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Unit 5: Quantitative Techniques for Business
POPCORN
Aim of Assignment
Produce forecasts based on formalised procedures
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This assignment will enable you to produce forecasts based on formalised procedures.
- use formalised methods to forecast results
- assess the reliability of the forecasts made
The First Task ~ Time Series
A time series is a set of data collected over a period of time. The purpose of generating time series is to allow the accurate prediction of future events. Time series contains short term variations and a long term trend. The short term variations are seasonal, cyclical or residual. A trend line can be fitted to a scattergraph of the time series data by eye or by regression formulae. A third method using moving averages is generally used to obtain the trend line. The trend line represents the time series data with the short term variations averaged out.
1. The quarterly retail sales (in thousands of pounds) for the Odeon Cinema retail outlet are recorded for four years. Calculate the trend and the four seasonal fluctuations.
|
Quarter |
1996 |
1997 |
1998 |
1999 |
|
I |
£295 |
£257 |
£255 |
£255 |
|
II |
£283 |
£284 |
£254 |
£260 |
|
III |
£266 |
£251 |
£255 |
£249 |
|
IV |
£265 |
£261 |
£258 |
£261 |
2. A small arts cinema in Bath recorded its quarterly ticket sales (in thousands of pounds) over three years.
|
Quarter |
1997 |
1998 |
1999 |
|
I |
£35 |
£42 |
£41 |
|
II |
£52 |
£57 |
£61 |
|
III |
£83 |
£85 |
£92 |
|
IV |
£43 |
£43 |
£45 |
(a) Find the trend and the seasonal adjustments.
(b) Forecast the value of sales for the first two quarters of 2000.
The Second Task ~ Correlation
When a change on one quantity is accompanied by change in another quantity then we say that both quantities are correlated.
3. The regional manager of the Odeon chain of cinemas has done some phoning around. She has the following information on the number of staff and the value of ticket sales in a particular week for each of the seven cinemas in her area:
|
|
Weston |
Bath |
Bristol |
Taunton |
Wells |
Stroud |
Yate |
|
Number of Staff |
6 |
11 |
12 |
9 |
11 |
7 |
9 |
|
Value of Sales (£) |
£21,000 |
£18,000 |
£33,000 |
£39,000 |
£25,000 |
£27,000 |
£21,000 |
 |
(a) Draw a suitable graph to illustrate this data.
(b) Calculate the value of the product moment correlation coefficient of the data.
(c) Without further calculations, interpret the value obtained in part (b)
4. The regional manager was head-hunted by a chain of fast-food restaurants she got more money and a better company car, so the Odeon advertised the position, there were thirty applicants for the position, and ten were invited to interview.
The interview panel consisted of Dipika Soor and Dexter Fletcher. The ten candidates were ranked by Dipika and Dexter (from one to ten) in the following manner:
|
Dipika Soor |
4 |
2 |
7 |
1 |
5 |
6 |
9 |
3 |
10 |
8 |
||
|
Dexter Fletcher |
3 |
2 |
5 |
1 |
4 |
9 |
6 |
7 |
8 |
10 |
||
Calculate Spearman's rank correlation coefficient and discuss whether it represents a measure of agreement between the two panel members.
5. Write an explanation on the usefulness of correlation to business. Look at the uses, advantages and disadvantages.
The Third Task ~ Regression
A scattergraph is a plot of two sets of data. Interpolation can be used to make an assumption about intermediate values. Graphs can be extended to predict future changes in the data. By drawing a straight line, we can express the line as a relationship
y = a + bx
By using a mathematical method called the "Method of Least Squares" it is possible to find the values of a and b for y = a + bx in order to find a regression line.
6. The following data shows the number of people in the Star Trek Fan Club.
|
Year |
Members |
|
1992 |
13,792 |
|
1993 |
No data available |
|
1994 |
13,801 |
|
1995 |
14,307 |
|
1996 |
14,772 |
|
1997 |
14,943 |
|
1998 |
15,303 |
|
1999 |
15,543 |
(a) Round the figures to the nearest hundred.
(b) Draw a graph, showing how the rounded figures change with the time, label your graph clearly).
(c) Draw, by eye, a regression line on the graph.
(d) Estimate which year the number in the Star Trek Fan Club exceeds 16,000.
(e) Comment on how reliable your estimate is likely to be.
7. The following table shows the number of production staff used in the first six episodes of Star Trek: The Next Generation and the production cost (in thousands of dollars) of each of the episodes.
|
Episode # |
1 |
2 |
3 |
4 |
5 |
6 |
||
|
Number of Production Staff |
16 |
10 |
8 |
14 |
11 |
7 |
||
|
Production Costs ($ thousands) |
1,360 |
1,020 |
930 |
1,430 |
1,210 |
990 |
||
(a) Draw a suitable graph to illustrate this information.
(b) Calculate the equation of the line of regression of total running costs on number of staff and draw it on your graph.
(c) Predict the production cost of an episode using 12 production staff.
8. Write a page on the usefulness of regression to business. Look at the uses, advantages and disadvantages.
The Assessment Criteria
In order to achieve a Pass for this assignment
The student has used formalised methods accurately to forecast results. The student has assessed the reliability of forecasts made.
In order to achieve a Merit for this assignment
The student has used a variety of formalised methods accurately to forecast results. The student has assessed the reliability in some detail of the forecasts made.
In order to achieve a Distinction for this assignment
The student has used a suitable range of formalised methods accurately to forecast results. The student has assessed and interpreted the reliability in some detail of the forecasts made.
Content
Time Series Analysis: derivation and use of moving averages, centred trend, seasonal variations and seasonally adjusted data using either the additive or multiplicative model
Correlation: scatter graphs. positive and negative correlation, coefficient. significance
Regression Analysis: derivation of regression equation
Forecasting Analysis: preparation of forecasts using time series analysis and regression. reliability
