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Phase-I
Corporate Overview
The brand Belvin is owned by Bless group of companies, one of the most advanced and professional engineering groups. The group is recognized as a professional, well-organized and efficient manufacturing organization among the industrial and consumer markets. Since its inception in 1989, the group evolved to encompass a diverse spectrum of production capabilities.
Every member of the Bless Group is full committed and thoroughly dedicated to provide the best quality products and excellent services to its customers by meeting with their defined specifications, quality standards and delivery deadlines.
The group’s production capability is divided into two core divisions:
Bless Electronics (PVT) Ltd
This division is responsible for assembling consumer electronic appliances which includes color televisions and DVD players.
Bless Engineering Company
Bless Engineering Company provides electronic and mechanical engineering services and provides complete solutions from tool designing to the development and manufacturing of electric fans.
Belvin Electronics 21'' Color TV Model # BL 2152 FSW
Features:
Raw Data of 21'' Color TV Model # BL 2152 FSW
Raw Data Analysis:
We officially visited the Belvin Electronics for our own interest for the Quantitative Measures to Facilitate in our Final Project of Business Quantitative Techniques. We collected the data on four Variables (Sales, Production Cost, Advertising Expense, and Profit). We specifically collected the data on their Color TV 21'' Model # BL 2152 FSW.
Our first considerations about Raw Data that it is based on Monthly bases and all Values we converted into Millions for our own Convenience and almost normally distributed or symmetrical, in graphical language we can say that it is Bell-Shaped in Cartesian plane.
Methodologies:
In first phase of our project we have to calculate the Descriptive statistics, Estimations, ANOVA, Inferences, & Multiple Regression & Correlation for our data.
- Mean (Average)
- Standard Deviation (Dispersion)
- Co-efficient of Variation (Per unit Deviation)
- Estimation
- ANOVA
- Inferences
- Multiple Regression & Correlation
Reasoning for using these Statistical tools:
Arithmetic mean is appropriate for our data because there are no out liars in the data so simple mean tells us the accurate answer. Though we move further for checking out the spread in the data for which we calculate the Standard deviation, it provides us with the variation in the data. Further more we calculate the C.V. for per unit deviation in the data.
Estimations for estimate our Population parameters. Analyze the variations & run hypothesis. Design Multiple Regression & Correlation when we have more then two Independent Variables.
Descriptive Analysis
Production Cost Descriptive Analysis
Arithmetic Mean:
= 529.92/25
= 21.1968 Rs. (Millions)
Interpretation:
We concluded that the average production cost is 21.1968 Rs. (Millions). Our data is normally distributed therefore arithmetic mean tells us the accurate answer.
Standard Deviation:
= 4.718670999 Rs. (Millions)
Interpretation:
Measure of the unpredictability of a , expressed as the average of a set of from its arithmetic mean and computed as the positive square root of the .
Standard deviation is the most appropriate tool yet to finding out the variation within the data and it covers the whole data as well. Here the S.D. of Cost of Production is 4.718670999 Rs. (Millions).
Co-efficient of Variation:
= 0.2226124226
= 22.3%
Interpretation:
The coefficient of variation is used to measure the consistency of the data across all experiments. The coefficient of variation (CV) is calculated as standard deviation divided by mean. A high CV value reflects inconsistency among the data. Here we have the Value of C.V. 22.3% for the Variable of cost of Production which shows that the spread involve when we moving from one unit to the other.
Advertising Exp. Descriptive Analysis
Arithmetic Mean:
= 40.2/25
= 1.608
Interpretation:
We concluded that the average Advertising Expenses is 1.608 Rs. (Millions). Our data is normally distributed therefore arithmetic mean tells us the accurate answer.
Standard Deviation:
= 0.2498866631
Interpretation:
Measure of the unpredictability of a , expressed as the average of a set of from its arithmetic mean and computed as the positive square root of the .
Standard deviation is the most appropriate tool yet to finding out the variation within the data and it covers the whole data as well. Here the S.D. of Advertising Expenses is 0.249886631 Rs. (Millions).
