hline 1986 & 8993 & 5810 & 1796 \\1987 & 7101 & 4328 & 1225 \\1988 & 6762 & 4313 & 1283 \\1989 & 6244 & 4255 & 1182 \\1990 & 7769 & 4934 & 1494 \\1991 & 8256 & 5585 & 2034 \\1992 & 9305 & 5551 & 2157\end{array}

-Using 1986 as the base year, and only using the Chrysler sales data, find the simple index for 1992.

A) 102.49
B) 83.26
C) 97.57
D) 120.10
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40
(Situation N) An economist wishes to study the monthly trend in the Dow Jones Industrial Average (DJIA). Data collected over the past 40 months were used to fit the model E(Yt)=β0+β1tE \left( Y _ { t } \right) = \beta _ { 0 } + \beta _ { 1 } t , where y=y = monthly close of the DJIA and t=t = month (1,2,3,,40)( 1,2,3 , \ldots , 40 ) . The regression results appear below:
y^=88+0.25tR2=0.37 MSE =144F=4.25 Durbin-Watson d=0.96\hat { y } = 88 + 0.25 t \quad R ^ { 2 } = 0.37 \quad \text { MSE } = 144 \quad F = 4.25 \quad \text { Durbin-Watson } \mathrm { d } = 0.96

-What is the value of the test statistic for testing whether autocorrelation exists in the data?

A) 4.25
B) 0.25
C) 0.37
D) 0.96
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41
It is common to use dummy variables to describe seasonal differences in a time series.
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42
(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985.  Year  Sales 19801.74019811.44419820.89619831.28919841.45519854.882\begin{array} { l l } \hline \text { Year } & \text { Sales } \\\hline 1980 & 1.740 \\1981 & 1.444 \\1982 & 0.896 \\1983 & 1.289 \\1984 & 1.455 \\1985 & 4.882 \\\hline\end{array}

-Using a smoothing constant of w=0.70,\mathrm { w } = 0.70 , 0.70, calculate the value of the exponentially smoothed series in 1985.
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43
The printout below shows a regression analysis for a time series that included 20 observations.

Regression Analysis: C2 versus C1

The regression equation is
C2=1.20+0.0362C1\mathrm { C } 2 = 1.20 + 0.0362 \mathrm { C } 1

 Predictor  Coef  SE Coef TP Constant 1.199470.0847114.160.000C10.0362410.0070725.120.000\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \mathrm { T } & \mathrm { P } \\ \text { Constant } & 1.19947 & 0.08471 & 14.16 & 0.000 \\ \mathrm { C } 1 & 0.036241 & 0.007072 & 5.12 & 0.000 \\\end{array}

 S=0.182361RSq=59.3%RSq( adj )=57.1%\begin{array} { lll}\mathrm {~S} = 0.182361 & \mathrm { R } - \mathrm { Sq } = 59.3 \% \quad \mathrm { R } - \mathrm { Sq } ( \text { adj } ) = 57.1 \% \end{array}

Analysis of Variance
 Source  DF  SS  MS  F  P  Regression 10.873400.8734026.260.000 Residual Error 180.598600.03326 Total 191.47200\begin{array}{lcrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\\text { Regression } & 1 & 0.87340 & 0.87340 & 26.26 & 0.000 \\\text { Residual Error } & 18 & 0.59860 & 0.03326 & & \\\text { Total } & 19 & 1.47200 & & &\end{array}

Locate the Durbin-Watson d-statistic and test the null hypothesis that there is no autocorrelation of residuals. Use α=0.10\alpha = 0.10 .
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44
Since the theoretical distributional properties of the forecast errors with smoothing methods are unknown, many analysts regard smoothing methods as descriptive procedures rather than inferential procedures.
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45
The Laspeyres index is a weighted index while the Paasche index is not weighted.
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46
A composite index number represents combinations of the prices or quantities of several commodities.
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47
Fourth-order autocorrelation in a quarterly time series may indicate seasonality.
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48
The table below shows the price of a commodity for each of ten consecutive years. The table below shows the price of a commodity for each of ten consecutive years.
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49
The d-test requires that the residuals be normally distributed.
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50
The table below shows the price of a commodity for each of ten consecutive years.  Year 12345678910 Price $1.19$1.22$1.23$1.45$1.39$1.42$1.47$1.55$1.62$1.65\begin{array}{l}\begin{array} { l l l l l l l l l l l } \hline \text { Year } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\hline\text { Price } & \$ 1.19 & \$ 1.22 & \$ 1.23 & \$ 1.45 & \$ 1.39 & \$ 1.42 & \$ 1.47 & \$ 1.55 & \$ 1.62 & \$ 1.65 \\\hline\end{array}\end{array}

