Answer :
Let's go through the steps to find the equation of the line of best fit and the correlation coefficient for the given data points using the linear regression method.
### Given Data:
- Data points: (2, 144.42) and (4, 146.26)
### Step 1: Calculate the Mean of x and y
We start by finding the means of the x-values and y-values.
[tex]\[ x_1 = 2, \, x_2 = 4 \][/tex]
[tex]\[ y_1 = 144.42, \, y_2 = 146.26 \][/tex]
[tex]\[ \text{Mean of x} = \overline{x} = \frac{2+4}{2} = 3 \][/tex]
[tex]\[ \text{Mean of y} = \overline{y} = \frac{144.42+146.26}{2} = 145.34 \][/tex]
### Step 2: Calculate the Terms Needed for the Slope (m)
We use the formula for the slope [tex]\( m \)[/tex]:
[tex]\[ m = \frac{\sum{(x_i - \overline{x})(y_i - \overline{y})}}{\sum{(x_i - \overline{x})^2}} \][/tex]
For the numerator:
[tex]\[ \sum{(x_i - \overline{x})(y_i - \overline{y})} = (2 - 3)(144.42 - 145.34) + (4 - 3)(146.26 - 145.34) \][/tex]
[tex]\[ = (-1)(-0.92) + (1)(0.92) \][/tex]
[tex]\[ = 0.92 + 0.92 \][/tex]
[tex]\[ = 1.84 \][/tex]
For the denominator:
[tex]\[ \sum{(x_i - \overline{x})^2} = (2 - 3)^2 + (4 - 3)^2 \][/tex]
[tex]\[ = (-1)^2 + (1)^2 \][/tex]
[tex]\[ = 1 + 1 \][/tex]
[tex]\[ = 2 \][/tex]
Therefore, the slope [tex]\( m \)[/tex] is:
[tex]\[ m = \frac{1.84}{2} = 0.92 \][/tex]
### Step 3: Calculate the Intercept (b)
The intercept [tex]\( b \)[/tex] can be found using the formula:
[tex]\[ b = \overline{y} - m\overline{x} \][/tex]
[tex]\[ b = 145.34 - 0.92 \times 3 \][/tex]
[tex]\[ b = 145.34 - 2.76 \][/tex]
[tex]\[ b = 142.58 \][/tex]
### Step 4: Write the Equation of the Line of Best Fit
Combining our slope [tex]\( m = 0.92 \)[/tex] and intercept [tex]\( b = 142.58 \)[/tex], we have the equation of the line of best fit:
[tex]\[ y = 0.92x + 142.58 \][/tex]
### Step 5: Calculate the Correlation Coefficient (r)
The correlation coefficient [tex]\( r \)[/tex] is calculated using the formula:
[tex]\[ r = \frac{\sum{(x_i - \overline{x})(y_i - \overline{y})}}{\sqrt{\sum{(x_i - \overline{x})^2} \cdot \sum{(y_i - \overline{y})^2}}} \][/tex]
We've already calculated:
[tex]\[ \sum{(x_i - \overline{x})(y_i - \overline{y})} = 1.84 \][/tex]
[tex]\[ \sum{(x_i - \overline{x})^2} = 2 \][/tex]
For:
[tex]\[ \sum{(y_i - \overline{y})^2} = (144.42 - 145.34)^2 + (146.26 - 145.34)^2 \][/tex]
[tex]\[ = (-0.92)^2 + (0.92)^2 \][/tex]
[tex]\[ = 0.8464 + 0.8464 \][/tex]
[tex]\[ = 1.6928 \][/tex]
Thus:
[tex]\[ r = \frac{1.84}{\sqrt{2 \cdot 1.6928}} \][/tex]
[tex]\[ r = \frac{1.84}{\sqrt{3.3856}} \][/tex]
[tex]\[ r = \frac{1.84}{1.84} \][/tex]
[tex]\[ r = 1.0 \][/tex]
### Summary
The equation of the line of best fit is:
[tex]\[ y = 0.92x + 142.58 \][/tex]
The correlation coefficient is:
[tex]\[ r = 1.0 \][/tex]
### Given Data:
- Data points: (2, 144.42) and (4, 146.26)
### Step 1: Calculate the Mean of x and y
We start by finding the means of the x-values and y-values.
