Answer :
### Individual Assignment (10%)
#### 1. Advertisement Cost and Sales Volume
Given a table with advertisement costs and sales volumes for a sample of 10 households:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Adv Cost (X)} & 2 & 6 & 8 & 8 & 12 & 16 & 20 & 20 & 22 & 26 \\ \hline \text{Sales Volume (Y)} & 58 & 105 & 88 & 188 & 117 & 137 & 157 & 169 & 149 & 202 \\ \hline \end{array} \][/tex]
Step-by-Step Solution:
#### a) Developing the Regression Model
A linear regression model is typically given by:
[tex]\[ Y = \alpha + \beta X + \epsilon \][/tex]
where:
- [tex]\( Y \)[/tex] is the dependent variable (Sales Volume),
- [tex]\( X \)[/tex] is the independent variable (Advertisement Cost),
- [tex]\( \alpha \)[/tex] (intercept) and [tex]\( \beta \)[/tex] (slope) are the parameters we need to compute,
- [tex]\( \epsilon \)[/tex] is the error term.
From computations, we have:
- [tex]\( \alpha = 77.35211267605634 \)[/tex]
- [tex]\( \beta = 4.26056338028169 \)[/tex]
Interpretation of [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex]:
- [tex]\( \alpha \)[/tex]: This is the intercept of the regression line. It represents the predicted sales volume when the advertisement cost is zero. In this case, [tex]\( \alpha = 77.35 \)[/tex] thousand birr, meaning if no money is spent on advertising, the expected sales volume is 77.35 thousand birr.
- [tex]\( \beta \)[/tex]: This is the slope of the regression line. It indicates the change in sales volume for each additional unit of advertisement cost. Here, [tex]\( \beta = 4.26 \)[/tex], meaning for each additional thousand birr spent on advertisement, the sales volume increases by approximately 4.26 thousand birr.
#### b) Predicted Sales Volume for an Advertisement Cost of 27 thousand birr
To predict the sales volume when the advertisement cost is 27 thousand birr, we use the regression equation:
[tex]\[ \hat{Y} = \alpha + \beta X \][/tex]
Plugging in the values:
[tex]\[ \hat{Y} = 77.35211267605634 + 4.26056338028169 \times 27 \][/tex]
The predicted sales volume [tex]\(\hat{Y}\)[/tex] is:
[tex]\[ \hat{Y} = 192.38732394366195 \text{ thousand birr} \][/tex]
Hence, if the advertisement cost is 27 thousand birr, the predicted sales volume will be 192.39 thousand birr.
#### c) Pearson Correlation Coefficient (r) and Coefficient of Determination ([tex]\(R^2\)[/tex])
- Pearson Correlation Coefficient (r):
From our computations, we have [tex]\( r = 0.7473522558883839 \)[/tex].
Interpretation: The Pearson correlation coefficient [tex]\( r \)[/tex] measures the strength and direction of the linear relationship between advertisement cost and sales volume. An [tex]\( r \)[/tex] of approximately 0.75 suggests a strong positive correlation, meaning as advertisement cost increases, sales volume tends to increase as well.
- Coefficient of Determination ([tex]\(R^2\)[/tex]):
From our computations, [tex]\( R^2 = 0.5585353943814565 \)[/tex].
Interpretation: The coefficient of determination, [tex]\( R^2 \)[/tex], represents the proportion of the variance in the dependent variable (sales volume) that is predictable from the independent variable (advertisement cost). An [tex]\( R^2 \)[/tex] of 0.56 means that approximately 56% of the variability in sales volume can be explained by the advertisement cost.
#### d) Error Term Calculation using Deviation Formula (Method 2)
The error term for each data point can be calculated as the difference between the actual sales volume and the predicted sales volume:
[tex]\[ \text{Errors} = Y - \hat{Y} \][/tex]
Based on our computations, the calculated error terms for each observation are:
[tex]\[ \begin{array}{|c|c|} \hline \text{Observation} & \text{Error Term} \\ \hline 1 & -27.873239436619713 \\ 2 & 2.08450704225352 \\ 3 & -23.436619718309856 \\ 4 & 76.56338028169014 \\ 5 & -11.47887323943661 \\ 6 & -8.52112676056339 \\ 7 & -5.563380281690115 \\ 8 & 6.436619718309885 \\ 9 & -22.084507042253506 \\ 10 & 13.873239436619713 \\ \hline \end{array} \][/tex]
### 2. Effect of Education on Salary
The econometric model is given by:
[tex]\[ \text{Salary} = 20 + 2.1 \, \text{edu} + e \][/tex]
Interpretation of "e":
- [tex]\( e \)[/tex] represents the error term or the residual in this regression model. It captures the effect of all other factors influencing Salary that are not included in the model. These could be things like work experience, skills, workplace conditions, industry of employment, and other variables not accounted for by the independent variable (education). Essentially, it represents the deviation of the observed salary from the salary predicted by the model.
