Construct A Time Series Plot. What Type Of Pattern Exists In The Data?

Problem i

Consider the post-obit time series data.
\[\brainstorm{assortment}{c|rrrrrr}\text { Week } & one & 2 & 3 & four & v & 6 \\
\hline \text { Value } & eighteen & 13 & 16 & xi & 17 & fourteen\finish{array}\]
Using the naive method (well-nigh recent value) as the forecast for the next calendar week, compute the following measures of forecast accuracy.
a. Hateful absolute error.
b. Hateful squared error.
c. Hateful absolute percentage error.
d. What is the forecast for week $7 ?$

Problem 2

Refer to the time series information in exercise $1 .$ Using the average of all the historical data as a forecast for the next menses, compute the following measures of forecast accuracy.
a. Mean accented mistake.
b. Mean squared fault.
c. Mean accented percentage error.
d. What is the forecast for week $7 ?$

Problem iii

Exercises 1 and 2 used different forecasting methods. Which method appears to provide the more accurate forecasts for the historical data? Explicate.

Problem 4

Consider the post-obit time serial data.
\[\brainstorm{array}{l|rrrrrrr}\text { Month } & 1 & 2 & three & four & 5 & 6 & 7 \\\hline \text { Value } & 24 & thirteen & 20 & 12 & 19 & 23 & 15
\cease{array}\]
a. Compute MSE using the most recent value equally the forecast for the next menstruation. What is the forecast for calendar month $8 ?$
b. Compute MSE using the average of all the data available as the forecast for the side by side period. What is the forecast for month $8 ?$
c. Which method appears to provide the better forecast?

Problem five

Consider the following time serial data.
\[\begin{assortment}{c|rrrrrr}\text { Calendar week } & 1 & two & 3 & 4 & v & vi \\
\hline \text { Value } & xviii & 13 & 16 & 11 & 17 & 14\end{assortment}\]
a. Construct a fourth dimension series plot. What type of pattern exists in the data?
b. Develop the three-week moving average forecasts for this fourth dimension series. Compute MSE and a forecast for week 7
c. Use $\alpha=.two$ to compute the exponential smoothing forecasts for the time series. Compute MSE and a forecast for week 7
d. Compare the three-week moving average approach with the exponential smoothing approach using $\alpha=.ii .$ Which appears to provide more accurate forecasts based on MSE? Explain.
east. Employ a smoothing constant of $\blastoff=.iv$ to compute the exponential smoothing forecasts. Does a smoothing constant of .2 or .4 announced to provide more accurate forecasts based on MSE? Explain.

Trouble 6

Consider the following time series data.
\[\begin{array}{l|rrrrrrr}\text { Calendar month } & 1 & 2 & 3 & 4 & 5 & six & 7 \\
\hline \text { Value } & 24 & xiii & 20 & 12 & 19 & 23 & 15\end{assortment}\]
Construct a time serial plot. What blazon of design exists in the data?
a. Develop the three-week moving boilerplate forecasts for this time serial. Compute MSE and a forecast for calendar week eight
b. Apply $\alpha=.ii$ to compute the exponential smoothing forecasts for the fourth dimension series. Compute MSE and a forecast for week viii
c. Compare the three-week moving average approach with the exponential smoothing approach using $\alpha=.two .$ Which appears to provide more accurate forecasts based on MSE?
d. Use a smoothing constant of $\blastoff=.4$ to compute the exponential smoothing forecasts. Does a smoothing constant of .ii or .4 appear to provide more authentic forecasts based on MSE? Explicate.

Problem 7

Refer to the gasoline sales time series data in Table 18.1
a. Compute four-week and 5-week moving averages for the time series.
b. Compute the MSE for the four-week and five-week moving average forecasts.
c. What appears to be the all-time number of weeks of by information (3, four, or five) to employ in the moving average computation? Recall that MSE for the three-week moving average is x.22.

Problem 8

Refer again to the gasoline sales fourth dimension series data in Table xviii.i
a. Using a weight of $1 / 2$ for the most recent ascertainment, $ane / iii$ for the second most recent ascertainment, and $one / six$ for third most recent observation, compute a three-week weighted moving average for the time series.
b. Compute the MSE for the weighted moving boilerplate in part (a). Do you prefer this weighted moving boilerplate to the unweighted moving average? Recall that the MSE for the unweighted moving average is ten.22
c. Suppose yous are allowed to cull any weights as long every bit they sum to $i .$ Could you always detect a fix of weights that would make the MSE at least every bit modest for a weighted moving average than for an unweighted moving boilerplate? Why or why not?

Problem 9

With the gasoline fourth dimension series data from Table $18.1,$ show the exponential smoothing forecasts using $\alpha=.ane$
a. Applying the MSE measure of forecast accurateness, would you adopt a smoothing constant of $\alpha=.1$ or $\alpha=.2$ for the gasoline sales time series?
b. Are the results the same if you lot apply MAE as the measure out of accuracy?
c. What are the results if MAPE is used?

Problem x

With a smoothing constant of $\blastoff=.ii,$ equation (18.ii) shows that the forecast for calendar week 13 of the gasoline sales data from Table xviii.1 is given by $F_{thirteen}=.2 Y_{12}+.viii F_{12} .$ However, the forecast for week 12 is given by $F_{12}=.2 Y_{eleven}+.viii F_{xi} .$ Thus, nosotros could combine these two results to evidence that the forecast for week 13 can be written
\[F_{13}=.2 Y_{12}+.8\left(.two Y_{eleven}+.viii F_{11}\right)=.2 Y_{12}+.xvi Y_{11}+.64 Y_{xi}+.64 F_{11}\]
a. Making use of the fact that $F_{xi}=.2 Y_{10}+.8 F_{10}$ (and similarly for $F_{10}$ and $F_{9}$ ), continue to expand the expression for $F_{13}$ until it is written in terms of the past data values $Y_{12}$ $Y_{eleven}, Y_{x}, Y_{9}, Y_{eight},$ and the forecast for menstruum eight.
b. Refer to the coefficients or weights for the past values $Y_{12}, Y_{11}, Y_{10}, Y_{nine}, Y_{viii} .$ What observation can you make about how exponential smoothing weights by data values in arriving at new forecasts? Compare this weighting design with the weighting pattern of the moving averages method.

