To determine which table shows a negative correlation between [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we need to examine the correlation coefficients of each table. Here we have four tables with [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values given as follows:
Table 1:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 2 & 5 & 6 & 7 & 10 & 12 \\
\hline
y & -8 & -5 & -6 & -3 & -2 & -1 \\
\hline
\end{array}
\][/tex]
Table 2:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 2 & 5 & 6 & 7 & 10 & 12 \\
\hline
y & -5 & -5 & -5 & -5 & -5 & -5 \\
\hline
\end{array}
\][/tex]
Table 3:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 2 & 5 & 6 & 7 & 10 & 12 \\
\hline
y & 6 & 3 & 1 & 1 & 3 & 6 \\
\hline
\end{array}
\][/tex]
Table 4:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 2 & 5 & 6 & 7 & 10 & 12 \\
\hline
y & 4 & 2 & -4 & -3 & -11 & -12 \\
\hline
\end{array}
\][/tex]
The determined correlation coefficients for the given tables are:
- The correlation coefficient for Table 1 is 0.9530704598482771.
- The correlation coefficient for Table 2 is not a number (NaN), which indicates no meaningful correlation because all [tex]\( y \)[/tex]-values are equal and do not vary.
- The correlation coefficient for Table 3 is 0.049669963389939197.
- The correlation coefficient for Table 4 is -0.9655651219223368.
A negative correlation means that as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] tends to decrease. From the above coefficients, we can see that the correlation for Table 4 is -0.9655651219223368, which is negative and also the strongest in absolute value among the four tables.
Therefore, Table 4 shows a negative correlation.
