To find the value of the linear correlation coefficient [tex]\( r \)[/tex], we need to analyze the paired data for the costs of advertising and the number of products sold. Let's denote the cost of advertising as [tex]\( X \)[/tex] and the number of products sold as [tex]\( Y \)[/tex].
The provided data pairs are:
[tex]\[
X = [9, 2, 3, 4, 2, 5, 9, 10]
\][/tex]
[tex]\[
Y = [85, 52, 55, 68, 67, 86, 83, 73]
\][/tex]
The linear correlation coefficient [tex]\( r \)[/tex], also known as Pearson's correlation coefficient, measures the strength and direction of a linear relationship between two variables. It is computed using the formula:
[tex]\[
r = \frac{\text{cov}(X,Y)}{\sigma_X \sigma_Y}
\][/tex]
Here, [tex]\(\text{cov}(X,Y)\)[/tex] is the covariance of [tex]\(X\)[/tex] and [tex]\(Y\)[/tex], and [tex]\(\sigma_X\)[/tex] and [tex]\(\sigma_Y\)[/tex] are the standard deviations of [tex]\(X\)[/tex] and [tex]\(Y\)[/tex] respectively.
After calculating these values, we get the correlation coefficient. The result of this calculation indicates the linear relationship between the cost of the advertising and the number of products sold.
Given all the steps of computation, the value of the linear correlation coefficient [tex]\( r \)[/tex] is approximately [tex]\( 0.7077213602731203 \)[/tex].
So, the correct answer from the given choices is:
[tex]\[ 0.708 \][/tex]
This value indicates a positive correlation between the cost of advertising and the number of products sold, meaning that as advertising costs increase, the number of products sold tends to also increase.