Problem 11

For the Hawkins Company, the monthly percentages of all shipments received on time over the by 12 months are $80,82,84,83,83,84,85,84,82,83,84,$ and 83
a. Construct a time serial plot. What type of pattern exists in the data?
b. Compare the 3-month moving boilerplate approach with the exponential smoothing approach for $\alpha=.2 .$ Which provides more accurate forecasts using MSE as the measure of forecast accuracy?
c. What is the forecast for next month?

Trouble 12

Corporate triple-A bond interest rates for 12 consecutive months follow.
\[\begin{array}{lllllllllllll}ix.5 & 9.3 & 9.4 & nine.6 & 9.viii & 9.7 & 9.8 & 10.5 & 9.9 & 9.7 & 9.6 & 9.6\finish{array}\]
a. Construct a fourth dimension series plot. What type of pattern exists in the information?
b. Develop three-month and iv-month moving averages for this time series. Does the iii-month or four-month moving boilerplate provide more than accurate forecasts based on MSE? Explain.
c. What is the moving boilerplate forecast for the side by side calendar month?

Trouble 13

The values of Alabama building contracts (in $\$$ millions) for a 12 -month period follow.
\[
\begin{array}{lllllllll}
240 & 350 & 230 & 260 & 280 & 320 & 220 & 310 & 240 & 310 & 240 & 230
\stop{array}
\]
a. Construct a time series plot. What type of pattern exists in the data?
b. Compare the three-calendar month moving average arroyo with the exponential smoothing forecast using $\alpha=.2 .$ Which provides more than accurate forecasts based on MSE?
c. What is the forecast for the adjacent month?

Problem xiv

The following time series shows the sales of a particular product over the by 12 months.
$$\brainstorm{array}{cccc}
\text { Month } & \text { Sales } & \text { Month } & \text { Sales } \\
ane & 105 & 7 & 145 \\
2 & 135 & 8 & 140 \\
iii & 120 & nine & 100 \\
4 & 105 & 10 & 80 \\
v & 90 & 11 & 100 \\
half dozen & 120 & 12 & 110\finish{assortment}$$
a. Construct a time series plot. What blazon of blueprint exists in the data?
b. Employ $\alpha=.3$ to compute the exponential smoothing forecasts for the time serial.
c. Use a smoothing constant of $\alpha=.five$ to compute the exponential smoothing forecasts. Does a smoothing abiding of .3 or .5 announced to provide more accurate forecasts based on MSE?

Trouble 15

X weeks of information on the Article Futures Index are 7.35,7.forty,vii.55,7.56,7.60,seven.52 $7.52,7.70,7.62,$ and 7.55
a. Construct a fourth dimension series plot. What type of blueprint exists in the data?
b. Compute the exponential smoothing forecasts for $\blastoff=.2$
c. Compute the exponential smoothing forecasts for $\alpha=.three$
d. Which exponential smoothing constant provides more than accurate forecasts based on MSE? Forecast week 11.

Problem 16

The Nielsen ratings (percentage of U.Due south. households that tuned in) for the Masters Golf Tournament from 1997 through 2008 follow (Golf Mag, January 2009 ).
$$\brainstorm{array}{lr}
\text { Year } & \text { Rating } \\1997 & 11.2 \\1998 & 8.six \\1999 & 7.nine \\
2000 & 7.vi \\2001 & 10.7 \\2002 & eight.ane \\2003 & 6.9 \\2004 & 6.7 \\
2005 & 8.0 \\2006 & half dozen.9 \\2007 & vii.half-dozen \\2008 & vii.three\end{array}$$
The rating of 11.2 in 1997 indicates that $11.2 \%$ of U.S. households tuned in to watch Tiger Woods win his first major golf tournament and become the outset African American to win the Masters. Tiger Woods besides won the Masters in 2001 and 2005
a. Construct a time series plot. What blazon of pattern exists in the information? Hash out some of the factors that may take resulted in the pattern exhibited in the time series plot for this time series.
b. Given the pattern of the fourth dimension serial plot adult in part (a), exercise you call back the forecasting methods discussed in this section are advisable to develop forecasts for this time series? Explain.
c. Would y'all recommend using the Nielsen ratings for only $2002-2008$ to forecast the rating for 2009 , or should the entire time serial from $1997-2008$ be used? Explain.

Problem 17

Consider the following fourth dimension series data.
\[\brainstorm{array}{c|ccccc}\boldsymbol{t} & 1 & two & three & iv & 5 \\
\hline \boldsymbol{Y}_{\boldsymbol{t}} & half dozen & eleven & 9 & 14 & 15\cease{array}\]
a. Construct a time serial plot. What blazon of pattern exists in the information?
b. Develop the linear trend equation for this time serial.
c. What is the forecast for $t=6 ?$

Problem xviii

Refer to the time series in exercise $17 .$ Use Holt'southward linear exponential smoothing method with $\alpha=.iii$ and $\beta=.5$ to develop a forecast for $t=6$.

