Quantitative Decision

Introduction:

When an organisation is faced with options regarding decision making they opt to undertake quantitative data analysis that aid in decision making, statistical analysis in most cases is preferred for the purpose of further understanding of the current situation in the organisation. Statistical regression models are used for forecasting future values of sales, stocks required and future prices, this regression models contain variables that the organisation believes affects the dependent variable.

The methods of making *quantitative decision* can be used at the times when the objectives of the organisation are explicitly stated, there are a number of courses of action regarding objectives as an alternatives, and the worth or benefit for the various alternative options is calculable and finally there has to be some allowances for unforeseen circumstances that may affect the organisation negatively or positively in the near future.

In this paper we will focus on an organisation that is faced with the problem of making decisions regarding production, we will **take** an organisation that produces vehicles, The organisation therefore has to consider past production levels of cars to predict the future, this is done through statistical analysis of past data to come up with a model that helps in predicting future production values, for this reason the production level will depended on the various factors which include for example the consumers income levels and the price level.

In this paper we will consider data from the bureau of statistics available at www.statistics.gov.

uk which is a source of our data regarding production of vehicles in the UK, the price index of the vehicles and the income levels of employees, this will help us estimate our model which depict that the

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production of vehicles depend on the level of income and also the price level, this helps an organisation to know what levels to produce and at what prices it will charge its products.

Data:

Data was retrieved from the official UK bureau of statistics available at www.statistics.gov.uk , data collected include the production level over the years from 1991 to 2006 of passenger cars for the home market, the price index of vehicles in the UK from 1991 to 2006 and finally the income level or the total employee compensation from 19991 to 2006. This is time series data and this data is represented in the table below:

year

production

employee compensation

price index

1991

631515

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335704

85.2

1992

702590

347546

87

1993

842648

356595

90.2

1994

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848142

369146

92.4

1995

787473

386035

96.1

1996

777922

403887

98.9

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1997

736090

429967

100.2

1998

727531

466080

101.9

1999

648146

495793

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102.3

2000

578462

532179

100

2001

598111

564194

96.6

**2002**

582460

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587396

96.3

2003

513799

616893

96.1

2004

466994

648717

97.2

2005

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411194

686805

98.6

2006

335992

722414

99.5

The trends in the production, prices and compensation levels can be represented in a diagram as follows:

Trends for the production variable are as follows:

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From the above chart it is clear that the production of passenger vehicles in the UK has declined over the years, however there was an increase in the production of vehicles in the year 1991 to 1994 and from 1994 to 2006 there has been a decline in the production, the reason why there is a decline can be attributed to many factors which include increasing crude oil prices which is a complementary good to vehicles, increased competition from other producing companies that are located overseas, decline in demand for passenger vehicles, However generally the production levels have declined over the years as depicted by our chart.

Trends for the income levels are as follows:

The above chart shows the income levels of employees in the UK, from our chart we observe that there has been an increase in the level of income over the years; this is a steady increase in income levels.

Trends for the price index are as follows:

From the chart above the price index of vehicles in the UK has been generally steady over the years, this means that the prices have fluctuated only by a very negligible margin over the years from 19991 to 2006, therefore the price level has not really changed from 1991.

Mean and standard deviation:

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Having considered our variables as income, price and production levels it is then very easy to check the mean and the standard deviation of this data, our mean and other measures of central tendencies are shown in the following table:

year

production

employee compensation

price index

total

10189069

7949351

1538.5

mean

636816.8

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496834.4375

96.15625

SD

151752.4

128841.1628

5.041953

median

639830.5

480936.5

96.9

correlation

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Production and Income

-0.903366

Production and Price index

-0.276209

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income and price index

0.543306

From the analysis of correlation in our data above it is clear that there exist a very strong but negative correlation between income and production levels, also there is a moderate strong correlation between income and price index. The mean level of production for all these years is 636816.8 while the mean income levels for employees for all the years are 496834.4375 while the mean for price index is 96.15625. The mean helps us to locate the most possible outcome in future and this is because the mean is a central measure of given data.

The standard deviation for the production level is equal to 151752.4, the standard deviation for income levels is 128841.1628 and the standard deviation for the price index is 5.041953. The standard deviation is a measure of dispersion. It measures the deviations from the mean. It is an important measure to consider when undertaking statistical data as it gives us a rough idea of how expected values may deviate from the actual values in future. The higher the standard

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deviation then the higher the risk *that* we will have a large error and risk of predicted values.

Box and whisker diagram:

The box and whisker diagram is a form which represents the data provided using the quartile ranges, we consider the minimum value, the median and the maximum value, we consider the box and whisker diagram for each time series of the variables.

year

production

employee compensation

price index

1st quartile

562296.3

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381812.75

95.175

2nd quartile

639830.5

480936.5

96.9

3rd quartile

746548

594770.25

99.625

4th quartile

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848142

722414

102.3

min

335992

335704

85.2

max

848142

722414

102.3

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Min means the minimum value in our data and max means our maximum value in our data series, the box and whisker diagram are as follows for each variable that we have considered.

