chapter 3 forecasting. forecasting demand why is demand forecasting important? what is bad about...
TRANSCRIPT
Chapter 3Chapter 3
ForecastingForecasting
Forecasting DemandForecasting Demand
Why is demand forecasting important?Why is demand forecasting important?
What is bad about poor forecasting?What is bad about poor forecasting?
What do these organizations forecast:What do these organizations forecast: Sony (consumer products division)Sony (consumer products division) Foley’sFoley’s Dallas Area Rapid Transit (DART)Dallas Area Rapid Transit (DART) UTAUTA
Questions in Demand ForecastingQuestions in Demand Forecasting
For a particular product or service:For a particular product or service: What exactly is to be forecasted?What exactly is to be forecasted? What will the forecasts be used for?What will the forecasts be used for? What What forecasting periodforecasting period is most useful? is most useful? What What time horizontime horizon in the future is to be in the future is to be
forecasted?forecasted? How many periods of past data should be used?How many periods of past data should be used? What What patternspatterns would you expect to see? would you expect to see? How do you select a forecasting model?How do you select a forecasting model?
Demand ManagementDemand Management
Recognizing and planning for all sources of demandRecognizing and planning for all sources of demandCan demand be controlled or influenced?Can demand be controlled or influenced? appointment schedulesappointment schedules
doctor’s officedoctor’s office attorneyattorney SAM telephone registrationSAM telephone registration
sales promotionssales promotions restaurant discounts before 6pmrestaurant discounts before 6pm video rental store discounts on Tuesdaysvideo rental store discounts on Tuesdays golf course discounts if you start playing after 4pmgolf course discounts if you start playing after 4pm theater matinee movie discountstheater matinee movie discounts
Qualitative vs. QuantitativeQualitative vs. QuantitativeForecasting MethodsForecasting Methods
Some Qualitative Methods:Some Qualitative Methods: Experienced guess/judgmentExperienced guess/judgment Consensus of committeeConsensus of committee Survey of sales forceSurvey of sales force Survey of all customersSurvey of all customers Historical analogyHistorical analogy
new productsnew products Market researchMarket research
survey a sample of customerssurvey a sample of customers test market a producttest market a product
Steps for Quantitative Forecasting MethodsSteps for Quantitative Forecasting Methods
1.1. Collect past data—usually the more the betterCollect past data—usually the more the better
2.2. Identify patterns in past dataIdentify patterns in past data
3.3. Select one or more appropriate forecasting methodsSelect one or more appropriate forecasting methods
4.4. Forecast part of past data with each methodForecast part of past data with each method Determine best parameters for each methodDetermine best parameters for each method Compare forecasts with actual dataCompare forecasts with actual data
5.5. Select method that had smallest forecasting errors on Select method that had smallest forecasting errors on past datapast data
6.6. Forecast future time periodsForecast future time periods
7.7. Determine prediction interval (forecast range)Determine prediction interval (forecast range)
8.8. Monitor forecasting accuracy over timeMonitor forecasting accuracy over time Tracking signalTracking signal
Types of Quantitative Forecasting MethodsTypes of Quantitative Forecasting Methods
Pattern ProjectionPattern Projection– time series regressiontime series regression– trend or seasonal modelstrend or seasonal models
Data SmoothingData Smoothing– moving averagemoving average– exponential smoothingexponential smoothing
CausalCausal– multiple regressionmultiple regression
Data Pattern ComponentsData Pattern Components
Sales
Time
LEVEL
Sales
Time
TREND
Sales
Time
SEASONALITY
Sales
Time
CYCLICALITY
Sales
Time
NOISE
De
c
De
c
De
c
19
80
19
86
Identifying Data Patterns for Time SeriesIdentifying Data Patterns for Time Series
Always Plot Data FirstAlways Plot Data First After plotting data, patterns are often obvious.After plotting data, patterns are often obvious.