Trouble 19

Consider the following fourth dimension serial.
\[\begin{assortment}{c|rrrrrrr}t & i & 2 & three & 4 & 5 & vi & seven \\
\hline Y_{t} & 120 & 110 & 100 & 96 & 94 & 92 & 88
\end{array}\]
a. Construct a fourth dimension series plot. What type of pattern exists in the data?
b. Develop the linear trend equation for this time series.
c. What is the forecast for $t=viii ?$

Problem 20

Consider the post-obit fourth dimension serial.
\[\begin{array}{c|rrrrrrr}
t & 1 & ii & 3 & iv & five & vi & seven \\
\hline Y_{t} & 82 & lx & 44 & 35 & 30 & 29 & 35
\finish{array}\]
a. Construct a time series plot. What blazon of design exists in the data?
b. Using Minitab or Excel, develop the quadratic trend equation for the time serial.
c. What is the forecast for $t=8 ?$

Problem 21

Because of loftier tuition costs at state and private universities, enrollments at community colleges take increased dramatically in recent years. The following data show the enrollment (in thousands) for Jefferson Community College from $2001-2009$.
$$\begin{array}{ccc}\text { Twelvemonth } & \text { Menstruum }(t) & \text { Enrollment }(1000 \mathrm{south}) \\2001 & one & half dozen.v \\2002 & 2 & 8.i \\
2003 & 3 & viii.four \\2004 & 4 & 10.2 \\2005 & 5 & 12.5 \\2006 & 6 & thirteen.3 \\
2007 & 7 & 13.seven \\2008 & 8 & 17.2 \\2009 & 9 & eighteen.i\end{array}$$
a. Construct a time serial plot. What type of pattern exists in the data?
b. Develop the linear trend equation for this time series.
c. What is the forecast for $2010 ?$

Problem 22

The Seneca Children's Fund (SCF) is a local charity that runs a summer camp for disadvantaged children. The fund'due south board of directors has been working very hard in recent years to subtract the amount of overhead expenses, a major factor in how charities are rated past independent agencies. The following data show the percentage of the money SCF has raised that was spent on administrative and fund-raising expenses for $2003-2009$.

Problem 23

The president of a pocket-sized manufacturing house is concerned about the continual increase in manufacturing costs over the past several years. The following figures provide a time series of the toll per unit for the firm'southward leading product over the by eight years.
$$\begin{assortment}{cccc}
\text { Year } & \text { toll/Unit (\$) } & \text { Year } & \text { price/Unit (\$) } \\
1 & 20.00 & 5 & 26.60 \\
2 & 24.50 & vi & 30.00 \\
3 & 28.xx & 7 & 31.00 \\
4 & 27.50 & 8 & 36.00\end{array}$$
a. $\quad$ Construct a time series plot. What type of design exists in the data?
b. Develop the linear trend equation for this time series.
c. What is the average cost increase that the firm has been realizing per twelvemonth?
d. Compute an guess of the cost/unit of measurement for next year.

Trouble 24

FRED $^{\circ}$ (Federal Reserve Economic Data), a database of more than 3000 U.Due south. economic time series, contains historical data on foreign commutation rates. The post-obit data bear witness the foreign commutation rate for the The states and People's republic of china (Federal Reserve Bank of St.Louis website). The units for Rate are the number of Chinese yuan to i U.S. dollar.
$$\begin{assortment}{llc}
\text { Year } & \text { Month } & \text { Charge per unit } \\
2007 & \text { October } & 7.5019 \\
2007 & \text { November } & vii.4210 \\
2007 & \text { December } & 7.3682 \\
2008 & \text { January } & 7.2405 \\
2008 & \text { Feb } & 7.1644 \\
2008 & \text { March } & seven.0722 \\
2008 & \text { April } & 6.9997 \\
2008 & \text { May } & 6.9725 \\
2008 & \text { June } & 6.8993 \\
2008 & \text { July } & six.8355\stop{array}$$
a. Construct a time series plot. Does a linear trend appear to be present?
b. Using Minitab or Excel, develop the linear trend equation for this time series.
c. Employ the tendency equation to forecast the commutation charge per unit for August 2008
d. Would you lot feel comfortable using the trend equation to forecast the exchange rate for December $2008 ?$

Problem 25

Motorcar unit of measurement sales at $\mathrm{B}$. J. Scott Motors, Inc., provided the post-obit x -year time serial.
$$\brainstorm{assortment}{cccc}
\text { Year } & \text { Sales } & \text { Twelvemonth } & \text { Sales } \\
one & 400 & 6 & 260 \\
2 & 390 & seven & 300 \\
3 & 320 & 8 & 320 \\
4 & 340 & 9 & 340 \\
5 & 270 & x & 370\end{array}$$
a. Construct a time series plot. Comment on the ceremoniousness of a linear trend.
b. Using Minitab or Excel, develop a quadratic trend equation that tin can be used to forecast sales.
c. Using the tendency equation developed in part (b), forecast sales in year 11
d. Suggest an culling to using a quadratic trend equation to forecast sales. Explain.

Problem 26

Giovanni Food Products produces and sells frozen pizzas to public schools throughout the eastern United States. Using a very aggressive marketing strategy they accept been able to increase their annual acquirement past approximately $\$ x$ 1000000 over the past ten years. But increased competition has slowed their growth rate in the past few years. The almanac revenue, in millions of dollars, for the previous 10 years is shown.
$$\brainstorm{array}{cc}\text { Year } & \text { Revenue } \\1 & 8.53 \\2 & 10.84 \\
three & 12.98 \\iv & 14.11 \\5 & sixteen.31 \\vi & 17.21 \\7 & eighteen.37 \\viii & 18.45 \\9 & xviii.xl \\x & 18.43\end{array}$$
a. Construct a time series plot. Comment on the ceremoniousness of a linear tendency.
b. Using Minitab or Excel, develop a quadratic trend equation that can exist used to forecast revenue.
c. Using the tendency equation developed in role (b), forecast acquirement in year xi

Problem 27

Forbes magazine ranks NFL teams by value each year. The following data are the value of the Indianapolis Colts from 1998 to 2008 (Forbes website).
$$\begin{array}{lcc}
\text { Year } & \text { Catamenia } & \text { Value (\$millions) } \\
1998 & 1 & 227 \\
1999 & 2 & 305 \\
2000 & 3 & 332 \\
2001 & 4 & 367 \\
2002 & v & 419 \\
2003 & half dozen & 547 \\
2004 & vii & 609 \\
2005 & 8 & 715 \\
2006 & ix & 837 \\
2007 & x & 911 \\
2008 & 11 & 1076
\end{array}$$
a. Construct a time series plot. What type of design exists in the information?
b. Using Minitab or Excel, develop the quadratic trend equation that can exist used to forecast the team's value.
c. Using Minitab or Excel, develop the exponential trend equation that tin can be used to forecast the team's value.
d. Using Minitab or Excel, develop the linear trend equation that can be used to forecast the squad's value.
e. Which equation would you recommend using to estimate the squad's value in $2009 ?$
f. Use the model you recommended in part (e) to forecast the value of the Colts in 2009 .