Co-efficient of Variation:
= 0.155402153
= 15.5%
Interpretation:
The coefficient of variation is used to measure the consistency of the data across all experiments. The coefficient of variation (CV) is calculated as standard deviation divided by mean. A high CV value reflects inconsistency among the data. Here we have the Value of C.V. 15.5% for the Variable of Advertising Expenses which shows that the spread involve when we moving from one unit to the other.
Sales descriptive Analysis
Arithmetic Mean:
= 667.2/25
= 26.688
Interpretation:
We concluded that the average Sales is 26.688 Rs. (Millions). Our data is normally distributed therefore arithmetic mean tells us the accurate answer.
Standard Deviation:
= 6.692802104
Interpretation:
Measure of the unpredictability of a , expressed as the average of a set of from its arithmetic mean and computed as the positive square root of the .
Standard deviation is the most appropriate tool yet to finding out the variation within the data and it covers the whole data as well. Here the S.D. of Sales is 6.692802104 Rs. (Millions).
Co-efficient of Variation:
= 0.250779455
= 25.1%
Interpretation:
The coefficient of variation is used to measure the consistency of the data across all experiments. The coefficient of variation (CV) is calculated as standard deviation divided by mean. A high CV value reflects inconsistency among the data. Here we have the Value of C.V. 25.1% for the Variable of Sales which shows that the spread involve when we moving from one unit to the other.
Profit Descriptive Analysis
Arithmetic Mean:
= 133.44/25
= 5.3376
Interpretation:
We concluded that the average Profit is 5.3376 Rs. (Millions). Our data is normally distributed therefore arithmetic mean tells us the accurate answer.
Standard Deviation:
= 1.338560421
Interpretation:
Measure of the unpredictability of a , expressed as the average of a set of from its arithmetic mean and computed as the positive square root of the .
Standard deviation is the most appropriate tool yet to finding out the variation within the data and it covers the whole data as well. Here the S.D. of Profit is 1.338560421 Rs. (Millions). It is the Deviation involve when we move from one unit to the other.
Co-efficient of Variation:
= 0.250779455
= 25.07%
Interpretation:
The coefficient of variation is used to measure the consistency of the data across all experiments. The coefficient of variation (CV) is calculated as standard deviation divided by mean. A high CV value reflects inconsistency among the data. Here we have the Value of C.V. 25.07% for the Variable of Profit which shows that the spread involve when we moving from one unit to the other.
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Phase-II
Confidence Interval Estimation About Mean, Variance & Standard Deviation
Sales
Confidence interval about Mean:
Mean (Sales) =26.688
1 – α = 85% , 1 – α / 2 = 0.925 v = n - 1, v = 24
= (28.7 , 24.7)
Interpretation:
- In the probabilistic interpretation, we say that
- A 85% confidence interval for a population parameter means that, in repeated sampling, 85% of such confidence intervals will include the population parameter
- In the practical interpretation, we say that
- We are 85% confident that the 85% confidence interval will include the population parameter
Confidence interval about Variance:
Variance (Sales) = 44.796
χ 2 v , α / 2 = 34.5722 , χ 2 v , 1 - α / 2 = 14.8525
= (31.1, 72.4)
Interpretation:
This interval represents the most likely distribution of population Variance. We are 85% confidence the population's Variance will fall in this interval
Confidence interval about Standard Deviation:
= (5.6, 8.5)
Interpretation:
This interval represents the most likely distribution of population Standard Deviation. We are 85% confidence the population's Standard Deviation will fall in this interval
Cost of Production
Confidence interval about Mean:
Mean (Cost of Production) = 21.1968
1 – α = 90% , 1 – α / 2 = 0.950 v = n - 1, v = 24
= (22.8 , 19.6)
Interpretation:
- In the probabilistic interpretation, we say that
- A 90% confidence interval for a population parameter means that, in repeated sampling, 90% of such confidence intervals will include the population parameter
- In the practical interpretation, we say that
- We are 90% confident that the 90% confidence interval will include the population parameter
Confidence interval about Variance:
Variance (Cost of Production) = 22.265856
χ 2 v , α / 2 = 36.415 , χ 2 v , 1 - α / 2 = 13.848
= (14.675 , 38.589)
Interpretation:
This interval represents the most likely distribution of population Variance. We are 90% confidence the population's Variance will fall in this interval
Confidence interval about Standard Deviation:
= (3.83 , 6.21)
Interpretation:
This interval represents the most likely distribution of population Standard Deviation. We are 90% confidence the population's Standard Deviation will fall in this interval
Advertising Expenses
Confidence interval about Mean:
Mean (Advertising Expenses) = 1.608
1 – α = 95% , 1 – α / 2 = 0.975 v = n - 1, v = 24
= (1.71 , 1.51)
Interpretation:
- In the probabilistic interpretation, we say that
- A 95% confidence interval for a population parameter means that, in repeated sampling, 95% of such confidence intervals will include the population parameter
- In the practical interpretation, we say that
- We are 95% confident that the 95% confidence interval will include the population parameter
Confidence interval about Variance:
Variance (Advertising Expenses) = 0.062433333
χ 2 v , α / 2 = 39.364 , χ 2 v , 1 - α / 2 = 12.401
= (0.038 , 0.121)
Interpretation:
This interval represents the most likely distribution of population Variance. We are 95% confidence the population's Variance will fall in this interval
Confidence interval about Standard Deviation:
= (0.195 , 0.347)
Interpretation:
This interval represents the most likely distribution of population Standard Deviation. We are 95% confidence the population's Standard Deviation will fall in this interval
Profit
Confidence interval about Mean:
Mean (Profit) = 5.3376
1 – α = 99% , 1 – α / 2 = 0.995 v = n - 1, v = 24
= (6.08 , 4.59)
Interpretation:
- In the probabilistic interpretation, we say that
- A 99% confidence interval for a population parameter means that, in repeated sampling, 99% of such confidence intervals will include the population parameter
- In the practical interpretation, we say that
- We are 99% confident that the 99% confidence interval will include the population parameter
Confidence interval about Variance:
Variance (Profit) = 1.791744
χ 2 v , α / 2 = 45.559 , χ 2 v , 1 - α / 2 = 9.886
= (0.944 , 4.35)
Interpretation:
This interval represents the most likely distribution of population Variance. We are 99% confidence the population's Variance will fall in this interval
Confidence interval about Standard Deviation:
= (0.972 , 2.08)
Interpretation:
This interval represents the most likely distribution of population Standard Deviation. We are 99% confidence the population's Standard Deviation will fall in this interval.
Analysis of Variance
Hypothesis Testing at Significance Level = 0.05
(Null Hypothesis) H0 : μ1 = μ2 = μ3= μ4
(Alternative Hypothesis) H1 : μ1 ≠ μ2 ≠ μ3 ≠ μ4
Critical Region:
V1 = k – 1 → V1 = 3
V2 = n – k → V2 = 96
ƒ v1 , v2 , α → ƒ 3 , 96 , 0.05 = 2.70
If ƒ test > ƒ tabulated then we reject H0
Test Statistic:
ƒ = MSB / MSW
Where;
MSB = SSB / k – 1
MSW = SSW / n – k
Where;
SSB =
SSB = 11025.89
SSW =
SSW = 1653.93
ANOVA TABLE:
ƒ = MSB / MSW
= 213.3279
Decision:
So we rejected H0 Because ƒ test is Greater then ƒ tabulated
Interpretation:
What is ANOVA?
- Analysis of variation in an experimental outcome and especially of a statistical variance in order to determine the contributions of given factors or variables to the variance.
Usefulness;
- More versatile than t-test
- Compare one parameter (response variable) between two or more groups
Why no just use t-tests?
- Tedious when many groups are present
- Using all data increases stability
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Large number of comparisons→ some may appear significant by chance
F-Ratio = MSB / MSW
If
-
The ratio of Between-Groups MS: Within-Groups MS is LARGE→ reject H0→ there is a difference between groups
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The ratio of Between-Groups MS: Within-Groups MS is SMALL→do not reject H0→ there is no difference between groups
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Phase-III
Determination of Dependent & Independent Variables:
After strong considerations we came across with the conclusion of choosing the Variables to whom we select as the Dependent Variable & Independent Variable.
Thus we select “Profit” as Dependent Variable because this element is dependent on other factors such as Sales, In our Scenario (Belvin Electronics) 21” TV model we took the sales of particular Product if the sales increases then our profit will be increased as well after deductions of expenses including (Advertising Expenses & Production cost).