Average or levelAverage or level Use mean of all dataUse mean of all data
TrendTrend Use time series regression – slope is trend – time period is Use time series regression – slope is trend – time period is
independent variableindependent variable
SeasonalitySeasonality Deseasonalize the dataDeseasonalize the data
CyclicalityCyclicality Similar to deseasonalizingSimilar to deseasonalizing
Random noiseRandom noise No pattern – try to eliminate in forecastsNo pattern – try to eliminate in forecasts
Forecast AccuracyForecast Accuracy
n
EMAD
n
tt
1 Deviation AbsoluteMean
tperiodfor forecast
tperiodfor demand actual or
tperiodfor error forecast
t
ttttt
t
F
ADFDE
E
Forecast AccuracyForecast Accuracy
n
EME
n
EMSE
n
tt
n
tt
1
1
2
(Bias)Error Mean
Error SquaredMean
Forecast Accuracy ExampleForecast Accuracy Example
Period APeriod Att F Ftt E Ett |E |Ett| (E| (Ett))22
1 32 301 32 30
2 28 312 28 31
3 31 333 31 33
4 34 354 34 35
5 34 335 34 33
6 36 346 36 34
Totals:Totals:
2
-3
-2
-1
1
2
2 43
2
1
1
2
9
4
1
1
4
-1 11 23
Forecast Accuracy ExampleForecast Accuracy Example
Bias =Bias =
MAD =MAD =
MSE =MSE =
-1/6 = -0.17
11/6 = 1.83
23/6 = 3.83
Quantity of Electric Irons Shipped by U.S. Mfgs.Quantity of Electric Irons Shipped by U.S. Mfgs.
0
2
4
6
8
10
12
14
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
mill
ion
un
its
Electric Irons Example -- DataElectric Irons Example -- Data
Year QtyYear Qty Year QtyYear Qty
1979 12.079 1984 7.8431979 12.079 1984 7.843
1980 11.478 1985 6.8341980 11.478 1985 6.834
1981 11.013 1986 7.6601981 11.013 1986 7.660
1982 6.616 1987 5.9181982 6.616 1987 5.918
1983 7.279 1988 7.1151983 7.279 1988 7.115
10-year average = 10-year average =
Last-7-year average = Last-7-year average =
8.38
7.04
Do time series regression analysisDo time series regression analysis
Y = a + bXY = a + bX
Y = dependent variable (actual sales)Y = dependent variable (actual sales)
X = independent variable (time period in this case)X = independent variable (time period in this case)
a = y-intercept (value of Y when X=0)a = y-intercept (value of Y when X=0)
b = slope or trendb = slope or trend
where N = number of periods of datawhere N = number of periods of data
N
Xb
N
Ya
XXN
YXXYNb 22
Electric Irons ExampleElectric Irons Example
X Y XX Y X22 XY XY
4 6.616 16 26.464 6.616 16 26.46
5 7.279 25 36.405 7.279 25 36.40
6 7.843 : :6 7.843 : :
7 6.834 : :7 6.834 : :
8 7.660 : :8 7.660 : :
9 5.918 : :9 5.918 : :
10 7.115 : :10 7.115 : :
== ===== === ======= ===== === =====
49 49.265 371 343.4549 49.265 371 343.45
b =b =
a =a =
Y =Y =
YY1111 = =
YY1212 = =
-.05
7.388
7.388 - .05X
7.388 - .05(11) = 6.838
7.388 - .05(12) = 6.788
Moving Company Sales
0
50
100
150
200
250
Sp 19
88Sum Fall W
in
Sp 19
89Sum Fall W
in
Sp 19
90Sum Fall W
in
Sp 19
91Sum Fall W
in
Nu
mb
er
of
Tru
ck
s L
ea
se
d
Overlay the Years
0
50
100
150
200
250
Spring Summer Fall Winter
Nu
mb
er
of
Tru
cks
Le
ase
d
1988
1989
1990
1991
Seasonality and Trend Patterns Seasonality and Trend Patterns (Seasonalized Regression)(Seasonalized Regression)
Steps:Steps:
1. Deseasonalize the data to remove seasonality1. Deseasonalize the data to remove seasonality divide by seasonal index (SI)divide by seasonal index (SI)
2. Use regression to model trend2. Use regression to model trend
3. Make initial forecasts to project trend3. Make initial forecasts to project trend
4. Seasonalize the forecast4. Seasonalize the forecast multiply by SImultiply by SI
Moving Company ExampleMoving Company Example
Spring Summer Fall WinterSpring Summer Fall Winter
1988 90 160 70 120 1988 90 160 70 120 Overall Avg.Overall Avg.