Problem 28

Consider the following time series.
$$\begin{assortment}{cccc}
\text { Quarter } & \text { Year 1 } & \text { Yr two } & \text { Yr iii } \\
i & 71 & 68 & 62 \\
two & 49 & 41 & 51 \\
iii & 58 & sixty & 53 \\
four & 78 & 81 & 72
\end{array}$$
a. Construct a fourth dimension series plot. What type of pattern exists in the data?
b. Use the post-obit dummy variables to develop an estimated regression equation to account for seasonal effects in the data: $\mathrm{Qtr} 1=1$ if Quarter i,0 otherwise; $\mathrm{Qtr} 2=1 \mathrm{if}$ Quarter 2,0 otherwise; $\mathrm{Qtr} three=one$ if Quarter 3,0 otherwise.
c. Compute the quarterly forecasts for next year.

Problem 29

Consider the following time series data.
$$\brainstorm{array}{cccc}
\text { Quarter } & \text { Year 1 } & \text { Year 2 } & \text { Year iii } \\
i & 4 & 6 & 7 \\
two & 2 & 3 & 6 \\
3 & 3 & 5 & 6 \\
4 & 5 & 7 & 8
\end{array}$$
a. Construct a time series plot. What type of design exists in the data?
b. Use the following dummy variables to develop an estimated regression equation to business relationship for any seasonal and linear trend effects in the data: $\mathrm{Qtr} 1=one$ if Quarter ane,0 otherwise; $\mathrm{Qtr} 2=1$ if Quarter 2,0 otherwise; $\mathrm{Qtr} three=1$ if Quarter 3,0 otherwise.
c. Compute the quarterly forecasts for adjacent year.

Problem 30

The quarterly sales data (number of copies sold) for a higher textbook over the by iii years follow.
$$\brainstorm{assortment}{ccrr}
\text { Quarter } & \text { Twelvemonth one } & \text { Year ii } & \text { Twelvemonth three } \\
ane & 1690 & 1800 & 1850 \\
two & 940 & 900 & 1100 \\
3 & 2625 & 2900 & 2930 \\
4 & 2500 & 2360 & 2615
\stop{array}$$
a. Construct a time series plot. What blazon of pattern exists in the data?
b. Use the following dummy variables to develop an estimated regression equation to account for whatsoever seasonal furnishings in the information: $\mathrm{Qtr} 1=1$ if Quarter one,0 otherwise; $\mathrm{Qtr} 2=ane$ if Quarter 2,0 otherwise; $\mathrm{Qtr} 3=1$ if Quarter 3,0 otherwise.
c. Compute the quarterly forecasts for side by side year.
d. Allow $t=ane$ to refer to the observation in quarter 1 of year $one ; t=2$ to refer to the observation in quarter 2 of year $1 ; \ldots$ and $t=12$ to refer to the observation in quarter iv of yr $iii .$ Using the dummy variables defined in part (b) and $t,$ develop an estimated regression equation to business relationship for seasonal effects and whatsoever linear trend in the fourth dimension serial. Based upon the seasonal effects in the information and linear trend, compute the quarterly forecasts for next year.

Problem 31

Air pollution command 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 $15,sixteen,$ and $17,$ the following levels of nitrogen dioxide were observed for the 12 hours from 6: 00 a. . to 6: 00 ? . .
\[
\begin{array}{llllllllllll}
\text { July 15: } & 25 & 28 & 35 & 50 & threescore & sixty & 40 & 35 & 30 & 25 & 25 & 20 \\
\text { July 16: } & 28 & 30 & 35 & 48 & sixty & 65 & 50 & twoscore & 35 & 25 & 20 & 20 \\
\text { July 17: } & 35 & 42 & 45 & lxx & 72 & 75 & 60 & 45 & 40 & 25 & 25 & 25
\end{array}
\]
a. Construct a fourth dimension series plot. What type of design exists in the data?
b. Utilize the post-obit dummy variables to develop an estimated regression equation to business relationship for the seasonal furnishings in the data.
Hour $1=1$ if the reading was made betwixt 6: 00 a..... and seven: 00 A.M.; 0 otherwise 60 minutes $2=1$ if if the reading was made between 7: 00 A.yard. and eight: 00 A.Yard.; 0 otherwise
Hour11 $=i$ if the reading was made between 4: 00 p.
0 otherwise.
Notation that when the values of the 11 dummyivariables are equal to $0,$ the observation corresponds to the v: 00 P.M. to 6: 00 P.M. hour.
c. Using the estimated regression equation developed in part (a), compute estimates of the levels of nitrogen dioxide for July 18
d. Let $t=1$ to refer to the observation in hr 1 on July $xv ; t=two$ to refer to the ascertainment in hour 2 of July $15 ; \ldots$ and $t=36$ to refer to the ascertainment in hour 12 of July $17 .$ Using the dummy variables defined in part (b) and $t,$ develop an estimated regression equation to account for seasonal effects and any linear trend in the time series. Based upon the seasonal effects in the data and linear tendency, compute estimates of the levels of nitrogen dioxide for July eighteen.

