QUESTION 1
For this case a sample size of 1500 observation would represent the whole population. The sampling technique that is best for this study is a random sample
QUESTION 2
In statistics the larger the sample size the better the use of a small sample size will give unbiased results about the entire population. If possible it is better to use the entire population but the wastage of time, resources and the unavailability of the population makes it impossible to use a population. Therefore a good sample is supposed to represent the entire population without giving biased results. The sample size to consider therefore is that which has more variables to be observed over time. The best sample size would be that which gives a small standard error, the standard error of a sample is given by
Standard deviation of the sample divided by the square root of the sample size
Further as the central theorem predict that as the number of variables increase then they will assume a normal distribution. Therefore it is important to use a large sample size and in our case a sample size of 100 observations would give unbiassed results.
There exist many sampling techniques and they include:
Random sampling:
This involves the use of random numbers generated by either a calculator or a computer, form this point you give each observation in the population a number and select the observation that correspond to the random number
Stratified sampling:
This sampling technique involves grouping the data into categories whereby the sample that correspond with the study to be undertaken is selected.
Cluster sampling:
This sampling method involves the use of categories where one selects only the observations from the region required.
Systematic sampling:
Systematic sampling involves giving an interval example 5, from the population one is expected to pick the observation that comes 5th after the last observation selected.
In our case the best sampling technique to be used would be a random sample which will give best results.
QUESTION 3
Companies will use correlation, regression model, risk analysis, forecasting, sensitivity analysis, a decision trees and game theory to make decisions regarding their business performance and options provided, the best of the above to use is the use of regression models for forecasting, the model should be set in a way that it estimates both the sales expected, the expected revenue and profits. Also the model should consider the prices of competitors and the decisions that are likely to affect the market if price are changed by the competitors. A good model should have its parameters statistically significant and at the same time have good forecasting power regarding important indicators of the company.
QUESTION 4
in the burns auto scenario it is clear that the problem that exist is that the company financed its 300 million inventory through a local bank whose interest rate amounted to 9,000,000 per year, recently the inventory has increased to 360millionm and the cost of then finance from the local bank have increased to 10,800,000. For thus reason the company is loosing a lot of funds because the inventory level has risen because the company does not have a forecast for their sale.
the solutions available to the company is that of hiring a consultant who will help develop a forecasting model, the model incorporates interest rates, income, competitors prices, seasonal promotions and the monthlydummyvariable. despite the estimation of the model the coefficients of the model are not statistically significant however the model has a possibility of reducing inventory by 10%.
from the model there are possible solutions to the problem:
increase promotions to higher levels than competitors will increase sales
the assumptions of this model provided are that :
the factors that affect sales level include income, interest rates, competitor’s prices, seasonal promotions and monthly dummy variable
the other assumption is that gas mileage does not affect sales
the constraint for this model is that the estimated coefficients in the model are not statistically significant.
john the consultant brought by Richard uses a sales forecast using time and monthly dummy variables to predict the sales level of one model of car. the estimation model is also helpful in predicting the sales forecast for models of the cars because its forecasting power results into very close results compared to the actual results.
assumptions of this model by john include:
there exist no competitors in the market
income levels and interest rates do not affect the sales level
from these assumptions we cannot conclude that this is the best model because there are many factors that tend to affect the sales level of vehicles in the company.
the best solution to the problem is the estimation of a sales forecasting model that incorporates all the factors that affect sales level, all the independent variables used by the consultant to explain the sales level should be incorporated to one model and the estimated model should be used to forecast if the estimated coefficients are statistically significant, these independent variables should include:
income level
interest rates
Prices of the vehicle
Price of competitors
Seasonal promotion
Monthly dummy
Gas mileage
And time
For the assumptions and constraint faced by the model in the above case study you cant ignore them, this is because for a model to have the best forecasting power the estimates must be statistically significant and also include all the independent variables that affect the dependent variable.
QUESTION 5
the solutions to the problem still make further problems, the aim of the company is to save money and the loss incurred as a result of extra inventory adding to the cost of financing this, the solutions bring on board other expenses of hiring a consultant, also the fact that the models parameters are not easy to estimate for the future example determining what the interest rates will be in the future brings more complications.