Production:

Employee compensation:

Price index:

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Estimation of our models:

Income and production:

Lets assume that our organisation that produces vehicles is indifferent on at what level it will produce, next we assume that the only factor that will influence the demand of vehicles is the income level, having stated our assumption we will therefore have our production level as the dependent variable and the income level as the independent variable then we can state our model as follows:

Y = a + b X

Where Y is the production level and X is the level of income, a is the autonomous value and b is the coefficient of X which depict the slope of our model. After estimation the following is our results:

From the above scatter diagram which also depicts the linear regression line shows that as the level of income increases then production decreases, for this reason therefore due to the high correlation value between these two variables it shows that if income increases then the organisation should reduce production levels. Our estimated model is as follows:

Y=1X106–1.064X

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This shows that if we hold all other factors constant and let the value of income to be zero then the level of production will be 1 X 106 , also if we hold all other factors constant and increase the level of income by one unit then the level of production will decrease by 1.064 units, for this *reason* therefore it is clear that production and income have an inverse relationship.

Our correlation of determination R2 gives us the strength of the relationship that exist between two variables, in our case its value is equal to 0.8161, this in statistical terms means that

81.61% of deviations Y are explained by the independent variable X, in other words this means that 81.61% of deviations in the production levels is explained by the income level.

We undertake a statistical test of the forecasting power of this model, testing forecasting power of this model involves the use of data available and then undertaking a test of whether the model could be used for forecasting, in our case we will consider the data point where the year is

1996, the production level is 777922 and our income level is 403887, we substitute X in this model which states Y = 1 X 106 – 1.064 X with 403887 and find out if we will achieve the production level of this year, the value of Y in this case is 770264.2 and for this reason our model has a high forecasting power due to the low deviations of results from our actual value, this model therefore can be used to forecast on production leve

Price and production:

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Lets assume that our organisation that produces vehicles is indifferent on at what level it will produce, next we assume that the only factor that will influence the demand of vehicles is the price level, having stated our assumption we will therefore have our production level as the dependent variable and the price index level as the independent variable then we can state our model as follows:

Y = a + b X

Where Y is the production level and X is the price index, a is the autonomous value and b is the coefficient of X which depict the slope of our model. After estimation the following is our results:

From the above chart the trend line shows a negatively sloped linear regression line, it also shows that as the price increase which is a sign of increased inflation the production level also decreases, this can be attributed to the fact that production will reduce as the demand of products reduce due to the price increase, the correlation of determination level depict that only 4.78% deviations in Y are explained by X, the model estimated is as follows:

Y=1X106–5677X

This model states that if we hold all other factors constant and let the level of prices to be zero then the level of production will be 1 X 106 , also if we hold all other factors constant and increase the prices by one unit then the level of production will decrease by 5677 units, for this reason therefore it is clear that production and price have an inverse relationship.

We undertake a statistical test of the forecasting power of this model, testing forecasting power

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of this model involves the use of data available and then undertaking a test of whether the model could be used for forecasting, in our case we will consider the data point where the year is 1996, the production level is 777922 and our price index is 98.9, we substitute X in this model which states Y = 1 X 106 – 5677 X with 98.9 and find out if we will achieve the production level of this year, the value of Y in this case is 561465.2 and for this reason our model has low forecasting power due to the high deviations of results from our actual value, this can be attributed to the high levels of the stochastic variable in the estimated model.

Results:

From the above results on the regression models, it is clear that both variables we considered will negatively affect the production levels, however there is need to consider other variables in our analysis, this would involve the use of the multiple regression model to estimate a model with more than one independent variable, this will also involve undertaking t statistics which will give us the statistical significance of our estimated parameters.

The models estimated however can still be used to undertake forecasting on the organisations future production levels, these models will therefore be of much significance to the organisation. Apart from estimating models that depict the future production levels the organisation can also use the data at its disposal to estimate other models that forecast on prices, growth levels and demand for their products, this however will require that the organisation come up with a hypothesis on what variables influence one variable and then undertake forecasting measures.

Conclusion:

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Our data was retrieved from the bureau of statistics available at www.statistics.gov.uk , this source of data is a reliable source which enabled us to retrieve data regarding production of vehicles in the UK, the price index of the vehicles and the income levels of employees, this aided in undertaking statistical inferences and also in the estimation of linear regression models.

From our above discussion it is clear that the use of statistics is an important factor to consider when making decision, mean and median measures of data measure central tendencies in the data, they are important in that we can depict our expected values in any given situation, measures such as the standard deviation shows the deviations of the data from the mean, this will provide an estimate the probability of deviation in the expected value from the actual outcome.

Our estimated model will also provide the organisation with a forecasting value of what to produce, this will dependent on what they state as the independent value, in our case we considered income as the independent variable and there was a very high correlation value in this case, also there was a high correlation of determination where over 80% of deviations of production were explained by income, for this reason we conclude that our model estimated can be used to forecast future production levels, in this model it was found out that as income increases then the production level decreases.

Our second model included estimation of the production level using the price, however there is very low correlation between the two variables, also the correlation of determination is also low, however this model depict that as the as price increases then the level of production decreases, this can be attributed to inflationary pressure caused by increasing prices and as a result the production level goes down.

Quantitive data analysis will therefore aid the organisation in decision making, many organisations will keep records on prices, sales volume and other information that would be important in modelling and forecasting future values, and for this reason an organisation should

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start utilising this information for the future growth of the organisation. There are other factors that affect production levels that have not been discussed in this paper but this was limited by the unavailability of data regarding key variables that influence production.

References:

Levin Richard (1984) Statistics for Management, Prentice Hall publishers, New Jersey

UK bureau of statistics (2008) time series data on production, income and price index, retrieved on 1st February, available at www.statistics.gov.uk/statbase/TSDdownload2.asp

Bluman (2000) Elementary Statistics: A Step by Step Approach, McGraw Hill press, New York

D. Bridge (1993) Statistics: An Introduction to Quantitative and qualitative Research, Rand McNally publishers, Michigan

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