Problem 32

South Shore Structure builds permanent docks and seawalls along the southern shore of Long Island, New York. Although the business firm has been in business only five years, revenue has increased from $\$ 308,000$ in the get-go year of operation to $\$ 1,084,000$ in the most contempo yr. The post-obit data testify the quarterly sales revenue in thousands of dollars.
$$\brainstorm{array}{ccccc}
\text { Quarter } & \text { Year ane } & \text { Year ii } & \text { Yr 3 } & \text { Yr 4 } & \text { Year five } \\
ane & 20 & 37 & 75 & 92 & 176 \\
two & 100 & 136 & 155 & 202 & 282 \\
3 & 175 & 245 & 326 & 384 & 445 \\
4 & xiii & 26 & 48 & 82 & 181
\cease{array}$$
a. Construct a time series plot. What blazon of pattern exists in the information?
b. Employ the following dummy variables to develop an estimated regression equation to account for seasonal furnishings in the data. $\mathrm{Qtr} 1=1$ if Quarter 1,0 otherwise; $\mathrm{Qtr} two=ane$
if Quarter 2,0 otherwise; $\mathrm{Qtr} 3=1$ if Quarter iii,0 otherwise. Based merely on the seasonal effects in the information, compute estimates of quarterly sales for year 6
c. Allow Flow $=1$ to refer to the observation in quarter 1 of twelvemonth $1 ;$ Menstruation $=ii$ to refer to the observation in quarter 2 of year $1 ; \ldots$ and Period $=20$ to refer to the observation in quarter 4 of year
five. Using the dummy variables defined in office (b) and Period, develop an estimated regression equation to business relationship for seasonal effects and any linear tendency in the time series. Based upon the seasonal effects in the data and linear trend, compute estimates of quarterly sales for year 6.

Problem 33

Electric ability consumption is measured in kilowatt-hours (kWh). The local utility company offers an interrupt program whereby commercial customers that participate receive favorable rates just must agree to cut back consumption if the utility requests them to do so. Timko Products has agreed to cut back consumption from noon to eight: 00 p.M. on Thursday. To determine Timko's savings, the utility must estimate Timko'due south normal power usage for this period of time. Data on Timko's electric power consumption for the previous 72 hours are shown below.
$$\begin{assortment}{lcccc}
\text { Fourth dimension Period } & \text { Mon } & \text { Tuesday } & \text { Midweek } & \text { Thursday } \\
12-4 \text { a..... } & - & nineteen,281 & 31,209 & 27,330 \\
4-8 \mathrm{A} . \mathrm{M} & - & 33,195 & 37,014 & 32,715 \\
viii-12 \text { apex } & - & 99,516 & 119,968 & 152,465 \\
12-4 \mathrm{P.G} & 124,299 & 123,666 & 156,033 & \\
four-eight \mathrm{PM} . & 113,545 & 111,717 & 128,889 & \\
eight-12 \text { midnight } & 41,300 & 48,112 & 73,923 &\terminate{array}$$
a. Construct a time series plot. What type of pattern exists in the data?
b. Employ the following dummy variables to develop an estimated regression equation to account for whatsoever seasonal effects in the data.
Fourth dimension $1=one$ for time period $12-4$ a.n. $; 0$ otherwise Time $2=i$ for time period $4-eight$ a. $\mathrm{Grand} ; 0$ otherwise Time3 $=i$ for time period $eight-12$ noon; 0 otherwise Time4 $=one$ for time period $12-4$ p. $\mathrm{M} ; 0$ otherwise Time5 $=i$ for time catamenia $4-8$ P.Thousand; 0 otherwise
c. Employ the estimated regression equation developed in part (b) to estimate Timko'southward normal usage over the menses of interrupted service.
d. Allow Period $=ane$ to refer to the observation for Monday in the fourth dimension period $12-4$ P.M. Catamenia $=two$ to refer to the observation for Monday in the time period $four-8 \mathrm{P}, \mathrm{M} ; \ldots$ and Period $=18$ to refer to the observation for Thursday in the fourth dimension menstruation $8-12$ noon. Using the dummy variables defined in function (b) and Flow, develop an estimated regression equation to account for seasonal effects and any linear trend in the time series.
eastward. Using the estimated regression equation developed in function
(d), estimate Timko's normal usage over the period of interrupted service.

Problem 34

Three years of monthly backyard-maintenance expenses ( $\$$ ) for a six-unit apartment business firm in southern Florida follow.
a. Construct a time serial plot. What type of blueprint exists in the data?
b. Develop an estimated regression equation that tin exist used to business relationship for whatsoever seasonal and linear trend effects in the data. Utilize the following dummy variables to account for the seasonal effects in the data: Jan $=one$ if January, 0 otherwise; $\mathrm{Feb}=1$ if February, 0 otherwise; $\mathrm{Mar}=1$ if March, 0 otherwise $; \ldots \mathrm{Nov}=i$ if Nov, 0 otherwise. Note that using this coding method, when all the 11 dummy variables are 0 , the ascertainment corresponds to an expense in December.
c. Compute the monthly forecasts for side by side year based upon both trend and seasonal effects.

Problem 35

Consider the post-obit time series data.
$$\begin{assortment}{cccc}
\text { Quarter } & \text { Year 1 } & \text { Year 2 } & \text { Year iii } \\
i & four & vi & vii \\
ii & 2 & 3 & 6 \\
three & iii & 5 & half dozen \\
4 & five & vii & 8
\end{array}$$
a. Construct a time serial plot. What type of design exists in the data?
b. Show the four-quarter and centered moving average values for this time series.
c. Compute seasonal indexes and adjusted seasonal indexes for the four quarters.

