To find the equation for the least squares regression line, we first need to compute the necessary statistics such as means, sums, and sum of products. The regression line has the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
We need to calculate the following:
- The means of \( x \) and \( y \): \[ \overline{x} = \frac{\sum x}{n}, \quad \overline{y} = \frac{\sum y}{n} \]
- The slope \( m \): \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]
- The y-intercept \( b \): \[ b = \overline{y} - m\overline{x} \]
Let's calculate the necessary sums for this dataset.
Data summary
- Volume (x): [2.92, 2.98, 3.05, 3.49, 3.89, 4.98]
- Weight (y): [3.33, 3.42, 4.70, 3.51, 5.06, 7.98]
Calculations
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Sums and means:
\[ n = 6 \] \[ \sum x = 2.92 + 2.98 + 3.05 + 3.49 + 3.89 + 4.98 = 21.31 \] \[ \sum y = 3.33 + 3.42 + 4.70 + 3.51 + 5.06 + 7.98 = 27.00 \] \[ \sum xy = (2.92 \cdot 3.33) + (2.98 \cdot 3.42) + (3.05 \cdot 4.70) + (3.49 \cdot 3.51) + (3.89 \cdot 5.06) + (4.98 \cdot 7.98) = 9.7336 + 10.1936 + 14.355 + 12.25149 + 19.6874 + 39.6996 = 106.4201 \] \[ \sum x^2 = (2.92)^2 + (2.98)^2 + (3.05)^2 + (3.49)^2 + (3.89)^2 + (4.98)^2 = 8.5344 + 8.8804 + 9.3025 + 12.1801 + 15.1521 + 24.8004 = 79.8499 \]
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Compute means:
\[ \overline{x} = \frac{21.31}{6} \approx 3.55167 \] \[ \overline{y} = \frac{27.00}{6} = 4.50 \]
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Calculate slope \( m \): \[ m = \frac{6(106.4201) - (21.31)(27.00)}{6(79.8499) - (21.31)^2} \] Calculate \( (21.31)(27.00) \): \[ 21.31 \times 27.00 \approx 575.37 \] So, \[ m = \frac{638.5206 - 575.37}{479.0994 - 453.4161} \approx \frac{63.1506}{25.6833} \approx 2.459 \]
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Calculate y-intercept \( b \): \[ b = 4.50 - (2.459)(3.55167) \approx 4.50 - 8.747 \approx -4.247 \]
Final Equation
Now we round our values to the nearest thousandth:
\[ y \approx 2.459x - 4.247 \]
Thus, the equation of the least squares regression line is:
\[ y = 2.459x - 4.247 \]