MIDDLE EAST TECHNICAL UNIVERSITY I E 2 66
–
E N GI N EE RI N G
S T A T I S T I C S II
CASE STUDY 2 Burcu YÜZÜAK Cem YOĞURTCU Onur YILMAZ
Instructor:
YASEMİN SERİN
May 2010, Ankara
Part I:
In the first part, as our manager assigned we are trying to find out whether or not there is a relationship between sales amounts and “seasonality” seasonality”, “price of one bottle of `DAZNEDAR RAKI'” RAKI'”, “average rakı price per bottle in the national market” and “average beer price in the national market market””. We carried out a multiple linear regression analysis to investigate these possible relationships. In order to do multiple regression analysis, we set our variables as:
X1:
average rakı price per bottle in the national market (market price)
X2:
price of one bottle of `DAZNEDAR RAKI' („Daznedar Rakı‟ price)
X3:
average beer price in the national market (beer price)
And in order to control the effects of seasonality on the sales, we assigned seasonal variables with respect to months of seasons:
X4:
spring months
X5 :
summer months
X6 :
fall months
We used only three variables because the rest of the months, which are in the winter season, will be simultaneously set. And our model: Y = β0 + β1X1 + β2X2 + β3X3 + β4X4 + β5X5 + β6X6 1
Following this, we applied regression analysis according to this h ypothesis: H0: β1= β2= β3= β4= β5= β6= 0 HA: otherwise
And results of the analysis are given as Minitab outputs:
The regression equation is sales bottles = - 34867 + 11538 market price - 8347 daznedar raki price + 170 beer market price - 70 spring - 55 summer - 25 fall
Predictor Constant market price daznedar raki price beer market price spring summer fall
Coef -34867 11537.7 -8346.6 169.7 -69.6 -55.3 -25.4
S = 1188.79
SE Coef 3901 205.0 252.9 483.3 592.2 616.3 632.2
T -8.94 56.27 -33.00 0.35 -0.12 -0.09 -0.04
P 0.000 0.000 0.000 0.728 0.907 0.929 0.968
R-Sq = 99.5%
R-Sq(adj) = 99.4%
SS 8263146690 39570015 8302716706
MS 1377191115 1413215
Analysis of Variance Source Regression Residual Error Total
D 6 28 34
F 974.51
P 0.000
Residual Plots for sales sal es bottles bottles No rmal Pro b abilit y Plot
Versus Fit s l a u 3 d i s e 2 R d e 1 z i d r 0 a d n -1 a t S
99 90 t n e c 50 r e P
10 1 -2
0
2
4
20000
40000
Standardized Residual
Hist o g ram
80000
Versus Ord er l a u 3 d i s e 2 R d e 1 z i d r 0 a d n a -1 t S
10.0
y c n e u q e r F
60000
Fitted Value
7.5 5.0 2.5 0.0 -1
0
1
2
Standardized Residual
3
1
5
10
15
20
25
30
35
Observation Order
Residual Plots for sales bottles (Retrieved from Minitab Output) 2
These given Minitab outputs show that, only “average market price in the national market” and “price of one bottle Daznedar Raki” are significant since only related p values equal to zero. However, when we looked at the residual plots, we saw there is a pattern. Therefore, we decided to use
rather than Y to eliminate this pattern.
Now our model became:
=β0+β1X1+ β2X2+ β3X3+ β4X4+ β5X5+ β6X6
Following this change, when we tested the same hypothesis again with the new model we acquired these outputs:
The regression equation is kök bottles = 23.9 + 24.6 market price - 17.3 daznedar raki price - 0.115 beer market price - 0.204 spring - 0.350 summer - 0.065 fall
Predictor Constant market price daznedar raki price beer market price spring summer fall
S = 0.601078
Coef 23.877 24.6433 -17.2593 -0.1155 -0.2038 -0.3503 -0.0654
R-Sq = 100.0%
SE Coef 1.972 0.1037 0.1279 0.2443 0.2994 0.3116 0.3197
T 12.11 237.71 -134.96 -0.47 -0.68 -1.12 -0.20
P 0.000 0.000 0.000 0.640 0.502 0.270 0.839
R-Sq(adj) = 100.0%
Analysis of Variance Source Regression Residual Error Total
DF 6 28 34
SS 39420.1 39420.1 10.1 39430.2
MS 6570.0 0.4
F 18184.62
P 0.000
3
Residual Plots for kök bottles No rmal Pro b abilit y Plo t
Versus Fit s
99
l a u 2 d i s e R 1 d e z 0 i d r a -1 d n a t -2 S
90 t n e c 50 r e P 10 1 -2
-1
0
1
2
150
200
Standardized Residual
250
300
Fitted Value
Hist o gram
Versus Ord er l a u 2 d i s e R 1 d e z 0 i d r a -1 d n a t S -2
8
y 6 c n e u 4 q e r F 2 0 -2
-1
0
1
Standardized Residual
2
1
5
10
15
20
25
30
35
Observation Order
Residual Plots for kök bottles (Retrieved from Minitab Output)
When we check the new residual plots it is seen that the pattern is gone and still only “market price” price” and “Daznedar raki price” price” are significant since only their p-values are equal to zero as highlighted in the Minitab output above. P-values of “beer market price” price ”, “summer” summer ”, “spring” spring” and “fall” fall” seasons are 0.640, 0.502, 0.270 and 0.839 respectively. These values indicate that there is no linear relationship between these conditions and sales volume on the other hand it is evident from the output that there is a relationship between “average raki price per bottle in the national market” market” and “price of one bottle of `DAZNEDAR RAKI'” RAKI' ” and sales volume.
