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Task 1A brewing company wants to launch a new canned larger to the market. Company has close
links with a major supermarket chain which will only permit a promotion of the larger in two of
its stores. Store A and Store B, which has been selected, monitor their sales of al brand of
canned lager over a weekend period with the following results:
Sales value of
lager (USD)
Frequency
Store A Store B
0-2.5 27 1
2.5-5 114 3
5-7.5 333 31
7.5-10 530 142
10-12.5 504 328
12.5-15 334 498
15-17.5 121 504
17.5-20 29 351
20-22.5 5 110
22.5-25 2 29
25-27.5 1 3
Total 2,000 2,000
To make a comparison between 2 stores A and B, we can calculate the mean
consumer expenditure in each store. Mean is calculated from the sum of values of
items divided by the number of items. After calculating, we find that the mean
consumer expenditure for store A is 10.07 and for store B is 14.93 . Based on the
frequency distribution below, for store A, the mean are affected by minority of times
with very high sales values. Or in other word, we can say that because of the
frequency for high sales values in store A is not high so the mean are not affected
much. More than 50% of times (1000 times) in which the sales values are less than
$10 in store A whereas in store B there is a half of times in which the sales value is
higher than $15. The mean in store B is more reliable because it is effected both bylow and high sales values.
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(To know how to calculate the mean, see Appendix2.1).
As we have already seen, the mean is representative for the average of population. In order
to understand more about the variability of customer expenditure, we can compute the
quartile range , variance, standard deviation and the coefficient of variation .
First, quartiles are one means of identifying the range within which most of the values in the
population occur. We calculate 2 quartiles: Q1 (the 25th
percentile), Q3 (the 75th
percentile)and then take Q3 minus Q1 to get the quartile range . For store A, the quartile range is 4.84
while in store B is 5. From this, we can know that the range of values of the middle half of
the population for store A is 4,84 units and for store B is 5.01 units.
Variance shows the extent to which a set of values depart from uniformity. The variance is
the average of the square mean deviation for each value in a distribution. After calculating,
the variance for store A is 12.86 and for store B is 13.25. This means that the sales value of
larger are closely clustered around 12,86 in store A and 13,25 in store B.
Standard deviation is the square root of the variance . To avoid the negative number when
minus the mid-point and the mean, we have to square the differences. Then, we need to
take the square root to get back the original data. Standard deviation is the most widely used
measure of dispersion.
Finally, to compare the dispersion of two distributions with similar mean, we use the
coefficient of variation . The coefficient of variation indicates how large the standard
deviation is in relation to the mean. For store A, the coefficient of variation is 0.36 and for
store B is 0.24. That means the store B has wider dispersion than store A.
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(To see the formula, see Appendix2.2; 2.3; 2.4; 2.5)
Based on the calculation above, store B seems will have a better opportunity for a
successful promotion of the new larger. First, the skewness of store B, which is symmetrical
frequency distribution with the shape of a bell, is one that can be divided into two halves.
The mean, median and the mode have nearly the same value whereas in store A, they are
skewed.
Next, the mean of store B is higher than in store A (store A: 10.07; store B: 14.93). This
means the average values of store B is higher so choosing store B will create more
opportunity to have a higher volume of sales. Moreover, the mean of store A is less accurate
than store B. Because the mean of store A is affected by the extreme value while in store B,
mean is affected by both low and high values. Last, the coefficient of variation of store B is
0.24 and in store A is 0.36. That means from the central tendency of store A is more spread
out all other values of the distribution than in store B. From these reasons, the revenue in
store B is generally higher than that in store A. Hence, it seems more suitable to promote
new lager in store B.
Task 2The manager of a club wanted to know how the temperature affected beer sales in order to
determine how many beers to stock. Accordingly, she took ten records of beer sales at
different temperature and listed below:
Temperature ( oC) Sale volume of beer
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(litres)
27 20.533
20 1.439
26 13.82926 21.286
31 30.985
23 17.187
30 30.240
33 37.596
25 9.610
29 28.742
From these data above, we can draw the scatterplot. In scatterplot, it is easy to see a trend
that the higher the temperature is, the lager volumes of sales are sold. It can be said that
temperature and the sale volume is partial and positive correlated. The hot weather lead to
the change in demand for beer, therefore this is a cause-effect relationship.