1989 130 200 90 100 2020/161989 130 200 90 100 2020/16
1990 80 170 130 140 = 126.251990 80 170 130 140 = 126.25
1991 1991 130130 210210 80 80 120120
Total: 430 740 370 480Total: 430 740 370 480
Avg: 107.5 185 92.5 120Avg: 107.5 185 92.5 120
SI:SI: 0.85 1.47 0.73 0.95
(107.5/126.25) (120/126.25)
Deseasonalize the DataDeseasonalize the Data
Spring Summer Fall WinterSpring Summer Fall Winter
1988 105.71988 105.7** 109.2 95.5 126.3 109.2 95.5 126.3
1989 152.7 136.5 122.8 105.21989 152.7 136.5 122.8 105.2
1990 94.0 116.01990 94.0 116.0++ 177.4 147.3 177.4 147.3
1991 152.7 143.3 109.2 126.31991 152.7 143.3 109.2 126.3
** Spring 1988: 90/.851 = 105.7 Spring 1988: 90/.851 = 105.7++ Summer 1990: 170/1.465 = 116.0 Summer 1990: 170/1.465 = 116.0
Perform Time Series RegressionPerform Time Series Regression
X Y XX Y X22 XY XY
1 105.7 1 105.71 105.7 1 105.7
2 109.2 4 218.42 109.2 4 218.4
3 95.5 9 286.63 95.5 9 286.6
4 : : :4 : : :
: : : :: : : :
16 126.3 256 2020.016 126.3 256 2020.0
=== ====== ===== ========== ====== ===== =======
136 2,020.0 1,496 17,773.5 Totals136 2,020.0 1,496 17,773.5 Totals
b =b =
a =a =
Y =Y =
N
Xb
N
Ya
XXN
YXXYNb 22
1.779
111.216
111.216 + 1.779(X)
Make initial forecasts:Make initial forecasts:
YY1717 = =
YY1818 = =
YY1919 = =
YY2020 = =
Make final forecasts: (Seasonalize F = Y x SI)Make final forecasts: (Seasonalize F = Y x SI)
FF1717 = =
FF1818 = =
FF1919 = =
FF2020 = =
111.216 + 1.779(17) = 141.46
(18) = 143.24
(19) = 145.02
(20) = 146.80
141.46 x 0.85 = 120.24
143.24 x 1.47 = 210.56
145.02 x 0.73 = 105.87
146.80 x 0.95 = 139.46
Gasoline Service Station Monthly Sales
6
7
8
9
10
11
12
13
0 6 12 18 24 30 36 42 48 54 60 66 72 78
month
bill
ion
$
Gasoline Service Station Monthly Sales
6
7
8
9
10
11
12
13
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
month
bill
ion
$
1985
1986
1987
1988
1989
1990
Deseasonalized Sales
6
7
8
9
10
11
12
13
0 6 12 18 24 30 36 42 48 54 60 66 72 78
month
bill
ion
$
Regression Line
6
7
8
9
10
11
12
13
0 6 12 18 24 30 36 42 48 54 60 66 72 78
month
bill
ion
$
Final Forecasts
6
7
8
9
10
11
12
13
0 6 12 18 24 30 36 42 48 54 60 66 72 78
month
bill
ion
$
Actual Sales
6
7
8
9
10
11
12
13
0 6 12 18 24 30 36 42 48 54 60 66 72 78
month
bill
ion
$
Past Sales Forecasts Actuals
Forecast RangingForecast Ranging
Forecasts are rarely perfect!Forecasts are rarely perfect!
A forecast range reflects the degree of confidence A forecast range reflects the degree of confidence that you have in your forecasts.that you have in your forecasts.
Forecast ranging allows you to estimate a Forecast ranging allows you to estimate a prediction interval for actual demandprediction interval for actual demand
““There is a ___% probability that actual demand There is a ___% probability that actual demand will be within the upper and lower limits of the will be within the upper and lower limits of the forecast range.”forecast range.”
Standard Error of the ForecastStandard Error of the Forecast(a measure of dispersion of the forecast errors)(a measure of dispersion of the forecast errors)
Upper Limit = FUpper Limit = Fii + t(s + t(syxyx))
Lower Limit = FLower Limit = Fii - t(s - t(syxyx))
Need desired Need desired level of significance (level of significance (αα)) and and degrees degrees of freedom (df)of freedom (df) to look up to look up tt in table in table
2n
xybyays
2
yx
Forecast Confidence Intervals(Forecast Ranging)
LowerLimit
UpperLimit
Ft
t(Syx)α/2 α/2
(1 – α)
Actual Salesin Future
tt-statistic and -statistic and degrees of freedomdegrees of freedom
For a confidence interval of 95%,For a confidence interval of 95%,αα = .05 (.025 in = .05 (.025 in each tail), and df=16each tail), and df=16
From table, t = From table, t =
Why does df = n-2 for simple regression?Why does df = n-2 for simple regression?