a. Using Year 1 as the base period, calculate the simple index for the price of the commodity for each year.
b. Plot the simple indexes for years 1-10.
c. Use the simple index to interpret the trend in the price of the commodity.
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51
The table below shows the prices and quantities of three commodities for six consecutive
years.
\quad \quad \quad \quad Commodity A\text {Commodity A}\quad \quad Commodity B\text {Commodity B}\quad Commodity C\text {Commodity C}
 Year  Price  Quantity  Price  Quantity  Price  Quantity 125012001213200675180022551500115350070019003253270012824007142100425518001262800721250052592100129270072531006261200013525007343900\begin{array} { c c c c c c c } \hline \text { Year } & \text { Price } & \text { Quantity } & \text { Price } & \text { Quantity } & \text { Price } & \text { Quantity } \\\hline 1 & 250 & 1200 & 121 & 3200 & 675 & 1800 \\2 & 255 & 1500 & 115 & 3500 & 700 & 1900 \\3 & 253 & 2700 & 128 & 2400 & 714 & 2100 \\4 & 255 & 1800 & 126 & 2800 & 721 & 2500 \\5 & 259 & 2100 & 129 & 2700 & 725 & 3100 \\6 & 261 & 2000 & 135 & 2500 & 734 & 3900 \\\hline\end{array} a. Compute the Laspeyres price index for the six-year period, using Year 1 as the base period.
b. Compute the Paasche price index for the six-year period, using Year 1 as the base period.
c. Plot the Laspeyres and Paasche indexes on the same graph. Comment on the
differences.
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52
Consider the monthly time series shown in the table.  Month  t Y January 1185 February 2192 March 3189 April 4201 May 5195 June 6199 July 7206 August 8203 September 9208 October 10209 November 11218 December 12216\begin{array} { l | c | c } \hline \text { Month } & \text { t } & \mathrm { Y } \\\hline \text { January } & 1 & 185 \\\hline \text { February } & 2 & 192 \\\hline \text { March } & 3 & 189 \\\hline \text { April } & 4 & 201 \\\hline \text { May } & 5 & 195 \\\hline \text { June } & 6 & 199 \\\hline \text { July } & 7 & 206 \\\hline \text { August } & 8 & 203 \\\hline \text { September } & 9 & 208 \\\hline \text { October } & 10 & 209 \\\hline \text { November } & 11 & 218 \\\hline \text { December } & 12 & 216 \\\hline\end{array} a. Calculate the values in the exponentially smoothed series using w w=0.6w = 0.6 b. Graph the time series and the exponentially smoothed series on the same graph.
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53
Consider the monthly time series shown in the table.  Month tY January 1185 February 2192 March 3189 April 4201 May 5195 June 6199 July 7206 August 8203 September 9208 October 10209 November 11218 December 12216\begin{array} { l | c | c } \hline \text { Month } & \mathrm { t } & \mathrm { Y } \\\hline \text { January } & 1 & 185 \\\hline \text { February } & 2 & 192 \\\hline \text { March } & 3 & 189 \\\hline \text { April } & 4 & 201 \\\hline \text { May } & 5 & 195 \\\hline \text { June } & 6 & 199 \\\hline \text { July } & 7 & 206 \\\hline \text { August } & 8 & 203 \\\hline \text { September } & 9 & 208 \\\hline \text { October } & 10 & 209 \\\hline \text { November } & 11 & 218 \\\hline \text { December } & 12 & 216 \\\hline\end{array} a. Use the method of least squares to fit the mo E(Yt)=β0+β1\mathrm { E } \left( \mathrm { Y } _ { \mathrm { t } } \right) = \beta _ { 0 } + \beta _ { 1 } t to the data. Write the prediction equation.
b. Use the prediction equation to obtain forecasts for the next two months.
c. Find 95% forecast intervals for the next two months.
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54
The exponential smoothing constant can be any number between 0 and 100.
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55
We plot time series residuals against observed values of Y to determine whether a cyclical component is apparent.
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56
Consider the table below which displays the price of a commodity for six consecutive years.  Year  Price (dollars) 125022553253425552596261\begin{array} { c | c } \hline \text { Year } & \text { Price (dollars) } \\\hline 1 & 250 \\\hline 2 & 255 \\\hline 3 & 253 \\\hline 4 & 255 \\\hline 5 & 259 \\\hline 6 & 261 \\\hline\end{array} a. Use the Holt model to forecast values for Years 7-10 using w=0.6\mathrm { w } = 0.6 and v=0.5\mathrm { v } = 0.5 .
b. Calculate the forecast errors for Years 7-10 if the actual values in those years are 263, 267, 269, 268 respectively.
c. Calculate MAD, MAPE, and RMSE, using the forecast errors for Years 7-10.
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57
(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986 through 1992 are shown in the table below.  Year  G.M.  Ford  Chrysler 1986899358101796198771014328122519886762431312831989624442551182199077694934149419918256558520341992930555512157\begin{array} { l r c r } \hline \text { Year } & \text { G.M. } & \text { Ford } & \text { Chrysler } \\\hline 1986 & 8993 & 5810 & 1796 \\1987 & 7101 & 4328 & 1225 \\1988 & 6762 & 4313 & 1283 \\1989 & 6244 & 4255 & 1182 \\1990 & 7769 & 4934 & 1494 \\1991 & 8256 & 5585 & 2034 \\1992 & 9305 & 5551 & 2157 \\\hline\end{array}

