To determine which table shows no correlation, we need to calculate the Pearson correlation coefficient for each table. The Pearson correlation coefficient is a measure of the strength and direction of the relationship between two variables.
Here are the steps:
1. Identify the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values for each table.
2. Calculate the Pearson correlation coefficient for each table.
3. Determine which correlation coefficient is closest to zero, indicating no correlation.
Now, let's consider the given correlation coefficients for the tables:
- The correlation coefficient for the first table is [tex]\(-0.8548\)[/tex].
- The correlation coefficient for the second table is [tex]\(0.9031\)[/tex].
- The correlation coefficient for the third table is [tex]\(-0.1036\)[/tex].
- The correlation coefficient for the fourth table is [tex]\(-0.9666\)[/tex].
The coefficient closest to zero is [tex]\(-0.1036\)[/tex], which corresponds to the third table. This indicates that the third table exhibits the least correlation, closest to none.
Therefore, the table showing no correlation is the third table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $x$ & 3 & 5 & 6 & 8 & 10 & 14 & 15 \\
\hline $y$ & -2 & -4 & 6 & 8 & 12 & 10 & -16 \\
\hline
\end{tabular}
\][/tex]
