Questions: Run a regression analysis on the following bivariate set of data with y as the response variable.
x y
16.9 86.6
67.6 52.5
54.9 70.8
76.4 50.6
40.9 70.8
33.7 78.4
68.7 68.2
19.1 76.1
49.0 65.9
37.2 71.8
Find the correlation coefficient and report it accurate to three decimal places.
r=
Transcript text: Run a regression analysis on the following bivariate set of data with $y$ as the response variable.
\begin{tabular}{|c|c|}
\hline $\mathbf{x}$ & $\mathbf{y}$ \\
\hline 16.9 & 86.6 \\
\hline 67.6 & 52.5 \\
\hline 54.9 & 70.8 \\
\hline 76.4 & 50.6 \\
\hline 40.9 & 70.8 \\
\hline 33.7 & 78.4 \\
\hline 68.7 & 68.2 \\
\hline 19.1 & 76.1 \\
\hline 49. & 65.9 \\
\hline 37.2 & 71.8 \\
\hline
\end{tabular}
Find the correlation coefficient and report it accurate to three decimal places.
$r=$ $\square$
Solution
Solution Steps
Step 1: Calculate Covariance
The covariance between the variables \( X \) and \( Y \) is calculated as follows:
\[
\text{Cov}(X,Y) = -197.838
\]
Step 2: Calculate Standard Deviations
The standard deviation of \( X \) is given by:
\[
\sigma_X = 20.595
\]
The standard deviation of \( Y \) is given by:
\[
\sigma_Y = 10.973
\]
Step 3: Calculate Correlation Coefficient
The correlation coefficient \( r \) is calculated using the formula:
\[
r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}
\]
Substituting the values:
\[
r = \frac{-197.838}{20.595 \times 10.973} = -0.875
\]