4. Alfonso surveyed several students in his school. He asked them to give the grade that they are in and the number of pets they have at home. The results are shown in the table below.
\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{ Survey Results } \\
\hline Grade & \begin{tabular}{c}
Number of \\
Pets
\end{tabular} \\
\hline 5 & 0 \\
\hline 6 & 4 \\
\hline 2 & 2 \\
\hline 1 & 3 \\
\hline 4 & 5 \\
\hline 2 & 1 \\
\hline 4 & 2 \\
\hline 1 & 2 \\
\hline 4 & 0 \\
\hline 3 & 5 \\
\hline 5 & 2 \\
\hline
\end{tabular}

Find the correlation coefficient for the data. Round to the nearest hundredth.

Step 1 ：Alfonso surveyed several students in his school. He asked them to give the grade that they are in and the number of pets they have at home. The results are shown in the table below.

Step 2 ：\begin{tabular}{|c|c|} \hline \multicolumn{2}{|c|}{ Survey Results } \\ \hline Grade & \begin{tabular}{c} Number of \\ Pets \end{tabular} \\ \hline 5 & 0 \\ \hline 6 & 4 \\ \hline 2 & 2 \\ \hline 1 & 3 \\ \hline 4 & 5 \\ \hline 2 & 1 \\ \hline 4 & 2 \\ \hline 1 & 2 \\ \hline 4 & 0 \\ \hline 3 & 5 \\ \hline 5 & 2 \\ \hline \end{tabular}

Step 3 ：We are asked to find the correlation coefficient for the data. The correlation coefficient, often denoted by r, is a measure of the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where -1 indicates a strong negative relationship, 1 indicates a strong positive relationship, and 0 indicates no relationship.

Step 4 ：To calculate the correlation coefficient, we need to use the formula: \(r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}\) where: - n is the number of pairs of data - \(\Sigma xy\) is the sum of the products of paired data - \(\Sigma x\) and \(\Sigma y\) are the sums of the x and y data respectively - \(\Sigma x^2\) and \(\Sigma y^2\) are the sums of the squares of the x and y data respectively

Step 5 ：We can calculate these values using the data from the table. The grades are [5, 6, 2, 1, 4, 2, 4, 1, 4, 3, 5] and the number of pets are [0, 4, 2, 3, 5, 1, 2, 2, 0, 5, 2].

Step 6 ：Calculating the sums, we get \(\Sigma x = 37\), \(\Sigma y = 26\), \(\Sigma x^2 = 153\), \(\Sigma y^2 = 92\), and \(\Sigma xy = 88\).

Step 7 ：Substituting these values into the formula, we get \(r = \frac{11(88) - (37)(26)}{\sqrt{[11*153 - (37)^2][11*92 - (26)^2]}}\)

Step 8 ：Solving the above expression, we get \(r = 0.02\)

Step 9 ：Final Answer: The correlation coefficient for the data is \(\boxed{0.02}\)