Sampling
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
1/39
Sampling
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:
2/39
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.
3/39
Sampling
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
4/39
Sampling
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:
1. income level
2. interest rates
3. Prices of the vehicle
4. Price of competitors
5. Seasonal promotion
6. Monthly dummy
7. Gas mileage
8. 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
5/39
Sampling
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
1. 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:
2. 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
6/39
Sampling
Alternative three
3. 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
4. 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.
7/39
Sampling
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 YShare,
xi
8/39
Sampling
yi
xiyi
xi2
yi2
1
9
10.8
-70.5867
-0.10667
7.529244
4982.478
0.011378
9/39
Sampling
2
94.4
11.3
14.81333
0.393333
5.826578
219.4348
0.154711
3
27.3
11.2
10/39
Sampling
-52.2867
0.293333
-15.3374
2733.896
0.086044
4
179.2
11.1
99.61333
0.193333
19.25858
11/39
Sampling
9922.816
0.037378
5
71.9
11.1
-7.68667
0.193333
-1.48609
59.08484
0.037378
6
12/39
Sampling
97.9
11.2
18.31333
0.293333
5.371911
335.3782
0.086044
7
93.5
11
13.91333
0.093333
13/39
Sampling
1.298578
193.5808
0.008711
8
70
10.7
-9.58667
-0.20667
1.981244
91.90418
0.042711
14/39
Sampling
9
160.7
11.3
81.11333
0.393333
31.90458
6579.373
0.154711
10
96.5
10.6
15/39
Sampling
16.91333
-0.30667
-5.18676
286.0608
0.094044
11
83
10.5
3.413333
-0.40667
-1.38809
11.65084
16/39
Sampling
0.165378
12
23.5
10.3
-56.0867
-0.60667
34.02591
3145.714
0.368044
13
58.7
17/39
Sampling
10.7
-20.8867
-0.20667
4.316578
436.2528
0.042711
14
93.8
11
14.21333
0.093333
18/39
Sampling
1.326578
202.0188
0.008711
15
34.4
10.8
-45.1867
-0.10667
4.819911
2041.835
0.011378
19/39
Sampling
TOTAL
1193.8
163.6
1.07E-13
5.33E-15
94.26133
31241.48
1.309333
MEAN
79.58667
10.90667
20/39
Sampling
Y = b |
0 |
^ _ _
21/39
Sampling
b 0 = Y – b 1 . X
22/39
Sampling
^
b 1 = x
23/39
Sampling
24/39
Sampling
b1 =
0.003017
b0 =
10.66654
25/39
Sampling
26/39
Sampling
Y = 10.667 + 0.003017 X
27/39
Sampling
28/39
Sampling
coefficient of determination
R 2 = b i
29/39
Sampling
R2 =
0.217213
30/39
Sampling
a. the regression equation Y = 10.667 + 0.003017 X
b. coefficient of determination
R2 = 0.217213
31/39
Sampling
From the above regression estimate the autonomous value of price of share is positive10.667, if we hold all other thins constant and increase the size in millions by one unit then the price per share will increase by0.003017, the correlation of determination shows the strength of the relationship that exist between two variables, in our case the value is 0.2172 and this value indicates that there exist a very weak relationship between the size and the price per share, therefore using the size as the independent variable to determine the price of shares will not be a goods option or estimation of share price.
Question 8
Correlation between wage rate and years of education
The correlation coefficient of the wage rate and years of education is equal to 0.408090683
To test for statistical significance whether the correlation coefficient is positive then we will first state the null and alternative hypothesis
Null hypothesis:
H0: b = 0
Alternative hypothesis
Ha: b ≠ 0
32/39
Sampling
Where b is the correlation value
Our calculated t will be given by
Z calculated = (x1-x2)/ ((σ12/ n1) + (σ22/ n2))1/2
WAGE and education
t calculated
t critical at 0.05
33/39
Sampling
0.031345359
1.644854
t calculated < t critical
34/39
Sampling
If t calculated is less than t critical we accept the null hypothesis and therefore this means that the correlation coefficient between the wage rate and education is not positive.
Correlation between wage rate and experience
The correlation coefficient of the wage rate and years of education is equal to 0.070538612
To test for statistical significance whether the correlation coefficient is positive then we will first state the null and alternative hypothesis
Null hypothesis:
H0: b = 0
Alternative hypothesis
Ha: b ≠ 0
35/39
Sampling
Where b is the correlation value
wage and experience
t calculated
t critical at 0.05
0.005416337
1.644854
36/39
Sampling
t calculated < t critical
If t calculated is less than t critical we accept the null hypothesis and therefore this means that the correlation coefficient between the wage rate and education is not positive.
37/39
Sampling
Question 9
We use t statistics to either reject of accept the null hypothesis, we first state the null hypothesis and the alternative hypothesis, if we reject the null hypothesis then the alternative hypothesis is true, if we the null hypothesis is accepted then the alternative hypothesis is rejected.
The test of t statistics involves comparing the t critical value that ifs derived from the normal table and the t calculated value which depends on the nature of the tast, when the t calculated is greater than the t critical then we reject the nullhypothesis, if the t calculated is less than the t critical then we accept the null hypothesis.
Below shows a two tail and a one tail test and regions of acceptance and rejection:
Question 10
P-value
The values of P in statistics can also be retrieved from tables that are published, these tables
38/39
Sampling
include the Z table available in various publications including almost on all statistic books and literature.
Further the P rep which is a new proposed method of getting the p value is calculated as follows:
P rep = (1 +( ( p/(1-p)2/3)) -1
References:
Lind and Mason (2003) Statistical Techniques in Business and Economics,
McGraw Hill, New York
39/39
- Academic Writing
- Accounting
- Anthropology
- Article
- Blog
- Business
- Career
- Case Study
- Critical Thinking
- Culture
- Dissertation
- Education
- Education Questions
- Essay Tips
- Essay Writing
- Finance
- Free Essay Samples
- Free Essay Templates
- Free Essay Topics
- Health
- History
- Human Resources
- Law
- Literature
- Management
- Marketing
- Nursing
- other
- Politics
- Problem Solving
- Psychology
- Report
- Research Paper
- Review Writing
- Social Issues
- Speech Writing
- Term Paper
- Thesis Writing
- Writing Styles