To determine which table shows no correlation, we'll calculate the correlation coefficients for each combined set of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values. The correlation coefficient, [tex]\( r \)[/tex], is a measure of the linear relationship between two variables. It ranges from -1 (perfect negative correlation) to 1 (perfect positive correlation). A correlation of 0 indicates no linear relationship.
Let's examine the correlation coefficients for each table:
1. Table 1:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & 3 & 5 & 6 & 8 & 10 & 14 & 15 \\
\hline
y & -1 & -2 & -3 & -2 & -5 & -4 & -8 \\
\hline
\end{array}
\][/tex]
The correlation coefficient [tex]\( r \)[/tex] for this table is approximately [tex]\(-0.85\)[/tex].
2. Table 2:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & 3 & 5 & 6 & 8 & 10 & 14 & 15 \\
\hline
y & -6 & -7 & -4 & -2 & 0 & -1 & 3 \\
\hline
\end{array}
\][/tex]
The correlation coefficient [tex]\( r \)[/tex] for this table is approximately [tex]\(0.90\)[/tex].
3. Table 3:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & 3 & 5 & 6 & 8 & 10 & 14 & 15 \\
\hline
y & -2 & -4 & 6 & 8 & 12 & 10 & -16 \\
\hline
\end{array}
\][/tex]
The correlation coefficient [tex]\( r \)[/tex] for this table is approximately [tex]\(-0.10\)[/tex].
4. Table 4:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & 3 & 5 & 6 & 8 & 10 & 14 & 15 \\
\hline
y & -3 & -5 & -9 & -11 & -13 & -15 & -17 \\
\hline
\end{array}
\][/tex]
The correlation coefficient [tex]\( r \)[/tex] for this table is approximately [tex]\(-0.97\)[/tex].
We are looking for the table that shows no correlation, which implies a correlation coefficient close to [tex]\(0\)[/tex]. Among the given tables, Table 3 has the correlation coefficient closest to [tex]\(0\)[/tex] ([tex]\(-0.10\)[/tex]), indicating no significant linear relationship.
Therefore, Table 3 shows no correlation.