If the forecast was from a multiple regression If the forecast was from a multiple regression model with 3 independent variables, what would model with 3 independent variables, what would be the degrees of freedom? df = n - __be the degrees of freedom? df = n - __
2.120
Y = a + bX estimate 2 parameters from data
4Y = a + b1X1 + b2X2 + b3X3 estimate 4 parameters
ExampleExample: Judy manages a large used car dealership that has : Judy manages a large used car dealership that has experienced a steady growth in sales during the last few years. experienced a steady growth in sales during the last few years. Using time series regression and sales data for the last 20 quarters, Using time series regression and sales data for the last 20 quarters, Judy obtained a forecast of 800 car sales for next quarter. With her Judy obtained a forecast of 800 car sales for next quarter. With her model and the past data the standard error of the forecast was 50 model and the past data the standard error of the forecast was 50 cars. What are the limits for a 95% forecast range? for an 80% cars. What are the limits for a 95% forecast range? for an 80% forecast range?forecast range?
95% range: α = .05 n = 20 df = n – 2 = 18 t = 2.101
UL, LL = Ft ± t(Syx) = 800 ± 2.101(50)
UL = 905LL = 695
= 800 ± 105
80% range: α = .20 n = 20 df = 18 t = 1.330
UL, LL = Ft ± t(Syx) = 800 ± 1.330(50) = 800 ± 66.5
UL = 866.5LL = 733.5
ExampleExample: A manager’s forecast of next month’s sales of product Q : A manager’s forecast of next month’s sales of product Q was 1500 units using time series regression based on the last 24 was 1500 units using time series regression based on the last 24 months of sales, which had a standard forecast error of 29 units. months of sales, which had a standard forecast error of 29 units. Her boss asked how sure she was that actual sales would be within Her boss asked how sure she was that actual sales would be within 50 units of her forecast.50 units of her forecast.
n = 24 df = n – 2 = 22
UL = Ft + t(Syx)
1550 = 1500 + t(29)
t = 50/29 = 1.724
From Appendix B, α = .1 (closest column to 1.724 in row 22)
So, she is 90% sure that actual sales will be within 50 units of the forecast.
Short Range ForecastingShort Range Forecasting
A few days to a few monthsA few days to a few months Assumes there are no patterns in the dataAssumes there are no patterns in the data Random noise has a greater impact in the short Random noise has a greater impact in the short
termterm These approaches try to eliminate some of the These approaches try to eliminate some of the
random noiserandom noise Random walk, moving average, weighted Random walk, moving average, weighted
moving average, exponential smoothingmoving average, exponential smoothing
Random WalkRandom Walk
The next forecast is equal to the last period’s The next forecast is equal to the last period’s actual valueactual value
PeriodPeriod SalesSales ForecastForecast
1 1 2121
22 3030
33 2727
44 ? ? 27
Moving Average MethodMoving Average Method
The next forecast is equal to an average of the last The next forecast is equal to an average of the last AP periods of actual dataAP periods of actual data
PeriodPeriod SalesSales AP=4AP=4 AP=3AP=3 AP=2AP=2
11 21 21
22 28 28
33 35 35
44 30 30
55 ? ? 28.5 31.0 32.5
AP=1
30
Impulse ResponseImpulse Response – how fast the forecasts react to – how fast the forecasts react to changes in the datachanges in the data
The higher the value of AP, the less the forecast will react The higher the value of AP, the less the forecast will react to changes in the data, so the lower the impulse to changes in the data, so the lower the impulse response is.response is.
Noise DampeningNoise Dampening – how much the forecasts are smoothed – how much the forecasts are smoothed
Noise dampening is the opposite of impulse response.Noise dampening is the opposite of impulse response.
A moving average model with AP=1 has high impulse A moving average model with AP=1 has high impulse response and low noise dampening characteristics.response and low noise dampening characteristics.
Weighted Moving Average methodWeighted Moving Average method
Like the moving average method except that each of the Like the moving average method except that each of the AP periods can have a different weightAP periods can have a different weight
Actual Actual AP=4 AP=4PeriodPeriod SalesSales WeightWeight 11 21 21 .1 .1 22 28 28 .15 .15 33 35 35 .25 .25 44 30 30 .5 .5 55 ? ?