Problem 36

Refer to do 35
a. Deseasonalize the time series using the adapted seasonal indexes computed in function (c) of exercise 35
b. Using Minitab or Excel, compute the linear tendency regression equation for the deseasonalized information.
c. Compute the deseasonalized quarterly trend forecast for Year 4.
d. Use the seasonal indexes to arrange the deseasonalized trend forecasts computed in function (c)

Trouble 37

The quarterly sales information (number of copies sold) for a college textbook over the past three years follow.
$$\brainstorm{assortment}{cccc}
\text { Quarter } & \text { Year 1 } & \text { Yr 2 } & \text { Yr 3 } \\
1 & 1690 & 1800 & 1850 \\
2 & 940 & 900 & 1100 \\
three & 2625 & 2900 & 2930 \\
four & 2500 & 2360 & 2615
\end{array}$$
a. Construct a fourth dimension series plot. What type of pattern exists in the data?
b. Prove the iv-quarter and centered moving average values for this time series.
c. Compute the seasonal and adjusted seasonal indexes for the four quarters.
d. When does the publisher take the largest seasonal index? Does this result appear reasonable? Explain.
east. Deseasonalize the time serial.
f. Compute the linear trend equation for the deseasonalized information and forecast sales using the linear trend equation.
grand. Adjust the linear trend forecasts using the adjusted seasonal indexes computed in part (c)

Trouble 38

Three years of monthly lawn-maintenance expenses (\$) for a vi-unit apartment business firm in southern Florida follow.
$$\brainstorm{array}{lccc}
\text { Month } & \text { Yr one } & \text { Twelvemonth 2 } & \text { Year three } \\
\text { Jan } & 170 & 180 & 195 \\
\text { February } & 180 & 205 & 210 \\
\text { March } & 205 & 215 & 230 \\
\text { April } & 230 & 245 & 280 \\
\text { May } & 240 & 265 & 290 \\
\text { June } & 315 & 330 & 390 \\
\text { July } & 360 & 400 & 420 \\
\text { Baronial } & 290 & 335 & 330 \\
\text { September } & 240 & 260 & 290 \\
\text { Oct } & 240 & 270 & 295 \\
\text { Nov } & 230 & 255 & 280 \\
\text { December } & 195 & 220 & 250\end{array}$$
a. Construct a time series plot. What type of blueprint exists in the data?
b. Place the monthly seasonal indexes for the 3 years of backyard-maintenance expenses for the apartment firm in southern Florida as given hither. Use a 12 -calendar month moving boilerplate calculation.
c. Deseasonalize the fourth dimension series.
d. Compute the linear trend equation for the deseasonalized data.
e. Compute the deseasonalized trend forecasts and so adjust the trend forecasts using the seasonal indexes to provide a forecast for monthly expenses in year 4.

Problem 39

Air pollution command specialists in southern California monitor the amount of ozone, carbon dioxide, and nitrogen dioxide in the air on an hourly basis. The hourly time serial data exhibit seasonality, with the levels of pollutants showing patterns over the hours in the day. On July $fifteen,16,$ and $17,$ the following levels of nitrogen dioxide were observed in the downtown area for the 12 hours from six: 00 A.One thousand. to 6: 00 P.One thousand.
\[\brainstorm{array}{lllllllllllll}
\text { July 15: } & 25 & 28 & 35 & l & lx & 60 & twoscore & 35 & 30 & 25 & 25 & 20 \\
\text { July 16: } & 28 & 30 & 35 & 48 & 60 & 65 & 50 & 40 & 35 & 25 & twenty & xx \\
\text { July 17: } & 35 & 42 & 45 & 70 & 72 & 75 & sixty & 45 & 40 & 25 & 25 & 25\end{assortment}\]
a. Construct a time series plot. What type of pattern exists in the data?
b. Identify the hourly seasonal indexes for the 12 readings each mean solar day.
c. Deseasonalize the time serial.
d. Using Minitab or Excel, compute the linear tendency equation for the deseasonalized data.
e. Compute the deseasonalized trend forecasts for the 12 hours for July xviii and so arrange the trend forecasts using the seasonal indexes computed in part (b).

Problem 40

Electric ability consumption is measured in kilowatt-hours (kWh). The local utility company offers an interrupt plan whereby commercial customers that participate receive favorable rates just must concur to cut back consumption if the utility requests them to practise so. Timko Products cut back consumption at 12: 00 noon Th. To assess the savings, the utility must estimate Timko's usage without the interrupt. The period of interrupted service was from noon to viii: 00 p.Thousand. Data on electric ability consumption for the previous 72 hours are available.
$$\begin{array}{lrrr}
\text { Time Catamenia } & \text { Monday } & \text { Tuesday } & \text { Wednesday } & \text { Thursday } \\
12-4 \mathrm{A} . \mathrm{M} . & - & 19,281 & 31,209 & 27,330 \\
iv-viii \mathrm{A} . \mathrm{Thou} & - & 33,195 & 37,014 & 32,715 \\
eight-12 \text { noon } & -99,516 & 119,968 & 152,465 \\
12-4 \mathrm{P.M.} & 124,299 & 123,666 & 156,033 & \\
4-8 \mathrm{P.Yard} & 113,545 & 111,717 & 128,889 & \\
viii-12 \text { midnight } & 41,300 & 48,112 & 73,923 &
\end{assortment}$$
a. Is there a seasonal effect over the 24 -hour catamenia?
b. Compute seasonal indexes for the six iv-hour periods.
c. Use trend adjusted for seasonal indexes to gauge Timko's normal usage over the period of interrupted service.

Trouble 41

The weekly demand (in cases) for a particular make of automatic dishwasher detergent for a chain of grocery stores located in Columbus, Ohio, follows.
$$\begin{assortment}{cccc}
\text { Week } & \text { Demand } & \text { Week } & \text { Demand } \\
one & 22 & six & 24 \\
2 & 18 & seven & 20 \\
3 & 23 & 8 & 19 \\
4 & 21 & 9 & 18 \\
v & 17 & x & 21\end{array}$$
a. Construct a fourth dimension series plot. What blazon of design exists in the information?
b. Apply a three-calendar week moving average to develop a forecast for week 11
c. Employ exponential smoothing with a smoothing constant of $\blastoff=.2$ to develop a forecast for week 11
d. Which of the two methods do y'all prefer? Why?