-Using 1986 as the base year, find the simple composite index for 1992.
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58
Price indexes measure changes in the price of a commodity or group of commodities over time.
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59
Consider the table below which displays the price of a commodity for six consecutive years.
 Year  Price (dollars) 125022553253425552596261\begin{array}{c|c}\hline \text { Year } & \text { Price (dollars) } \\\hline 1 & 250 \\\hline 2 & 255 \\\hline 3 & 253 \\\hline 4 & 255 \\\hline 5 & 259 \\\hline 6 & 261 \\\hline\end{array}

a. Use the method of least squares to fit the model E(Yt)=β0+β1tE \left( Y _ { t } \right) = \beta _ { 0 } + \beta _ { 1 } t to the data. Write the prediction equation.
b. Use the prediction equation to obtain forecasts of the prices in years 7 and 8 .
c. Find 95%95 \% prediction intervals for years 7 and 8 .
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60
Retail sales for a home improvement store in quarters 1-4 over a five-year period are shown (in millions of dollars) in the table below.  Quarter  Year 123411.21.41.51.121.31.61.51.231.41.81.81.641.41.71.91.651.62.02.11.9\begin{array} { c | c | c | c | c } \hline { \text { Quarter } } \\\hline \text { Year } & 1 & 2 & 3 & 4 \\\hline 1 & 1.2 & 1.4 & 1.5 & 1.1 \\\hline 2 & 1.3 & 1.6 & 1.5 & 1.2 \\\hline 3 & 1.4 & 1.8 & 1.8 & 1.6 \\\hline 4 & 1.4 & 1.7 & 1.9 & 1.6 \\\hline 5 & 1.6 & 2.0 & 2.1 & 1.9 \\\hline\end{array} a. Write a regression model that contains trend and seasonal components to describe the sales data.
b. Use least squares regression to fit the model.
c. Use the regression model to forecast the quarterly sales during Year 6. Give 95% prediction intervals for the forecasts.
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61
The least squares model is an excellent choice for forecasting time series since it works particularly well outside the region of known observations.
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62
Smaller values of the trend smoothing constant v assign more weight to the most recent trend of the series and less to past trends.
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63
One of the major weaknesses of exponential smoothing is that it is not easily adapted to forecasting.
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64
Smoothing techniques are used to remove rapid fluctuations in a time series so the general trend can be seen.
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65
With N time periods in your data, a good rule of thumb is to forecast ahead no more than 2N time periods.
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66
A major advantage of forecasting with smoothing techniques is that the standard deviation of the forecast errors is known prior to observing the future values.
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67
The Holt forecasting model consists of both an exponentially smoothed component and a trend component.
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68
The straight-line regression model accounts for both the secular trend and the cyclical effect in a time series.
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69
The exponentially smoothed forecast takes into account both changes in trend and seasonality.
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70
The value of the Durbin-Watson d-statistic always falls in the interval from 0 to 1.
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71
The choice of exponential smoothing constant w has little or no effect on forecast values found using exponential smoothing.
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72
The Laspeyres index uses the purchase quantities of the period as weights.
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73
Smaller choices of the exponential smoothing constant w assign more weight to the current value of the series and yield a smoother series.
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