Calculate $\Sigma x$ : To find the coefficients of the linear regression equation $y = mx + b$ , we need to use the method of least squares. This involves calculating the slope ( $m$ ) and the y-intercept ( $b$ ) using the given data points. The formulas for the slope ( $m$ ) and y-intercept ( $b$ ) are: $\newline$ $m = \frac{N\Sigma(xy) - (\Sigma x)(\Sigma y)}{N\Sigma(x^2) - (\Sigma x)^2}$ $\newline$ $b = \frac{\Sigma y - m(\Sigma x)}{N}$ $\newline$ where $N$ is the number of data points, $\Sigma$ denotes the sum over all data points, $y = mx + b$ $0$ and $y = mx + b$ $1$ are the individual data points, and $y = mx + b$ $2$ is the product of $y = mx + b$ $0$ and $y = mx + b$ $1$ for each data point. $\newline$ First, we will calculate the sums needed for these formulas: $\newline$ $\Sigma x$ , $y = mx + b$ $6$ , $y = mx + b$ $7$ , $y = mx + b$ $8$ , and $N$ . $\newline$ Let's start by calculating $\Sigma x$ (the sum of all x-values).
Calculate $\Sigma y$ : $\Sigma x = 3 + 7.5 + 12 + 16.5 + 21 + 25.5 + 30$ $\newline$ $\Sigma x = 115.5$
Calculate $\Sigma(xy)$ : Next, we calculate $\Sigma y$ (the sum of all $y$ -values).
Calculate $\Sigma(x^2)$ : $\Sigma y = 19.701 + 20.825 + 22.826 + 24.19 + 25.806 + 26.65 + 27.44$ $\newline$ $\Sigma y = 167.438$
Calculate slope $m$ : Now, we calculate $\Sigma(xy)$ (the sum of the product of each $x$ and $y$ pair).
Calculate y-intercept (b): $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $\newline$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $\newline$ $\Sigma(xy) = 2932.9385$
Calculate y-intercept (b): $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $\newline$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $\newline$ $\Sigma(xy) = 2932.9385$ Next, we calculate $\Sigma(x^2)$ (the sum of the squares of each x-value).
Calculate y-intercept (b): $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $\newline$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $\newline$ $\Sigma(xy) = 2932.9385$ Next, we calculate $\Sigma(x^2)$ (the sum of the squares of each x-value). $\Sigma(x^2) = (3^2) + (7.5^2) + (12^2) + (16.5^2) + (21^2) + (25.5^2) + (30^2)$ $\newline$ $\Sigma(x^2) = 9 + 56.25 + 144 + 272.25 + 441 + 650.25 + 900$ $\newline$ $\Sigma(x^2) = 2472.75$
Calculate y-intercept (b): $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $\newline$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $\newline$ $\Sigma(xy) = 2932.9385$ Next, we calculate $\Sigma(x^2)$ (the sum of the squares of each x-value). $\Sigma(x^2) = (3^2) + (7.5^2) + (12^2) + (16.5^2) + (21^2) + (25.5^2) + (30^2)$ $\newline$ $\Sigma(x^2) = 9 + 56.25 + 144 + 272.25 + 441 + 650.25 + 900$ $\newline$ $\Sigma(x^2) = 2472.75$ Now we have all the sums needed to calculate the slope (m). Let's calculate it using the formula provided.
Calculate y-intercept (b): $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $\newline$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $\newline$ $\Sigma(xy) = 2932.9385$ Next, we calculate $\Sigma(x^2)$ (the sum of the squares of each x-value). $\Sigma(x^2) = (3^2) + (7.5^2) + (12^2) + (16.5^2) + (21^2) + (25.5^2) + (30^2)$ $\newline$ $\Sigma(x^2) = 9 + 56.25 + 144 + 272.25 + 441 + 650.25 + 900$ $\newline$ $\Sigma(x^2) = 2472.75$ Now we have all the sums needed to calculate the slope (m). Let's calculate it using the formula provided. $N = 7$ (since there are $7$ data points) $\newline$ $m = \frac{N\Sigma(xy) - (\Sigma x)(\Sigma y)}{N\Sigma(x^2) - (\Sigma x)^2}$ $\newline$ $m = \frac{(7 \times 2932.9385) - (115.5)(167.438)}{(7 \times 2472.75) - (115.5)^2}$ $\newline$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $0$ $\newline$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $1$ $\newline$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $2$
Calculate y-intercept (b): $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $\newline$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $\newline$ $\Sigma(xy) = 2932.9385$ Next, we calculate $\Sigma(x^2)$ (the sum of the squares of each x-value). $\Sigma(x^2) = (3^2) + (7.5^2) + (12^2) + (16.5^2) + (21^2) + (25.5^2) + (30^2)$ $\newline$ $\Sigma(x^2) = 9 + 56.25 + 144 + 272.25 + 441 + 650.25 + 900$ $\newline$ $\Sigma(x^2) = 2472.75$ Now we have all the sums needed to calculate the slope (m). Let's calculate it using the formula provided. $N = 7$ (since there are $7$ data points) $\newline$ $m = \frac{N\Sigma(xy) - (\Sigma x)(\Sigma y)}{N\Sigma(x^2) - (\Sigma x)^2}$ $\newline$ $m = \frac{7 \times 2932.9385 - (115.5)(167.438)}{7 \times 2472.75 - (115.5)^2}$ $\newline$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $0$ $\newline$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $1$ $\newline$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $2$ Finally, we calculate the y-intercept (b) using the slope we just found and the formula for b.
Calculate y-intercept ( $b$ ): $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $\Sigma(xy) = 2932.9385$ Next, we calculate $\Sigma(x^2)$ (the sum of the squares of each x-value). $\Sigma(x^2) = (3^2) + (7.5^2) + (12^2) + (16.5^2) + (21^2) + (25.5^2) + (30^2)$ $\Sigma(x^2) = 9 + 56.25 + 144 + 272.25 + 441 + 650.25 + 900$ $\Sigma(x^2) = 2472.75$ Now we have all the sums needed to calculate the slope ( $m$ ). Let's calculate it using the formula provided. $N = 7$ (since there are $7$ data points) $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $0$ $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $1$ $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $2$ $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $3$ $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $4$ Finally, we calculate the y-intercept ( $b$ ) using the slope we just found and the formula for $b$ . $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $7$ $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $8$ $\Sigma(xy) = (3 \times 19.701) + (7.5 \times 20.825) + (12 \times 22.826) + (16.5 \times 24.19) + (21 \times 25.806) + (25.5 \times 26.65) + (30 \times 27.44)$ $9$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $0$ $\Sigma(xy) = 59.103 + 156.1875 + 273.912 + 399.135 + 541.926 + 679.575 + 823.2$ $1$