[tex]\[ x_1 = 2, \, x_2 = 4 \][/tex]
[tex]\[ y_1 = 144.42, \, y_2 = 146.26 \][/tex]
[tex]\[ \text{Mean of x} = \overline{x} = \frac{2+4}{2} = 3 \][/tex]
[tex]\[ \text{Mean of y} = \overline{y} = \frac{144.42+146.26}{2} = 145.34 \][/tex]
### Step 2: Calculate the Terms Needed for the Slope (m)
We use the formula for the slope [tex]\( m \)[/tex]:
[tex]\[ m = \frac{\sum{(x_i - \overline{x})(y_i - \overline{y})}}{\sum{(x_i - \overline{x})^2}} \][/tex]
For the numerator:
[tex]\[ \sum{(x_i - \overline{x})(y_i - \overline{y})} = (2 - 3)(144.42 - 145.34) + (4 - 3)(146.26 - 145.34) \][/tex]
[tex]\[ = (-1)(-0.92) + (1)(0.92) \][/tex]
[tex]\[ = 0.92 + 0.92 \][/tex]
[tex]\[ = 1.84 \][/tex]
For the denominator:
[tex]\[ \sum{(x_i - \overline{x})^2} = (2 - 3)^2 + (4 - 3)^2 \][/tex]
[tex]\[ = (-1)^2 + (1)^2 \][/tex]
[tex]\[ = 1 + 1 \][/tex]
[tex]\[ = 2 \][/tex]
Therefore, the slope [tex]\( m \)[/tex] is:
[tex]\[ m = \frac{1.84}{2} = 0.92 \][/tex]
### Step 3: Calculate the Intercept (b)
The intercept [tex]\( b \)[/tex] can be found using the formula:
[tex]\[ b = \overline{y} - m\overline{x} \][/tex]
[tex]\[ b = 145.34 - 0.92 \times 3 \][/tex]
[tex]\[ b = 145.34 - 2.76 \][/tex]
[tex]\[ b = 142.58 \][/tex]
### Step 4: Write the Equation of the Line of Best Fit
Combining our slope [tex]\( m = 0.92 \)[/tex] and intercept [tex]\( b = 142.58 \)[/tex], we have the equation of the line of best fit:
[tex]\[ y = 0.92x + 142.58 \][/tex]
### Step 5: Calculate the Correlation Coefficient (r)
The correlation coefficient [tex]\( r \)[/tex] is calculated using the formula:
[tex]\[ r = \frac{\sum{(x_i - \overline{x})(y_i - \overline{y})}}{\sqrt{\sum{(x_i - \overline{x})^2} \cdot \sum{(y_i - \overline{y})^2}}} \][/tex]
We've already calculated:
[tex]\[ \sum{(x_i - \overline{x})(y_i - \overline{y})} = 1.84 \][/tex]
[tex]\[ \sum{(x_i - \overline{x})^2} = 2 \][/tex]
For:
[tex]\[ \sum{(y_i - \overline{y})^2} = (144.42 - 145.34)^2 + (146.26 - 145.34)^2 \][/tex]
[tex]\[ = (-0.92)^2 + (0.92)^2 \][/tex]
[tex]\[ = 0.8464 + 0.8464 \][/tex]
[tex]\[ = 1.6928 \][/tex]
Thus:
[tex]\[ r = \frac{1.84}{\sqrt{2 \cdot 1.6928}} \][/tex]
[tex]\[ r = \frac{1.84}{\sqrt{3.3856}} \][/tex]
[tex]\[ r = \frac{1.84}{1.84} \][/tex]
[tex]\[ r = 1.0 \][/tex]
### Summary
The equation of the line of best fit is:
[tex]\[ y = 0.92x + 142.58 \][/tex]
The correlation coefficient is:
[tex]\[ r = 1.0 \][/tex]