#### 1. Advertisement Cost and Sales Volume
Given a table with advertisement costs and sales volumes for a sample of 10 households:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Adv Cost (X)} & 2 & 6 & 8 & 8 & 12 & 16 & 20 & 20 & 22 & 26 \\ \hline \text{Sales Volume (Y)} & 58 & 105 & 88 & 188 & 117 & 137 & 157 & 169 & 149 & 202 \\ \hline \end{array} \][/tex]
Step-by-Step Solution:
#### a) Developing the Regression Model
A linear regression model is typically given by:
[tex]\[ Y = \alpha + \beta X + \epsilon \][/tex]
where:
- [tex]\( Y \)[/tex] is the dependent variable (Sales Volume),
- [tex]\( X \)[/tex] is the independent variable (Advertisement Cost),
- [tex]\( \alpha \)[/tex] (intercept) and [tex]\( \beta \)[/tex] (slope) are the parameters we need to compute,
- [tex]\( \epsilon \)[/tex] is the error term.
From computations, we have:
- [tex]\( \alpha = 77.35211267605634 \)[/tex]
- [tex]\( \beta = 4.26056338028169 \)[/tex]
Interpretation of [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex]:
- [tex]\( \alpha \)[/tex]: This is the intercept of the regression line. It represents the predicted sales volume when the advertisement cost is zero. In this case, [tex]\( \alpha = 77.35 \)[/tex] thousand birr, meaning if no money is spent on advertising, the expected sales volume is 77.35 thousand birr.
- [tex]\( \beta \)[/tex]: This is the slope of the regression line. It indicates the change in sales volume for each additional unit of advertisement cost. Here, [tex]\( \beta = 4.26 \)[/tex], meaning for each additional thousand birr spent on advertisement, the sales volume increases by approximately 4.26 thousand birr.
#### b) Predicted Sales Volume for an Advertisement Cost of 27 thousand birr
To predict the sales volume when the advertisement cost is 27 thousand birr, we use the regression equation:
[tex]\[ \hat{Y} = \alpha + \beta X \][/tex]
Plugging in the values:
[tex]\[ \hat{Y} = 77.35211267605634 + 4.26056338028169 \times 27 \][/tex]
The predicted sales volume [tex]\(\hat{Y}\)[/tex] is:
[tex]\[ \hat{Y} = 192.38732394366195 \text{ thousand birr} \][/tex]
Hence, if the advertisement cost is 27 thousand birr, the predicted sales volume will be 192.39 thousand birr.
#### c) Pearson Correlation Coefficient (r) and Coefficient of Determination ([tex]\(R^2\)[/tex])
- Pearson Correlation Coefficient (r):
From our computations, we have [tex]\( r = 0.7473522558883839 \)[/tex].
Interpretation: The Pearson correlation coefficient [tex]\( r \)[/tex] measures the strength and direction of the linear relationship between advertisement cost and sales volume. An [tex]\( r \)[/tex] of approximately 0.75 suggests a strong positive correlation, meaning as advertisement cost increases, sales volume tends to increase as well.
- Coefficient of Determination ([tex]\(R^2\)[/tex]):
From our computations, [tex]\( R^2 = 0.5585353943814565 \)[/tex].
Interpretation: The coefficient of determination, [tex]\( R^2 \)[/tex], represents the proportion of the variance in the dependent variable (sales volume) that is predictable from the independent variable (advertisement cost). An [tex]\( R^2 \)[/tex] of 0.56 means that approximately 56% of the variability in sales volume can be explained by the advertisement cost.
#### d) Error Term Calculation using Deviation Formula (Method 2)
The error term for each data point can be calculated as the difference between the actual sales volume and the predicted sales volume:
[tex]\[ \text{Errors} = Y - \hat{Y} \][/tex]
Based on our computations, the calculated error terms for each observation are:
[tex]\[ \begin{array}{|c|c|} \hline \text{Observation} & \text{Error Term} \\ \hline 1 & -27.873239436619713 \\ 2 & 2.08450704225352 \\ 3 & -23.436619718309856 \\ 4 & 76.56338028169014 \\ 5 & -11.47887323943661 \\ 6 & -8.52112676056339 \\ 7 & -5.563380281690115 \\ 8 & 6.436619718309885 \\ 9 & -22.084507042253506 \\ 10 & 13.873239436619713 \\ \hline \end{array} \][/tex]
### 2. Effect of Education on Salary
The econometric model is given by:
[tex]\[ \text{Salary} = 20 + 2.1 \, \text{edu} + e \][/tex]
Interpretation of "e":
- [tex]\( e \)[/tex] represents the error term or the residual in this regression model. It captures the effect of all other factors influencing Salary that are not included in the model. These could be things like work experience, skills, workplace conditions, industry of employment, and other variables not accounted for by the independent variable (education). Essentially, it represents the deviation of the observed salary from the salary predicted by the model.
