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Sagot :
Sure, let's tackle each part of the question step by step using the provided data.
### Part (a): Develop the Regression Model
To develop the regression model, we need to find the values of the parameters [tex]\(\alpha\)[/tex] (intercept) and [tex]\(\beta\)[/tex] (slope).
The formula for the slope [tex]\(\beta\)[/tex] is:
[tex]\[ \beta = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \][/tex]
The formula for the intercept [tex]\(\alpha\)[/tex] is:
[tex]\[ \alpha = \bar{y} - \beta \bar{x} \][/tex]
Given the calculation outcomes:
- [tex]\(\alpha = 77.3521\)[/tex]
- [tex]\(\beta = 4.2606\)[/tex]
Interpretation:
- [tex]\(\alpha\)[/tex] (intercept) = 77.3521, meaning that if the advertisement cost is 0, the model predicts a base sales volume of 77.3521 thousand birr.
- [tex]\(\beta\)[/tex] (slope) = 4.2606, meaning that for each additional thousand birr spent on advertisement, the sales volume is predicted to increase by 4.2606 thousand birr.
### Part (b): Predicted Sales Volume for Advertisement Cost of 27 Thousand Birr
Using the regression model, the sales volume [tex]\( y \)[/tex] can be predicted as follows:
[tex]\[ y = \alpha + \beta \cdot x \][/tex]
For an advertisement cost [tex]\( x = 27 \)[/tex] thousand birr:
[tex]\[ y = 77.3521 + 4.2606 \cdot 27 \][/tex]
Thus, the predicted sales volume is:
[tex]\[ y = 192.3873 \][/tex]
So, the predicted sales volume for an advertisement cost of 27 thousand birr is approximately 192.3873 thousand birr.
### Part (c): Pearson Correlation Coefficient and Coefficient of Determination
The Pearson correlation coefficient [tex]\( r \)[/tex] measures the strength and direction of the linear relationship between advertisement cost and sales volume.
Given:
- [tex]\( r = 0.7474 \)[/tex]
The coefficient of determination [tex]\( R^2 \)[/tex] indicates the proportion of the variance in the dependent variable (sales volume) that is predictable from the independent variable (advertisement cost).
[tex]\[ R^2 = r^2 = (0.7474)^2 = 0.5585 \][/tex]
Interpretation:
- The Pearson correlation coefficient [tex]\( r = 0.7474 \)[/tex] signifies a strong positive linear relationship between advertisement cost and sales volume.
- The coefficient of determination [tex]\( R^2 = 0.5585 \)[/tex] means that approximately 55.85% of the variation in sales volume can be explained by the variation in advertisement cost.
### Part (d): Error Terms Using Deviation Formula (Method 2)
The error term for each data point can be calculated as the observed value minus the predicted value.
Given the error terms:
[tex]\[ \text{error terms} = \left[ -27.8732, 2.0845, -23.4366, 76.5634, -11.4789, -8.5211, -5.5634, 6.4366, -22.0845, 13.8732 \right] \][/tex]
These values represent the deviations of the observed sales volumes from the predicted sales volumes based on the regression model.
### Summary
1. Regression Model:
- Intercept [tex]\(\alpha = 77.3521\)[/tex]
- Slope [tex]\(\beta = 4.2606\)[/tex]
- Interpretation: [tex]\(\alpha\)[/tex] represents the base sales volume when advertisement cost is zero, [tex]\(\beta\)[/tex] represents the change in sales volume for each additional thousand birr spent on advertisement.
2. Predicted Sales Volume for [tex]\( x = 27 \)[/tex]:
- [tex]\( 192.3873 \)[/tex] thousand birr
3. Pearson Correlation Coefficient and Coefficient of Determination:
- [tex]\( r = 0.7474 \)[/tex]
- [tex]\( R^2 = 0.5585 \)[/tex]
- Interpretation: Strong positive linear relationship, 55.85% variability explained.