Trouble 42

The following tabular array reports the pct of stocks in a portfolio for nine quarters from 2007 to 2009.
$$\begin{assortment}{cc}
\text { Quarter } & \text { Stock \% } \\
\text { 1st-2007 } & 29.8 \\
\text { 2nd-2007 } & 31.0 \\
\text { 3rd-2007 } & 29.ix \\
\text { 4th-2007 } & 30.1 \\
\text { 1st-2008 } & 32.2 \\
\text { 2d-2008 } & 31.5 \\
\text { 3rd-2008 } & 32.0 \\
\text { fourth-2008 } & 31.nine \\
\text { 1st-2009 } & thirty.0
\stop{array}$$
a. Construct a time series plot. What type of pattern exists in the data?
b. Use exponential smoothing to forecast this fourth dimension series. Consider smoothing constants of $\alpha=.2, .3,$ and.four. What value of the smoothing abiding provides the virtually accurate forecasts?
c. What is the forecast of the percentage of stocks in a typical portfolio for the 2nd quarter of $2009 ?$

Problem 43

United Dairies, Inc., supplies milk to several independent grocers throughout Dade Canton, Florida. Managers at United Dairies want to develop a forecast of the number of half-gallons of milk sold per week. Sales information for the past 12 weeks follow.
$$\begin{assortment}{cccc}
\text { Week } & \text { Sales } & \text { Week } & \text { Sales } \\
ane & 2750 & seven & 3300 \\
2 & 3100 & 8 & 3100 \\
3 & 3250 & ix & 2950 \\
4 & 2800 & x & 3000 \\
5 & 2900 & 11 & 3200 \\
vi & 3050 & 12 & 3150\finish{array}$$
a. Construct a time series plot. What type of blueprint exists in the information?
b. Use exponential smoothing withf $\alpha=.4$ to develop a forecast of demand for calendar week 13

Problem 44

To avoid a monthly service fee in an interest-bearing checking account, customers must maintain a minimum boilerplate daily residual. Bankrate's 2008 survey of 249 banks and thrifts in the height 25 metropolitan areas showed that you demand to maintain an average remainder of $\$ three,462$ to avert a monthly service fee. With an average fee of $\$ 11.97$ and an average interest rate of only 0.24 percentage, customers with interest-begetting checking accounts are not getting much value for basically providing the banking concern with a line of credit equal to the average monthly balance required to avert the monthly service fee (Bankrate website, October 27,2008 ). The post-obit table shows the minimum average remainder required to avoid paying a monthly service fee from $2001-2008$.
$$\begin{array}{cc}\text { Yr } & \text { Balance (\$) } \\2001 & 2435 \\2002 & 2593 \\
2003 & 2258 \\2004 & 2087 \\2005 & 2294 \\2006 & 2660 \\
2007 & 3317 \\2008 & 3462\end{assortment}$$
a. Construct a time series plot. What type of design exists in the data?
b. Using Minitab or Excel, develop a linear trend equation for the time series. Compute an estimate of the average balance required to avert a monthly service fee for 2009
c. Using Minitab or Excel, develop a quadratic tendency equation for the time series. Compute an estimate of the average remainder required to avoid a monthly service fee for 2009
d. Using MSE, which approach provides the about accurate forecasts for the historical data?
e. For these information would you recommend that the forecast for 2009 be developed using the linear tendency equation or the quadratic tendency equation? Explain.

Problem 45

The Garden Avenue Seven sells CDs of its musical performances. The following table reports sales (in units) for the by 18 months. The group'southward manager wants an accurate method for forecasting future sales.
$$\brainstorm{aligned}
&-\\
&\begin{array}{ccccc}
\text { Month } & \text { Sales } & \text { Month } & \text { Sales } & \text { Month } & \text { Sales } \\
i & 293 & 7 & 381 & thirteen & 549 \\
2 & 283 & viii & 431 & 14 & 544 \\
3 & 322 & 9 & 424 & 15 & 601 \\
4 & 355 & ten & 433 & 16 & 587 \\
5 & 346 & 11 & 470 & 17 & 644 \\
6 & 379 & 12 & 481 & 18 & 660
\finish{assortment}\\&\text { the }\terminate{aligned}$$
a. Construct a fourth dimension series plot. What blazon of pattern exists in the information?
b. Use exponential smoothing with $\blastoff=.iii, .iv,$ and .5. Which value of $\alpha$ provides the most authentic forecasts?
c. Apply trend projection to provide a forecast. What is the value of MSE?
d. Which method of forecasting would you recommend to the manager? Why?

Problem 46

The Mayfair Section Shop in Davenport, Iowa, is trying to determine the corporeality of sales lost while it was shut down during July and August because of impairment caused by the Mississippi River flood. Sales information for January through June follow.
$$\begin{array}{lccc}
\text { Calendar month } & \text { Sales (\$1000s) } & \text { Month } & \text { Sales (\$1000s) } \\
\text { January } & 185.72 & \text { April } & 210.36 \\
\text { February } & 167.84 & \text { May } & 255.57 \\
\text { March } & 205.11 & \text { June } & 261.nineteen\end{assortment}$$
a. Utilise exponential smoothing, with $\alpha=.4,$ to develop a forecast for July and August. (Hint: Use the forecast for July every bit the bodily sales in July in developing the Baronial forecast.) Comment on the use of exponential smoothing for forecasts more than one period into the future.
b. Use trend projection to forecast sales for July and Baronial.
c. Mayfair'due south insurance company proposed a settlement based on lost sales of $\$ 240,000$ in July and Baronial. Is this amount fair? If not, what amount would you recommend as a counteroffer?

