Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
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.
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