To know how to predict values for one variable (y) given values for the other
variable (x), in this case is to predict the sale volume from the temperature; we need to find a
line which is best fits for all the points on a scattergraph. After calculating, the linear
regression equation that best fits the data is Y = -51.23 + 2.68 * X
(To know how to calculate, see Appendix2.6).
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1 = 2.68 means that when the temperature increase 1 oC, the sale volume of beer will
increases by an average of 2.68 litters. Moreover, 1 have a positive number, which
correspond to a positive relation between sale volume and temperature.
o=-51.23 means that other reasons unrelated to the temperature make the salevolume change equal to -51.23. o have negative number, so, to have a positive volume of
sales, 1 * X should have the positive number. Hence, if the temperature is less than 20 0C,
based on the linear regression, no one will drink beer.
Use the regression equation to estimate sale volumes of beer when the temperature
changes, we have a table below:
Temperature Sale volume
28 23.8332 34.5535 42.5939 53.31
These forecast is very reliable because that the relationship between temperature and beer
sales has strong linear relationship . The correlation coefficient (R) between temperature
and sales is 0.94. (To know how to calculate, see Appendix2.7.) The correlation coefficient
measures the degree of correlation between two variables. The more R is closer to 1, the
stronger linear relationship between X and Y. Hence, the variables are closed to perfectly
positively correlation . It means that this relationship doesnt have function between X and
Y but the trend is nearly the same.
Task 3This report aims to forecast the value of sale for 4 quarters in 2008. The report includes
two parts. Part 1 show audience that the time series contain two clear components: trend
and season. Part 2 forecasts the value of sales for 4 quarters in 2008.
Part 1:
The sale volume is recorded from 2004 and 2007. It means that sale volume has been
observed for a number of periods so this is a time series . A time series can consist of 4
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difference components: long-term trend, seasonal variation, cyclical variation, random
(irregular) variation. In this part of the report, we will prove that the time series contain two
clear components: trend and season by using the technique of three-quarter moving
averages and plotting the original data and the trend values on a scatter diagram .
The trend is the underlying long-term movement over time in the values of the data
recorded. From quarterly figures on unit sales for the last four years, we can have the
scatterplot below:
Long-term trend shows a long-term pattern or direction exhibited by a series. There is an
upward trend in the sales over time. Although sales went down sometimes, the basic
movement of sales from 2004 to 2007 is one of raising sale.
Seasonal variations are short-term fluctuations in recorded values, due to difference
circumstances which affect results at different time of the year. This is typically due to 4
seasons in the year or particular patterns during a month, a week, or a day. In this case, sale
will be divided by 4 quarters per year.
There are three methods of finding a trend: inspection, regression analysis by the least
squares method and moving average. In this case, we will use the moving average
method . This method attempts to remove seasonal (or cyclical) variations by a process of
averaging.
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From the graph, it seems that the trend to rise overtime. However, seasonal component is
not clear, moving average will help to solve that.
Year Quarter Time period sales(Y)moving
average(T)
2004
1 1 441.102 2 397.70 411.633 3 396.10 422.204 4 472.80 448.43
2005
1 5 476.40 467.872 6 454.40 460.533 7 450.80 486.234 8 553.50 528.33
2006
1 9 580.70 569.132 10 573.20 575.173 11 571.60 616.134 12 703.60 655.73
2007
1 13 692.00 690.702 14 676.50 676.133 15 659.90 696.404 16 752.80
(To know how to calculate moving average, see Appendix2.8.)
After calculating three-quarter moving averages, we can draw a moving average plot for
sales:
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The trend is detected more clearly after using moving average. This is an upward trend .
The sales increase over period of time.
