To create a linear model for the given data, we can first identify the independent variable \( x \) and the dependent variable \( y \), and then use the method of least squares to find the slope (\( m \)) and y-intercept (\( b \)) of the linear equation in the form \( y = mx + b \).
Given Data:
\[ \begin{array}{|c|c|} \hline x & y \ \hline 5 & 7 \ 7 & 16 \ 9 & 20 \ 11 & 28 \ 13 & 38 \ 15 & 45 \ \hline \end{array} \]
Step 1: Calculate the Means
Calculate the means of \( x \) and \( y \):
\[ \bar{x} = \frac{5 + 7 + 9 + 11 + 13 + 15}{6} = \frac{60}{6} = 10 \]
\[ \bar{y} = \frac{7 + 16 + 20 + 28 + 38 + 45}{6} = \frac{154}{6} \approx 25.67 \]
Step 2: Calculate the Slope (\( m \))
To calculate the slope \( m \):
\[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]
Calculating the sums:
- \( \sum (x_i - \bar{x})(y_i - \bar{y}) \)
\[ = (5 - 10)(7 - 25.67) + (7 - 10)(16 - 25.67) + (9 - 10)(20 - 25.67) + (11 - 10)(28 - 25.67) + (13 - 10)(38 - 25.67) + (15 - 10)(45 - 25.67) \]
\[ = (-5)(-18.67) + (-3)(-9.67) + (-1)(-5.67) + (1)(2.33) + (3)(12.33) + (5)(19.33) \]
\[ = 93.35 + 29.01 + 5.67 + 2.33 + 37.00 + 96.65 = 263.01 \]
- \( \sum (x_i - \bar{x})^2 \)
\[ = (5 - 10)^2 + (7 - 10)^2 + (9 - 10)^2 + (11 - 10)^2 + (13 - 10)^2 + (15 - 10)^2 \]
\[ = 25 + 9 + 1 + 1 + 9 + 25 = 70 \]
Now substitute these sums into the formula for \( m \):
\[ m = \frac{263.01}{70} \approx 3.77 \]
Step 3: Calculate the Intercept (\( b \))
We can calculate \( b \) using the formula:
\[ b = \bar{y} - m\bar{x} \]
Substituting the values:
\[ b = 25.67 - (3.77 \times 10) = 25.67 - 37.7 \approx -12.03 \]
Final Model
Combining the slope and intercept, the linear model for the data is:
\[ y = 3.77x - 12.03 \]
This linear equation models the relationship between \( x \) and \( y \) based on the provided dataset.
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