1. The following table gives the advertisement cost and sales volume in thousands of birr for a sample of 10 HHs.

\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
Adv cost & [tex]$2$[/tex] & [tex]$6$[/tex] & [tex]$8$[/tex] & [tex]$8$[/tex] & [tex]$12$[/tex] & [tex]$16$[/tex] & [tex]$20$[/tex] & [tex]$20$[/tex] & [tex]$22$[/tex] & [tex]$26$[/tex] \\
\hline
Sales volume & 58 & 105 & 88 & 188 & 117 & 137 & 157 & 169 & 149 & 202 \\
\hline
\end{tabular}

a) Develop the regression model and compute the values of parameters [tex]$\alpha$[/tex] and [tex]$\beta$[/tex]. Interpret the values of [tex]$\alpha$[/tex] and [tex]$\beta$[/tex].

b) If the level of advertisement cost is 27 thousand birr, what will be the predicted sales volume?

c) Compute the Pearson correlation coefficient [tex]$r$[/tex] and the coefficient of determination [tex]$\left( R^2 \right)$[/tex]. Interpret their results.

d) Use the deviation formula (Method 2) and calculate the value of the error term in the model.



Answer :

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.