Then, to find the seasonal variations , we can use two methods : additive model and
multiplicative model . Additive model assumes that the components of series are
independent of each other and multiplicative model expresses each actual figure as a
proportion of the trend. Hence, to calculate seasonal components, we can assume no C in a
time series model. The additive model for a time series is Y = T + S + R. Therefore, Y T =
S + R while the multiplicative model is Y = T x S x I. Therefore, Y/T = S x I
Year Quarter Time period sales(Y)moving
average(T) Additive (Y-T) Multi ( Y/T)
2004
1 1 441.102 2 397.70 411.63 -13.93 -29.543 3 396.10 422.20 -26.10 -16.184 4 472.80 448.43 24.37 18.40
2005
1 5 476.40 467.87 8.53 54.832 6 454.40 460.53 -6.13 -75.093 7 450.80 486.23 -35.43 -13.724 8 553.50 528.33 25.17 20.99
2006
1 9 580.70 569.13 11.57 49.202 10 573.20 575.17 -1.97 -292.463 11 571.60 616.13 -44.53 -13.844 12 703.60 655.73 47.87 13.70
2007
1 13 692.00 690.70 1.30 531.312 14 676.50 676.13 0.37 1844.003 15 659.90 696.40 -36.50 -19.084 16 752.80
The average process will remove I, leaving S only. Take Additive ( S+I) and Multiplicative
(Sx I) from the above table and rewritten it as follows:
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Additive Q1 Q2 Q3 Q4
2004 -13.93 -26.10 24.37
`2005 8.53 -6.13 -35.43 25.17
2006 11.,57 -1.97 -44.53 47.87
2007 1.30 0.37 -36.50
Total 21.40 -21.66 -142.56 97.41
Average(S) 7.13 -5.42 -35.64 32.47
The average value of S above is unadjusted. Variations around the trend should be
cancelled out. In other words, for additive model , sum of S must equal to zero while for multiplicative model , it must equal to 4 .
Additive Q1 Q2 Q3 Q4 TotalUnadjusted S 7.13 -5.42 -35.64 32.47 -1.45
Adjustment to zero 0.36 0.36 0.36 0.36 1.45Adjusted S 7.50 -5.05 -35.28 32.83
Multi Q1 Q2 Q3 Q4 TotalUnadjusted S 1.01 0.75 0.94 1.06 3.76
Adjustment to 4 0.06 0.06 0.06 0.06 0.24Adjustment S 1.07 0.81 1.00 1.12
The adjusted S for both models is seasonal variation component. For additive model , sales
are 7,50 above the trend for Q1, -5,05 below the trend for Q2, and so on. For multiplicative
model , in the 1 st quarter, sales is 7% (107% - 100%) above the trend and in the 2 nd quarter,
sales is 19% (100% - 81%) below the trend, and so on.
Using additive and multiplicative model to forecast may give us difference results.
However, we can use both models to predict values for one variable.
Multi Q1 Q2 Q3 Q42004 0.97 0.94 1.052005 1.02 0.99 0.93 1.052006 1.02 1.00 0.93 1.072007 1.00 1.00 0.95Total 3.04 2.99 3.75 3.17
Average(S) 1.01 0.75 0.94 1.06
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Part 2:
In part 2, we are going to forecast the value of sales for 4 quarters in 2008. The forecast will
be based on the line of best fit and seasonal variation components using both additive
and multiplicative models.
