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Here is a bivariate data set.

[tex]
\[
\begin{array}{|r|r|}
\hline
\multicolumn{1}{|c|}{$x$} & \multicolumn{1}{c|}{$y$} \\
\hline
39.6 & 56.2 \\
\hline
6.4 & -11 \\
\hline
66.8 & 127.8 \\
\hline
50.7 & 114.8 \\
\hline
27.2 & 32 \\
\hline
11.4 & 34.8 \\
\hline
43.5 & 82.7 \\
\hline
50.7 & 81.3 \\
\hline
71.3 & 97.8 \\
\hline
-13.2 & -26.9 \\
\hline
36.6 & 84.8 \\
\hline
72.8 & 139.8 \\
\hline
31 & 46.8 \\
\hline
\end{array}
\]
[/tex]

Find the correlation coefficient and report it accurate to three decimal places.

[tex]
r = \square
[/tex]

Answer :

To find the correlation coefficient for the given bivariate data set, follow these steps:

1. Understand the Data:
We have two variables, [tex]\(x\)[/tex] and [tex]\(y\)[/tex], each with a set of values. The values are paired as follows:

[tex]\[
\begin{array}{c|c}
x & y \\
\hline
39.6 & 56.2 \\
6.4 & -11 \\
66.8 & 127.8 \\
50.7 & 114.8 \\
27.2 & 32 \\
11.4 & 34.8 \\
43.5 & 82.7 \\
50.7 & 81.3 \\
71.3 & 97.8 \\
-13.2 & -26.9 \\
36.6 & 84.8 \\
72.8 & 139.8 \\
31 & 46.8 \\
\end{array}
\][/tex]

2. Calculate the Mean of Each Variable:
- Calculate the mean (average) of all [tex]\(x\)[/tex] values.
- Calculate the mean (average) of all [tex]\(y\)[/tex] values.

3. Calculate the Deviations:
- Find the deviation of each value from its mean for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex].

4. Compute the Covariance:
- Combine these deviations to calculate the covariance between [tex]\(x\)[/tex] and [tex]\(y\)[/tex].

5. Calculate the Standard Deviations:
- Compute the standard deviation for [tex]\(x\)[/tex] and for [tex]\(y\)[/tex].

6. Compute the Correlation Coefficient:
- The correlation coefficient, [tex]\(r\)[/tex], is found by dividing the covariance of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] by the product of their standard deviations:
[tex]\[
r = \frac{\text{covariance}(x, y)}{\text{standard deviation}(x) \times \text{standard deviation}(y)}
\][/tex]

7. Result:
Based on these calculations, the correlation coefficient for this data set is where [tex]\( r = 0.944 \)[/tex].

This result indicates a strong positive linear relationship between the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] variables in the data set.

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