The solutions provided by the consultants involve the use of independent variables where each has his own independent variables that affect the as ales level, the best solution would be to adopt the entire model provided in sale forecasting and comparing the results.
the following are the various alternatives and the risks associated;
Alternative one
Use of manufacturer forecast:
When the company uses the manufacturer’s data there is a possibility of high inventory levels that will add into the cost of finances per year
Alternative two:
Use of peters first model:
There will be a risk of reduced sales level which will result into low income and revenue resulting into greater loss
Alternative three
Peter’s second model
This model does not incorporate other factors that affect sales; there is a risk of over or under estimation of sales
Alternative four
Use of john’s model
This model only considers time as the independent variable where monthly sales are estimated from January to December; the risk is that there will be loss of income as a result of hiring the consultant and also a risk of over or under estimation of the sale.
QUESTION 6
Regression analysis may be used to make decision in a competitive business which is full of uncertainty; the models estimated in this situation will involve the use of independent factors that predict the competitor’s actions regarding products in the market. In uncertain situations whereby the firm does not have clear information about the future results in business performance the firm will formulate a model that boosts its sales level to a higher level in the future whereby the factors that lead to an increase in sales level will be initiated by the business.
Example a situations where if we reduce the price by one unit then the sales level increases by ten units therefore the firm will reduce the price to a certain level that is desired to make high profits.
Uncertainties will occur due to the risks that involves various strategies initiated
Uncertainties in the actions of the competitions in the future
Uncertainties of other external factors such as government policies that may lead toa decline or an increase in sales level
In such situations the regression and correlation analysis of data from the past will be used to predict the future and cater for these uncertainties.
QUESTION 7
|
Company |
Size ($ millions), X |
Price per Share, Y |
xi |
yi |
xiyi |
xi2 |
yi2 |
|
1 |
9 |
10.8 |
-70.5867 |
-0.10667 |
7.529244 |
4982.478 |
0.011378 |
|
2 |
94.4 |
11.3 |
14.81333 |
0.393333 |
5.826578 |
219.4348 |
0.154711 |
|
3 |
27.3 |
11.2 |
-52.2867 |
0.293333 |
-15.3374 |
2733.896 |
0.086044 |
|
4 |
179.2 |
11.1 |
99.61333 |
0.193333 |
19.25858 |
9922.816 |
0.037378 |
|
5 |
71.9 |
11.1 |
-7.68667 |
0.193333 |
-1.48609 |
59.08484 |
0.037378 |
|
6 |
97.9 |
11.2 |
18.31333 |
0.293333 |
5.371911 |
335.3782 |
0.086044 |
|
7 |
93.5 |
11 |
13.91333 |
0.093333 |
1.298578 |
193.5808 |
0.008711 |
|
8 |
70 |
10.7 |
-9.58667 |
-0.20667 |
1.981244 |
91.90418 |
0.042711 |
|
9 |
160.7 |
11.3 |
81.11333 |
0.393333 |
31.90458 |
6579.373 |
0.154711 |
|
10 |
96.5 |
10.6 |
16.91333 |
-0.30667 |
-5.18676 |
286.0608 |
0.094044 |
|
11 |
83 |
10.5 |
3.413333 |
-0.40667 |
-1.38809 |
11.65084 |
0.165378 |
|
12 |
23.5 |
10.3 |
-56.0867 |
-0.60667 |
34.02591 |
3145.714 |
0.368044 |
|
13 |
58.7 |
10.7 |
-20.8867 |
-0.20667 |
4.316578 |
436.2528 |
0.042711 |
|
14 |
93.8 |
11 |
14.21333 |
0.093333 |
1.326578 |
202.0188 |
0.008711 |
|
15 |
34.4 |
10.8 |
-45.1867 |
-0.10667 |
4.819911 |
2041.835 |
0.011378 |
|
TOTAL |
1193.8 |
163.6 |
1.07E-13 |
5.33E-15 |
94.26133 |
31241.48 |
1.309333 |
|
MEAN |
79.58667 |
10.90667 |
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Y = b0 + b1 X + ui
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