Usually the recent periods have more weightUsually the recent periods have more weight
xxxx
= 2.1= 4.2= 8.75= 15
30.05
Exponential SmoothingExponential Smoothing
Most common short-term quantitative forecasting method Most common short-term quantitative forecasting method (especially for forecasting inventory levels)(especially for forecasting inventory levels)
Why?Why? surprisingly accuratesurprisingly accurate easy to understandeasy to understand simple to usesimple to use very little data is storedvery little data is stored
Need 3 pieces of data to make forecastNeed 3 pieces of data to make forecast1. most recent forecast1. most recent forecast2. actual sales for that period2. actual sales for that period3. smoothing constant (3. smoothing constant (αα))
Exponential Smoothing methodExponential Smoothing method
– gives a different weight to each periodgives a different weight to each period
FFt t = F= Ft-1 t-1 + + αα(A(At-1t-1 – F – Ft-1t-1))
αα is the smoothing parameter and is between 0 and 1 is the smoothing parameter and is between 0 and 1
Interpretation: the next forecast equals last period’s Interpretation: the next forecast equals last period’s forecast plus a percentage of last period’s forecasting forecast plus a percentage of last period’s forecasting error.error.
Alternative formula:Alternative formula:
FFtt = = ααAAt-1t-1 + (1 - + (1 - αα)F)Ft-1t-1(rearranging terms)(rearranging terms)
Example: assume Example: assume αα = 0.3 = 0.3
We must We must assumeassume a forecast for an earlier period a forecast for an earlier period
PeriodPeriod SalesSales ForecastForecast
11 21 21
22 24 24
33 23 23
44 19 19
55 22 22
66 ? ?
Assume Period 1 forecast is 21
21
.3(21) + .7(21) = 21
.3(24) + .7(21) = 21.9
.3(23) + .7(21.9) = 22.23
.3(19) + .7(22.23) = 21.26
.3(22) + .7(21.26) = 21.48
Find best value for Find best value for αα by trial and error by trial and error
The larger The larger αα is, the more weight that is placed on is, the more weight that is placed on the more recent periods’ actual values, so the the more recent periods’ actual values, so the higher the impulse and the lower the noise higher the impulse and the lower the noise dampening.dampening.
Tracking SignalTracking Signal
After a forecasting method has been selected, After a forecasting method has been selected, tracking signaltracking signal is used to monitor accuracy of is used to monitor accuracy of the method as time passesthe method as time passes
Particularly good at identifying underforecasting or Particularly good at identifying underforecasting or overforecasting trendsoverforecasting trends
Tracking Signal =Tracking Signal =
Ideal value for tracking signal is ___Ideal value for tracking signal is ___
MAD
)(E Errors of Sum t
0
Guidelines would be used if the value exceeds specified Guidelines would be used if the value exceeds specified limitslimits
Example: Suppose exponential smoothing is used (Example: Suppose exponential smoothing is used (αα = .2) = .2)
If |TS| < 2.3 then do not change If |TS| < 2.3 then do not change αα
If |TS| > 2.3 then increase If |TS| > 2.3 then increase αα by .1 by .1
If |TS| > 3.0 then increase If |TS| > 3.0 then increase αα by .3 by .3
If |TS| > 3.6 then increase If |TS| > 3.6 then increase αα by .5 by .5
After tracking signal goes back down, restore original value After tracking signal goes back down, restore original value of of αα or calculate new or calculate new αα
Double Exponential SmoothingDouble Exponential Smoothing(Exponential Smoothing with Trend)(Exponential Smoothing with Trend)
Two smoothing constants are used:Two smoothing constants are used:
αα smoothes out random variations smoothes out random variations
ββ smoothes out trends smoothes out trends
An alternative to time series regressionAn alternative to time series regression
Especially useful if there is much random variationEspecially useful if there is much random variation
Winter’s Exponential SmoothingWinter’s Exponential Smoothing
Accounts for trend and seasonalityAccounts for trend and seasonality
Three smoothing constants are usedThree smoothing constants are usedαα smoothes out random variations smoothes out random variationsββ smoothes out trends smoothes out trendsγγ smoothes out seasonality smoothes out seasonality
There are many other variations of exponential There are many other variations of exponential smoothingsmoothing
Box-Jenkins Forecasting ApproachBox-Jenkins Forecasting Approach
Relatively accurate, but complex and time Relatively accurate, but complex and time consuming to useconsuming to use
Needs at least 60 pointsNeeds at least 60 points
Good choice if there are not many time series to Good choice if there are not many time series to forecast, and accuracy is very importantforecast, and accuracy is very important
Works best when random variation is a small Works best when random variation is a small componentcomponent
Example: monthly automobile registrations in U.S.Example: monthly automobile registrations in U.S.