Trouble 47

Canton Supplies, Inc., is a service house that employs approximately 100 individuals. Managers of Canton Supplies are concerned about coming together monthly cash obligations and want to develop a forecast of monthly greenbacks requirements. Because of a recent change in operating policy, but the by seven months of data that follow are considered to be relevant.
\[\begin{array}{l|rrrrrrr}
\text { Month } & 1 & 2 & three & iv & 5 & half-dozen & vii \\
\hline \text { Greenbacks Required (\$1000s) } & 205 & 212 & 218 & 224 & 230 & 240 & 246
\terminate{array}\]
a. Construct a time series plot. What type of blueprint exists in the data?
b. Apply Holt'southward linear exponential smoothing with $\alpha=.6$ and $\beta=.4$ to forecast cash requirements for each of the next two months.
c. Using Minitab or Excel, develop a linear trend equation to forecast cash requirements for each of the next two months.
d. Would you recommend using Holt's linear exponential smoothing with $\blastoff=.vi$ and $\beta=.iv$ to forecast cash requirements for each of the adjacent two months or the linear trend equation? Explicate.

Problem 48

The costello Music Company has been in concern for 5 years. During that time, sales of pianos increased from 12 units in the first twelvemonth to 76 units in the well-nigh recent year. Fred costello, the firm's owner, wants to develop a forecast of pianoforte sales for the coming year. The historical data follow.
\[\brainstorm{array}{l|rrrrr}\text { Yr } & i & 2 & iii & 4 & 5 \\
\hline \text { Sales } & 12 & 28 & 34 & fifty & 76\end{assortment}\]
a. Construct a time series plot. What type of pattern exists in the data?
b. Develop the linear trend equation for the time series. What is the average increase in sales that the firm has been realizing per year?
c. Forecast sales for years 6 and vii.

Problem 49

Consider the costello Music Company problem in exercise $48 .$ The quarterly sales data follow.
$$\begin{assortment}{ccccc}
& & & & \text { } \\
\text { Year } & \text { Quarter i } & \text { Quarter 2 } & \text { Quarter iii } & \text { Quarter 4 } & \text { Total Yearly Sales } \\
1 & 4 & 2 & 1 & 5 & 12 \\
2 & 6 & 4 & iv & xiv & 28 \\
three & 10 & 3 & 5 & sixteen & 34 \\
four & 12 & 9 & 7 & 22 & l \\
five & 18 & 10 & xiii & 35 & 76\end{array}$$
a. Use the following dummy variables to develop an estimated regression equation to account for whatsoever seasonal and linear tendency effects in the data: $\mathrm{Qtr} 1=1$ if Quarter 1,0 otherwise; $\mathrm{Qtr} two=i$ if Quarter two,0 otherwise; and $\mathrm{Qtr} three=1$ if Quarter 3,0 otherwise.
b. Compute the quarterly forecasts for next year.

Problem 50

Refer to the costello Music Company fourth dimension series in exercise 49
a. Deseasonalize the data and utilise the deseasonalized time series to identify the trend.
b. Use the results of office (a) to develop a quarterly forecast for side by side yr based on trend.
c. Use the seasonal indexes developed in exercise 50 to adjust the forecasts developed in office (b) to account for the event of season.

Trouble 51

Refer to the costello Music Company time series in do 49
a. Deseasonalize the data and employ the deseasonalized fourth dimension series to place the trend.
b. Use the results of office (a) to develop a quarterly forecast for next year based on trend.
c. Use the seasonal indexes developed in practice fifty to adjust the forecasts developed in part (b) to business relationship for the issue of flavor.

Problem 52

Hudson Marine has been an authorized dealer for C\&D marine radios for the past 7 years. The post-obit table reports the number of radios sold each year.
\[\begin{array}{l|rrrrrrr}
\text { Year } & 1 & ii & 3 & 4 & 5 & half-dozen & 7 \\
\hline \text { Number Sold } & 35 & 50 & 75 & 90 & 105 & 110 & 130
\cease{array}\]
a. Construct a time series plot. Does a linear tendency appear to be present?
b. Using Minitab or Excel, develop a linear trend equation for this fourth dimension series.
c. Use the linear trend equation developed in part (b) to develop a forecast for annual sales in year 8

Problem 53

Refer to the Hudson Marine problem in do $52 .$ Suppose the quarterly sales values for the seven years of historical data are as follows.
$$\begin{array}{cccccc}
& & & & \text { Total Yearly } \\
\text { Twelvemonth } & \text { Quarter 1 } & \text { Quarter 2 } & \text { Quarter 3 } & \text { Quarter 4 } & \text { Sales } \\
1 & 6 & fifteen & ten & iv & 35 \\
2 & 10 & 18 & fifteen & 7 & l \\
3 & fourteen & 26 & 23 & 12 & 75 \\
4 & 19 & 28 & 25 & 18 & 90 \\
5 & 22 & 34 & 28 & 21 & 105 \\
6 & 24 & 36 & 30 & 20 & 110 \\
seven & 28 & 40 & 35 & 27 & 130\end{array}$$
a. Use the following dummy variables to develop an estimated regression equation to account for any season and linear trend effects in the information: $\mathrm{Qtr} 1=1$ if Quarter 1,0 otherwise; $\mathrm{Qtr} 2=one$ if Quarter 2,0 otherwise; and $\mathrm{Qtr} 3=1$ if Quarter 3,0 otherwise.
b. Compute the quarterly forecasts for next year.

Problem 54

Refer to the Hudson Marine problem in exercise 53
a. Compute the centered moving average values for this time serial.
b. Construct a time series plot that also shows the centered moving boilerplate and original time series on the same graph. Discuss the differences between the original time serial plot and the centered moving average time serial.
c. Compute the seasonal indexes for the iv quarters.
d. When does Hudson Marine experience the largest seasonal event? Does this consequence seem reasonable? Explain.

Problem 55

Refer to the Hudson Marine data in practise 53
a. Deseasonalize the data and utilise the deseasonalized time series to identify the trend.
b. Use the results of role (a) to develop a quarterly forecast for adjacent twelvemonth based on trend.
c. Utilise the seasonal indexes developed in exercise 54 to arrange the forecasts developed in part (b) to business relationship for the result of season.