To find the line of least squares regression for the given data, we can use the formula for the linear regression line:
\[ y = mx + b \]
where \( m \) is the slope and \( b \) is the y-intercept. The formulas we will use to calculate these values are:
\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]
\[ b = \frac{\sum y - m(\sum x)}{n} \]
Where:
- \( n \) is the number of data points,
- \( \sum x \) is the sum of x-values,
- \( \sum y \) is the sum of y-values,
- \( \sum xy \) is the sum of the product of x and y,
- \( \sum x^2 \) is the sum of the squares of x-values.
Let's calculate the needed sums using the provided data:
| \( x \) | \( y \) | \( xy \) | \( x^2 \) | |---------|---------|----------|-----------| | 0 | 8 | 0 | 0 | | 4 | 9 | 36 | 16 | | 8 | 10 | 80 | 64 | | 12 | 11 | 132 | 144 | | 16 | 12 | 192 | 256 |
Now, we can calculate the sums:
\[ \sum x = 0 + 4 + 8 + 12 + 16 = 40 \]
\[ \sum y = 8 + 9 + 10 + 11 + 12 = 50 \]
\[ \sum xy = 0 + 36 + 80 + 132 + 192 = 440 \]
\[ \sum x^2 = 0 + 16 + 64 + 144 + 256 = 480 \]
Now, plug these values into the formulas.
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Calculate \( n \): \[ n = 5 \]
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Calculate \( m \): \[ m = \frac{5(440) - (40)(50)}{5(480) - (40)^2} \] \[ m = \frac{2200 - 2000}{2400 - 1600} \] \[ m = \frac{200}{800} = 0.25 \]
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Calculate \( b \): \[ b = \frac{50 - 0.25(40)}{5} \] \[ b = \frac{50 - 10}{5} \] \[ b = \frac{40}{5} = 8 \]
Therefore, the equation of the line of least squares regression is:
\[ y = 0.25x + 8 \]
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