After calculating, the line of best fit is: Y = 356.92 + 23.106 * t ( To know how to calculate the
line of best fit, see Appendix2.6). From the line of best fit , we will have the trend by replace t
equals to 1-16. Then, using the additive and multiplicative models, we have tables below:
Timeperiod sales(Y) Trend(T)
Additive (Y-T) Multi(Y/T)
1 441.1 380.0260 61.0740 1.1607102 397.7 403.1320 -5.4320 0.9865263 396.1 426.2380 -30.1380 0.9292934 472.8 449.3440 23.4560 1.0522015 476.4 472.4500 3.9500 1.0083616 454.4 495.5560 -41.1560 0.9169507 450.8 518.6620 -67.8620 0.8691598 553.5 541.7680 11.7320 1.0216559 580.7 564.8740 15.8260 1.028017
10 573.2 587.9800 -14.7800 0.97486311 571.6 611.0860 -39.4860 0.93538412 703.6 634.1920 69.4080 1.10944313 692 657.2980 34.7020 1.05279514 676.5 680.4040 -3.9040 0.99426215 659.9 703.5100 -43.6100 0.93801116 752.8 726.6160 26.1840 1.036036
Additive Q1 Q2 Q3 Q4 Totalunadjusted S 28.888 -16.318 -45.274 32.695 -0.009
Adjustment to zero 0.002 0.002 0.002 0.002 0.009Adjusted S 28.890 -16.316 -45.272 32.697
Multi Q1 Q2 Q3 Q4 Totalunadjusted S 1.0625 0.9682 0.9180 1.0548 4.0034
adjustment to zero -0.0009 -0.0009 -0.0009 -0.0009 -0.0034adjusted S 1.0616 0.9673 0.9171 1.0540
To forecast the value of sales for 4 quarters in 2008, first, we calculate the trend by using theline of best fits. Changing t = 17- 20 to the formula: Y = 356.92 + 23.106 * t. Then, for
Multi Q1 Q2 Q3 Q42004 1.1607 0.9865 0.9293 1.05222005 1.0084 0.9170 0.8692 1.02172006 1.0280 0.9749 0.9354 1.10942007 1.0528 0.9943 0.9380 1.0360Total 4.2499 3.8726 3.6718 4.2193
average 1.0625 0.9682 0.9180 1.0548
Additive Q1 Q2 Q3 Q42004 61.074 -5.432 -30.138 23.456
2005 3.950 -41.156 -67.862 11.7322006 15.826 -14.780 -39.486 69.4082007 34.702 -3.904 -43.610 26.184Total 115.552 -65.272 -181.096 130.780
average 28.888 -16.318 -45.274 32.695
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additive model , to calculate the sale, take the trend adds adjusted S. For multiplicative
model , take the trend times adjusted S.
2008
Additive Time period Trend Adjusted S Sale volumeQ1 17 749.722 28.890 778.612Q2 18 772.828 -16.316 756.512Q3 19 795.934 -45.272 750.662Q4 20 819.040 32.697 851.737
2008
Multi Time period Trend Adjusted S Sale volumeQ1 17 749.7220 1.0616 795.9172Q2 18 772.8280 0.9673 747.5534Q3 19 795.9340 0.9171 729.9570Q4 20 819.0400 1.0540 863.2514
The additive model adds absolute and unchanging seasonal variation to the trend figures
whereas the multiplicative mode l, by multiplying increasing or decreasing trend values by a
constant seasonal variation factor, tack account of changing seasonal variations. Hence, the
multiplicative model is better than the additive model for forecasting. However, the results for
2 models are not too much difference in this case, so we can use both of models to predictthe sale volume.
This forecast is quite reliable because the relationship between time period and sales
volume has strong linear relationship . The correlation coefficient (R) between temperature
and sales is 0.9 . (To know how to calculate, see Appendix). The more R is closer to 1, the
stronger linear relationship between time period and sales volume. Hence, we can believe in
this forecast.
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Appendix
Formula1.1. Mean:
=
==k
ii
k
iii
f
x f x
1
1.
