Let’s go through the detailed and step-by-step solution for both parts of the question.
(c) Calculate the value of [tex]\( a \)[/tex]:
To find the value of [tex]\( a \)[/tex] in the equation for the line of best fit, we use the equation:
[tex]\[ Y = a + bx \][/tex]
We are given:
[tex]\[ \Sigma x = 440 \][/tex]
[tex]\[ \Sigma y = 330 \][/tex]
[tex]\[ n = 11 \][/tex]
[tex]\[ b = 0.69171 \][/tex]
First, we need to calculate the mean values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ \text{mean}_x = \frac{\Sigma x}{n} = \frac{440}{11} = 40.0 \][/tex]
[tex]\[ \text{mean}_y = \frac{\Sigma y}{n} = \frac{330}{11} = 30.0 \][/tex]
Next, we use the formula for [tex]\( a \)[/tex]:
[tex]\[ a = \text{mean}_y - b \times \text{mean}_x \][/tex]
Substituting the values:
[tex]\[ a = 30.0 - 0.69171 \times 40.0 \][/tex]
[tex]\[ a = 30.0 - 27.6684 \][/tex]
[tex]\[ a = 2.33 \][/tex]
Therefore, the value of [tex]\( a \)[/tex] to two decimal places is:
[tex]\[ a = 2.33 \][/tex]
(d) Interpret the correlation coefficient:
We are given that the correlation coefficient [tex]\( r \)[/tex] between advertising expenditure and sales revenue is 0.85.
To understand what this means, let's calculate the coefficient of determination, [tex]\( r^2 \)[/tex], which measures the proportion of the variance in the dependent variable that is predictable from the independent variable.
[tex]\[ r^2 = (0.85)^2 \][/tex]
[tex]\[ r^2 = 0.7225 \][/tex]
The coefficient of determination [tex]\( r^2 = 0.7225 \)[/tex] can be interpreted as follows:
- 72.25% of the variation in sales revenue can be explained by the variation in advertising expenditure.
This means there is a strong positive correlation between advertising expenditure and sales revenue, indicating that as advertising expenditure increases, sales revenue tends to increase as well.
In conclusion:
- The value of [tex]\( a \)[/tex] in the regression equation is [tex]\( 2.33 \)[/tex].
- With a correlation coefficient [tex]\( r = 0.85 \)[/tex], the true statement is that 72.25% of the variance in sales revenue is explained by the advertising expenditure. This indicates a strong positive linear relationship between the two variables.