4. Error Terms:
- The deviation of observed sales volumes from predicted values.
### Part (a): Develop the Regression Model
To develop the regression model, we need to find the values of the parameters [tex]\(\alpha\)[/tex] (intercept) and [tex]\(\beta\)[/tex] (slope).
The formula for the slope [tex]\(\beta\)[/tex] is:
[tex]\[ \beta = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \][/tex]
The formula for the intercept [tex]\(\alpha\)[/tex] is:
[tex]\[ \alpha = \bar{y} - \beta \bar{x} \][/tex]
Given the calculation outcomes:
- [tex]\(\alpha = 77.3521\)[/tex]
- [tex]\(\beta = 4.2606\)[/tex]
Interpretation:
- [tex]\(\alpha\)[/tex] (intercept) = 77.3521, meaning that if the advertisement cost is 0, the model predicts a base sales volume of 77.3521 thousand birr.
- [tex]\(\beta\)[/tex] (slope) = 4.2606, meaning that for each additional thousand birr spent on advertisement, the sales volume is predicted to increase by 4.2606 thousand birr.
### Part (b): Predicted Sales Volume for Advertisement Cost of 27 Thousand Birr
Using the regression model, the sales volume [tex]\( y \)[/tex] can be predicted as follows:
[tex]\[ y = \alpha + \beta \cdot x \][/tex]
For an advertisement cost [tex]\( x = 27 \)[/tex] thousand birr:
[tex]\[ y = 77.3521 + 4.2606 \cdot 27 \][/tex]
Thus, the predicted sales volume is:
[tex]\[ y = 192.3873 \][/tex]
So, the predicted sales volume for an advertisement cost of 27 thousand birr is approximately 192.3873 thousand birr.
### Part (c): Pearson Correlation Coefficient and Coefficient of Determination
The Pearson correlation coefficient [tex]\( r \)[/tex] measures the strength and direction of the linear relationship between advertisement cost and sales volume.
Given:
- [tex]\( r = 0.7474 \)[/tex]
The coefficient of determination [tex]\( R^2 \)[/tex] indicates the proportion of the variance in the dependent variable (sales volume) that is predictable from the independent variable (advertisement cost).
[tex]\[ R^2 = r^2 = (0.7474)^2 = 0.5585 \][/tex]
Interpretation:
- The Pearson correlation coefficient [tex]\( r = 0.7474 \)[/tex] signifies a strong positive linear relationship between advertisement cost and sales volume.
- The coefficient of determination [tex]\( R^2 = 0.5585 \)[/tex] means that approximately 55.85% of the variation in sales volume can be explained by the variation in advertisement cost.
### Part (d): Error Terms Using Deviation Formula (Method 2)
The error term for each data point can be calculated as the observed value minus the predicted value.
Given the error terms:
[tex]\[ \text{error terms} = \left[ -27.8732, 2.0845, -23.4366, 76.5634, -11.4789, -8.5211, -5.5634, 6.4366, -22.0845, 13.8732 \right] \][/tex]
These values represent the deviations of the observed sales volumes from the predicted sales volumes based on the regression model.
### Summary
1. Regression Model:
- Intercept [tex]\(\alpha = 77.3521\)[/tex]
- Slope [tex]\(\beta = 4.2606\)[/tex]
- Interpretation: [tex]\(\alpha\)[/tex] represents the base sales volume when advertisement cost is zero, [tex]\(\beta\)[/tex] represents the change in sales volume for each additional thousand birr spent on advertisement.
2. Predicted Sales Volume for [tex]\( x = 27 \)[/tex]:
- [tex]\( 192.3873 \)[/tex] thousand birr
3. Pearson Correlation Coefficient and Coefficient of Determination:
- [tex]\( r = 0.7474 \)[/tex]
- [tex]\( R^2 = 0.5585 \)[/tex]
- Interpretation: Strong positive linear relationship, 55.85% variability explained.
4. Error Terms:
- The deviation of observed sales volumes from predicted values.
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