Sure, let's walk through the process step-by-step to find the correlation coefficient and the proportion of the variation in [tex]\( y \)[/tex] that can be explained by [tex]\( x \)[/tex].
### Step 1: Data Points
We start with the given bivariate data set:
[tex]\[
\begin{array}{|r|c|}
\hline
x & y \\
\hline
70.6 & 45.8 \\
\hline
17.3 & 46.1 \\
\hline
52.8 & 41.6 \\
\hline
87.9 & 42.2 \\
\hline
15.0 & 36.6 \\
\hline
47.1 & 43.4 \\
\hline
44.9 & 46.2 \\
\hline
28.9 & 39.3 \\
\hline
\end{array}
\][/tex]
### Step 2: Correlation Coefficient
1. Mean of each variable
Calculate the mean of [tex]\( x \)[/tex] ([tex]\( \overline{x} \)[/tex]) and [tex]\( y \)[/tex] ([tex]\( \overline{y} \)[/tex]).
2. Deviation Scores
Calculate the deviation of each value from the mean for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
3. Product of Deviations
Multiply the deviation of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for each pair.
4. Sum of the Products
Sum these products to get the numerator of the correlation coefficient formula.
5. Square Deviations & Sum
Calculate the square deviations for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex], sum them up separately for the denominator of the correlation formula.
6. Correlation Formula
The correlation coefficient formula is:
[tex]\[
r = \frac{\sum{(x_i - \overline{x})(y_i - \overline{y})}}{\sqrt{\sum{(x_i - \overline{x})^2} \sum{(y_i - \overline{y})^2}}}
\][/tex]
Using this process, we find that the correlation coefficient is [tex]\( r = 0.316 \)[/tex] (to three decimal places).
### Step 3: Proportion of Variation
1. Coefficient of Determination
The coefficient of determination, [tex]\( R^2 \)[/tex], represents the proportion of the variance in the dependent variable that is predictable from the independent variable. It is the square of the correlation coefficient:
[tex]\[
R^2 = r^2 = (0.316)^2 = 0.10
\][/tex]
2. Convert to Percentage
Convert the coefficient of determination to a percentage by multiplying by 100:
[tex]\[
\text{Percentage of variation} = R^2 \times 100 = 0.10 \times 100 = 10.0\%
\][/tex]
### Final Results
- The correlation coefficient [tex]\( r = 0.316 \)[/tex].
- The proportion of the variation in [tex]\( y \)[/tex] that can be explained by [tex]\( x \)[/tex] is [tex]\( 10.0\% \)[/tex].
Thus, the results are:
[tex]\[
r = 0.316
\][/tex]
[tex]\[
R^2 = 10.0\%
\][/tex]