“beer market price”, We realized realized that “beer price”, “summer”, “spring” and “fall” seasons resulted as insignificant but we need to check whether they are significant when we make regression analaysis for each of them indivudally. So we tested the hypothesis for four of insignificant variables, but results didn‟t change. change . They were still insignificant and an example from these individual significance analyses which is conducted for „beer market price‟ could be seen 4
below. As reported in the Minitab output below, beer market price has a p-value of 0,950 and it shows that “beer market price” is still insignificant for the volume of sales.
The regression equation is kök bottles = 23.3 + 24.6 market price - 17.2 daznedar raki price - 0.014 beer market price
Predictor Constant market price daznedar raki price beer market price S = 0.586642
Coef 23.301 24.6225 -17.2348 -0.0141
R-Sq = 100.0%
SE Coef 1.788 0.0962 0.1151 0.2231
T 13.03 256.03 -149.79 -0.06
P 0.000 0.000 0.000 0.950
R-Sq(adj) = 100.0%
Analysis of Variance Source Regression Residual Error Total
DF 3 31 34
SS 39420 11 39430
MS 13140 0
F 38180.70
P 0.000
Part II: In part II, Ms. Bahar is considering a price promotion to increase revenues and she wants us to determine the price that maximizes expected revenue for the next month. Moreover it is stated that the price could be decreased at most to 32 TL and we should determine the price level from {32, 33, 34, 35, 36}. Now we have only these variables: X1 :
average rakı price per bottle in the national market (market price)
X2 :
price of one bottle of `DAZNEDAR RAKI' („Daznedar Rakı‟ price)
With these variables our model become as
=β0+β1X1+ β2X2
Then we did a regression analysis on Minitab again to determine the correct model and the outputs are shown below: 5
The regression equation is kök bottles = 23.2 + 24.6 market price - 17.2 daznedar raki price
Predictor Constant market price daznedar raki price
S = 0.577440
Coef 23.226 24.6205 -17.2317
SE Coef 1.321 0.0891 0.1022
R-Sq = 100.0%
T 17.58 276.37 -168.59
P 0.000 0.000 0.000
R-Sq(adj) = 100.0%
Analysis of Variance Source Regression Residual Error Total
DF 2 32 34
SS 39420 11 39430
MS 19710 0
F 59110.93
P 0.000
From output we could state coefficients as β0=23.2
β1=24.6
β2=-17.2
and using these, our correct model is:
=23.2+24.6X1-17.2X2
Using our model, we calculated expected revenues for the prices in the given range.
Calculation for price = 32TL
P(
P(
P(
)=
)=
P(Y) =
)=
E(Revenue) = 0.3*
+ 0.6*
+ 0.1*
= 565215.488
6
Calculation for price = 33TL
P(
)=
P(
)=
P(
P(Y )=
)=
E(Revenue) = 0.3*
+ 0.6*
+ 0.1*
= 442703.976
Calculation for price = 34TL
P(
)=
P(
)=
P(
P(Y) =
)=
E(Revenue) = 0.3*
+ 0.6*
Price Level (TL) 32 33 34 Summary of calculations
+ 0.1*
= 331814.16
Expected Revenue (TL) 565215.488 442703.976 331814.16
7
As it can be seen from the summary of calculations above, total expected revenue is decreasing as the price level increases. The decrease of revenues in spite of the increase of price level could be interpreted as the decrease in the sales volume because the increase of prices affects buyers negatively. Moreover, considering this rising down effect, there is no need to for further calculations for higher prices such as 35 and 36 TL. To sum up, in order to increase the expected profits and also revenues, price promotion level should be 32 TL.
8
