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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