**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

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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**:

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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.

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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

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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

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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

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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.

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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

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Sampling

yi

xiyi

xi2

yi2

1

9

10.8

-70.5867

-0.10667

7.529244

4982.478

0.011378

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Sampling

2

94.4

11.3

14.81333

0.393333

5.826578

219.4348

0.154711

3

27.3

11.2

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Sampling

-52.2867

0.293333

-15.3374

2733.896

0.086044

4

179.2

11.1

99.61333

0.193333

19.25858

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*Sampling*

9922.816

0.037378

5

71.9

11.1

-7.68667

0.193333

-1.48609

59.08484

0.037378

6

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Sampling

97.9

11.2

18.31333

0.293333

5.371911

335.3782

0.086044

7

93.5

11

13.91333

0.093333

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Sampling

1.298578

193.5808

0.008711

8

70

10.7

-9.58667

-0.20667

1.981244

91.90418

0.042711

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9

160.7

11.3

81.11333

0.393333

31.90458

6579.373

0.154711

10

96.5

10.6

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Sampling

16.91333

-0.30667

-5.18676

286.0608

0.094044

11

83

10.5

3.413333

-0.40667

-1.38809

11.65084

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Sampling

0.165378

12

23.5

10.3

-56.0867

-0.60667

34.02591

3145.714

0.368044

13

58.7

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Sampling

10.7

-20.8867

-0.20667

4.316578

436.2528

0.042711

14

93.8

11

14.21333

0.093333

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Sampling

1.326578

202.0188

0.008711

15

34.4

10.8

-45.1867

-0.10667

4.819911

2041.835

0.011378

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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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Sampling

Y = b |
0 |

^ _ _

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b 0 = Y – b 1 . X

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^

b 1 = x

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b1 =

0.003017

b0 =

10.66654

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Sampling

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Y = 10.667 + 0.003017 X

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coefficient of determination

R 2 = b i

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R2 =

0.217213

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a. the regression equation Y = 10.667 + 0.003017 X

b. coefficient of determination

R2 = 0.217213

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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

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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

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0.031345359

1.644854

t calculated < t critical

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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

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Sampling

Where b is the __correlation__ value

wage and experience

t calculated

t critical at 0.05

0.005416337

1.**644854**

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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.

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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

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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

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