Answer :
Answer:
To find the values of \( r \) (the correlation coefficient) and \( r^2 \) (the coefficient of determination) for the given data, we can use the following steps:
1. Calculate the mean of \( x \) and \( y \).
2. Calculate the deviations from the mean for both \( x \) and \( y \).
3. Calculate the sum of the products of the deviations.
4. Calculate the square root of the sum of the squared deviations for both \( x \) and \( y \).
5. Calculate the correlation coefficient \( r \).
6. Calculate the coefficient of determination \( r^2 \).
Given data:
\[ \begin{array}{|c|c|}
\hline
x & y \\
\hline
5 & 22 \\
8 & 23.9 \\
9 & 14 \\
14 & 17 \\
20 & 5.2 \\
\hline
\end{array} \]
### Step 1: Calculate the mean of \( x \) and \( y \)
\[ \text{Mean of } x = \frac{5 + 8 + 9 + 14 + 20}{5} = \frac{56}{5} = 11.2 \]
\[ \text{Mean of } y = \frac{22 + 23.9 + 14 + 17 + 5.2}{5} = \frac{82.1}{5} = 16.42 \]
### Step 2: Calculate the deviations from the mean for both \( x \) and \( y \)
| \( x \) | \( y \) | \( x - \bar{x} \) | \( y - \bar{y} \) |
|---------|---------|------------------|------------------|
| 5 | 22 | -6.2 | 5.58 |
| 8 | 23.9 | -3.2 | 7.48 |
| 9 | 14 | -2.2 | -2.42 |
| 14 | 17 | 2.8 | 0.58 |
| 20 | 5.2 | 8.8 | -11.22 |
### Step 3: Calculate the sum of the products of the deviations
\[ \text{Sum of } (x - \bar{x})(y - \bar{y}) = (-6.2)(5.58) + (-3.2)(7.48) + (-2.2)(-2.42) + (2.8)(0.58) + (8.8)(-11.22) \]
\[ = -34.596 + -23.936 + 5.324 + 1.624 - 98.496 \]
\[ = -150.98 \]
### Step 4: Calculate the square root of the sum of the squared deviations for both \( x \) and \( y \)
\[ \text{Sum of } (x - \bar{x})^2 = (-6.2)^2 + (-3.2)^2 + (-2.2)^2 + (2.8)^2 + (8.8)^2 \]
\[ = 38.44 + 10.24 + 4.84 + 7.84 + 77.44 \]
\[ = 139.8 \]
\[ \text{Sum of } (y - \bar{y})^2 = (5.58)^2 + (7.48)^2 + (-2.42)^2 + (0.58)^2 + (-11.22)^2 \]
\[ = 31.2164 + 55.9504 + 5.8564 + 0.3364 + 125.8884 \]
\[ = 219.2476 \]
### Step 5: Calculate the correlation coefficient \( r \)
\[ r = \frac{\text{Sum of } (x - \bar{x})(y - \bar{y})}{\sqrt{\text{Sum of } (x - \bar{x})^2 \cdot \text{Sum of } (y - \bar{y})^2}} \]
\[ r = \frac{-150.98}{\sqrt{139.8 \cdot 219.2476}} \]
\[ r = \frac{-150.98}{\sqrt{30633.924}} \]
\[ r = \frac{-150.98}{174.9829} \]
\[ r \approx -0.862 \]
### Step 6: Calculate the coefficient of determination \( r^2 \)
\[ r^2 = (-0.862)^2 \]
\[ r^2 \approx 0.745 \]
Therefore, the correlation coefficient \( r \) is approximately -0.862, and the coefficient of determination \( r^2 \) is approximately 0.745.