xi: the mid point of class
f i: frequency of class
Sales value of lager (USD) Frequency Cumalative Frequency Mi Fi * Mi
Store A0-2.5 27,00 27,00 1,25 33,752.5-5 114,00 141,00 3,75 427,505-7.5 333,00 474,00 6,25 2081,25
7.5-10 530,00 1004,00 8,75 4637,5010-12.5 504,00 1508,00 11,25 5670,0012.5-15 334,00 1842,00 13,75 4592,5015-17.5 121,00 1963,00 16,25 1966,2517.5-20 29,00 1992,00 18,75 543,7520-22.5 5,00 1997,00 21,25 106,2522.5-25 2,00 1999,00 23,75 47,5025-27.5 1,00 2000,00 26,25 26,25
Total 2000,00 20132,50
Mean =2000
5,20132= 10,07
Sales value of lager (USD) Frequency Cumalative Frequency Mi Mi * Fi
Store B0-2.5 1,00 1,00 1,25 1,252.5-5 3,00 4,00 3,75 11,255-7.5 31,00 35,00 6,25 193,75
7.5-10 142,00 177,00 8,75 1242,50
10-12.5 328,00 505,00 11,25 3690,0012.5-15 498,00 1003,00 13,75 6847,50
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15-17.5 504,00 1507,00 16,25 8190,0017.5-20 351,00 1858,00 18,75 6581,2520-22.5 110,00 1968,00 21,25 2337,5022.5-25 29,00 1997,00 23,75 688,7525-27.5 3,00 2000,00 26,25 78,75
Total 2000,00 29862,50
Mean =2000
5,29862= 14,93
1.2. Quartile range:
Q = Q3 Q1
Q3: 75 th percentile (For store A,B is observation 1500 th)
Q1: 25 th percentile (For store A,B is observation 500 th)
Store A: Q1 = 7.5 + 26 x530
5.2= 7.62
Q3 = 10 + 496 x504
5.2= 12.46
Quartile range : Q = Q3 - Q1
= 12,46 7,62 = 4,84
Store B: Q1 = 10 + 323*328
5.2 = 12.46
Q3 = 10 + 497*504
5.2= 17.47
Quartile range : Q = Q3 - Q1
= 17,47 12,46 = 5,01
1.3. Variance :
( )n
x x x *2
encytotalfrequn
Mean x
frequency x
:
:
:
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Sales value of lager (USD Frequency (Fi) Mi Mi- Mean (Mi-Mean)^2 * FiStore A
0-2.5 27,00 1,25 -8,82 2098,612.5-5 114,00 3,75 -6,32 4548,035-7.5 333,00 6,25 -3,82 4849,73
7.5-10 530,00 8,75 -1,32 918,2310-12.5 504,00 11,25 1,18 706,2412.5-15 334,00 13,75 3,68 4532,3815-17.5 121,00 16,25 6,18 4626,8917.5-20 29,00 18,75 8,68 2186,8220-22.5 5,00 21,25 11,18 625,3822.5-25 2,00 23,75 13,68 374,4925-27.5 1,00 26,25 16,18 261,91
Total 2000,00 25728,72
Variance = 200072,25728
= 12,864
Sales value of lager (USD) Frequency Mi Mi- mean (Mi-Mean)^2 * FiStore B
0-2.5 1,00 1,25 -13,68 187,182.5-5 3,00 3,75 -11,18 375,065-7.5 31,00 6,25 -8,68 2336,29
7.5-10 142,00 8,75 -6,18 5425,5110-12.5 328,00 11,25 -3,68 4444,9312.5-15 498,00 13,75 -1,18 694,8915-17.5 504,00 16,25 1,32 876,51
17.5-20 351,00 18,75 3,82 5118,5820-22.5 110,00 21,25 6,32 4391,9322.5-25 29,00 23,75 8,82 2255,3425-27.5 3,00 26,25 11,32 384,34
Total 2000,00 26490,55
Variance = =2000
55,26490= 13,25
1.4. Standard deviation
Standard deviation =
Store A: Standard deviation = = 86,12 = 3,59
Store B: Standard deviation = = 25,13 = 3,64
1.5. Coefficient of variation:
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Coefficient of variation = x
Store A: Coefficient of variation = x
=
07.10
59.3= 0.36
Store B: Coefficient of variation = x
=
93.14
64.3= 0.24
1.6. Linear regression: Y = X o + 1
1 =2^2^
11
111
==
===n
i
n
i
n
i
n
i
n
i
Xi Xin
Yi Xi XiYin
X Y o 1 =
Y is the sale volume of beer (litres)
X is the temperature ( C )
729007426*10
447,211*270633,073,6101
= = 2,68
o = 21,14 2,68 x 27 = -51,23
1.7. Correlation coefficient:
R= ( )[ ] ( )[ ]
2^2^*2^2^ Y Y n X X n
Y X XY n
1.8. Moving average( three-period)
Year Quarter Time period sales(Y)moving
average(T)
2004
1 1 441,102 2 397,70 411,633 3 396,10 422,204 4 472,80 448,43
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Moving average =3
1,3967,39710,441 ++= 411,63
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