But if we took sales as dependent and other factors as independent then we does not be able to say that if our profits are increases then our sales also increases with some specific amount of slope which is multiplied with our profits, means that our sales are dependent on profits which is not true.
Ŷ = a + b1x + b2x + b3x ……………… (1)
Here ‘a’ expressed as the constant value means that if other factors (production cost, advertising expenses, sales) are all zero then there must be some value expressing some amount of profit which is very minor because of, if we have no sales then our profit generation will stop as well at a certain level. Therefore here we can also say that our profit strongly dependent on sales.
Raw picture of the Data showing Dependent & Independent Variables
Predicted Values, Square sum of Errors, Square sum of total errors
Calculations for Multiple Regression Equation
Multiple Regression Analysis
Ŷ = a + b1x1+ b2x2 + b3x3 ………………….. (1)
Now after performing manual calculations we go the values of involving
Co-efficients & a constant, as follow;
a = 0.00017752 , b1 = 0.1999997809 , b2 = 0.000002049142345
b3 = -0.000001026301752
Required Regression Equation:
Co-efficient of Determination:
Interpretation:
- 99.9999982% of variance in ‘Y’ is explained by the explained variable & remaining due to the unexplained variation, in our scenario our explained variables are Production cost, Advertisement expenses, sales. Deviation in these variables causing variance in our dependent variable.
Adjusted R Square:
Interpretation:
- 99.9999982% of variance in ‘Y’ is explained by the regression equation because this deviation is due to the explained variable & remaining due to the unexplained variation, in our scenario our explained variables are Production cost, Advertisement expenses, sales. Deviation in these variables causing variance in our dependent variable.
- We took this step further to correct down the value after calculating the co-efficient of determination, because previous value conclude that the increased trend in the variation but in the actual situation it is not that much higher that it shows, thus we move further to the r(Square) Adjusted.
Correlation:
Interpretation:
The correlation coefficient is a quantitative measure of the strength of the linear relationship between two variables. The correlation coefficient, r, ranges between -1.0 & +1.0.
- Our relationship is approximately equivalent to perfectly positive relation.
Hypothesis of Regression
Hypothesis Testing at Significance Level = 0.05
(Null Hypothesis) H0 : b1 = b2 = b3=
(Alternative Hypothesis) H1 : b1 ≠ b2 ≠ b3
Critical Region:
V1 = k → V1 = 3
V2 = n – (k + 1) → V2 = 21
ƒ v1 , v2 , α → ƒ 3 , 21 , 0.05 = 3.07
If ƒ test > ƒ tabulated then we reject H0
Test Statistic:
ƒ = MSR/ MSE
ANOVA Table
ƒ = MSR / MSE
= 477798392.7
Decision:
So we rejected H0 Because ƒ test is Greater then ƒ tabulated
Interpretation:
- Similarity With Single Variable Regression
- New Values related to finding optimal combination of variables that can predict response variable
- We cannot reject the hypothesis that variation explained by regression is higher than the variation in residuals.
- In a multiple regression, a case of more than one x variable, we conduct a statistical test about the overall model. The basic idea is do all the x variables as package has a relationship with the Y variable. The null hypothesis is that there is no relationship, Alternative hypothesis has a relationship between these variables
- Thus we reject H0 because we have a relationship between these variables.
- If null hypothesis is true the equation for the line would mean the x’s do not have an influence on Y. The alternative hypothesis is that at least one of the independent’s is not zero, written H1: not all x’s = 0. Rejecting the null means that the x’s as a group are related to Y.
Prediction of next 10 Months:
Explanation:
Here we predict from the Regression Model for the further 10 months for our Company, to determine the profit picture in the future.
Additions of more Independent Variables:
After performing all the calculations & research work we came across with the conclusion that if we increases the independent variables we have also increases the accuracy in the model, therefore if we adding more independent variable in our model as ‘Admin Expenses’ , ‘Salary’, ‘Electricity Bills’, ‘Freight Charges’ etc. then we can more consistently & accurately predict our Dependent Element.
Graphs:
References Websites:
References Books:
Introductory to Statistics by “Prem S. Mann”
Business Statistics by “Lind Marshall”
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