To determine if LaTasha is correct about the lack of correlation between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given data set, we'll analyze the correlation coefficient between these two sets of data. The correlation coefficient is a statistical measure that calculates the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where:
- 1 indicates a perfect positive linear relationship,
- -1 indicates a perfect negative linear relationship,
- 0 indicates no linear relationship.
Given the data:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
y & 4 & 5 & 4 & 5 & 4 & 5 & 4 \\
\hline
\end{array}
\][/tex]
When we calculate the correlation coefficient for this data set, the value obtained is [tex]\(0.0\)[/tex].
This result means that there is no linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Therefore, LaTasha is correct in asserting that there is no correlation between the two variables in the given data set.
To complement this, the regression equation can also be considered. However, with a correlation coefficient of 0.0, the regression line would essentially be horizontal or vertical, indicating that knowing the value of [tex]\( x \)[/tex] does not help in predicting [tex]\( y \)[/tex], further supporting the assertion that there is no linear correlation between the variables.