Forecast = DForecast = Dtt + D + Dt-11t-11 – D – Dt-12t-12 – 0.21E – 0.21Ett – 0.21E – 0.21Et-1t-1
– – 0.85E0.85Et-11t-11 + 0.18E + 0.18Et-12t-12 + 0.22E + 0.22Et-13t-13
wherewhere DDtt = Actual demand for time period t = Actual demand for time period t
EEtt = Error term for time period t = Error term for time period t
Focus ForecastingFocus Forecasting(Forecasting Simulation)(Forecasting Simulation)
Bernard Smith at American Hardware Supply developed Bernard Smith at American Hardware Supply developed this method to make forecasts for 100,000 itemsthis method to make forecasts for 100,000 items
Based on 2 principles:Based on 2 principles:– sophisticated methods don’t always work bettersophisticated methods don’t always work better– no single method works best for all itemsno single method works best for all items
Buyers tended not to use the previous exponential Buyers tended not to use the previous exponential smoothing model because they did not trust or smoothing model because they did not trust or understand it. Instead, they were making up their own understand it. Instead, they were making up their own simple rule-of-thumb approaches.simple rule-of-thumb approaches.
Smith selected 7 forecasting methods to use, such as Smith selected 7 forecasting methods to use, such as 1. sales = last month’s sales plus a percentage1. sales = last month’s sales plus a percentage2. sales = sales for same month last year plus a %2. sales = sales for same month last year plus a %3. 2-month moving average3. 2-month moving average4. exponential smoothing4. exponential smoothingetc. (most were relatively simple)etc. (most were relatively simple)
All methods were used to forecast each product.All methods were used to forecast each product.Whichever method worked best for the previous month, Whichever method worked best for the previous month,
that method was used to forecast the next month.that method was used to forecast the next month.
Approach worked very well, and people understood and Approach worked very well, and people understood and used it. Smith wrote a popular book describing his used it. Smith wrote a popular book describing his approach and success.approach and success.
Multiple Regression ForecastingMultiple Regression ForecastingSales = f($advertising, #salespeople, $price)Sales = f($advertising, #salespeople, $price)
SalesSales AdvAdv PeoplePeople PricePrice
52005200 350350 1818 5353
56005600 520520 1818 5252
51005100 400400 1515 5454
38003800 320320 1313 6464
52005200 410410 1616 5151
49004900 290290 1717 6060
52005200 390390 1717 5454
54005400 470470 2020 5555
47004700 450450 1414 6161
50005000 500500 1515 5858
51005100 470470 1818 6060
SUMMARY OUTPUTSUMMARY OUTPUT
Regression StatisticsRegression Statistics
Multiple RMultiple R 0.9520.952
R SquareR Square 0.9070.907
Adjusted R SquareAdjusted R Square 0.8670.867
Standard ErrorStandard Error 170.988170.988
ObservationsObservations 1111
ANOVAANOVA
dfdf SSSS MSMS FF Significance FSignificance F
RegressionRegression 33 1991704.9741991704.974 663901.7663901.7 22.70822.708 0.000550.00055
ResidualResidual 77 204658.662204658.662 29236.9529236.95
TotalTotal 1010 2196363.6362196363.636
CoefficientsCoefficients Standard ErrorStandard Error t Statt Stat P-valueP-value
InterceptIntercept 5839.3475839.347 1236.0031236.003 4.7244.724 0.0020.002
AdvAdv 1.7421.742 0.7650.765 2.2772.277 0.0570.057
PeoplePeople 100.207100.207 30.72330.723 3.2623.262 0.0140.014
PricePrice -56.478-56.478 14.99914.999 -3.765-3.765 0.0070.007
Multiple Regression ExampleMultiple Regression Example
Suppose the manager wants toSuppose the manager wants to
forecast sales if $430 in advertising, forecast sales if $430 in advertising,
19 salespeople, and a price of $64 19 salespeople, and a price of $64
per unit are planned.per unit are planned.
Forecasting equationForecasting equation::
Sales = 5839.347 + 1.742(adv) + 100.207(people) – 56.478(price)Sales = 5839.347 + 1.742(adv) + 100.207(people) – 56.478(price)
Sales = Sales =
Sales =Sales =
CoefficientsCoefficients
InterceptIntercept 5839.3475839.347
AdvAdv 1.7421.742
PeoplePeople 100.207100.207
PricePrice -56.478-56.478
5839.347 + 1.742(430) + 100.207(19) – 56.478(64)
4877.748
