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=

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. $r=$ $\square$
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Solution

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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:

\[ r = \frac{-197.838}{20.595 \times 10.973} = -0.875 \]

Final Answer

The correlation coefficient \( r \) is

\[ \boxed{-0.875} \]

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