To create a linear model for the given data, we can use the least squares method to determine the best-fitting line of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Given the data:
\[ \begin{align*} x: & \quad 3, 7, 10, 11, 12, 14, 18 \ y: & \quad 6, 10, 15, 16, 18, 21, 26 \ \end{align*} \]
Step 1: Calculate the necessary sums
- \( n = 7 \) (the number of data points)
- Sum of \( x \) values:
\( \sum x = 3 + 7 + 10 + 11 + 12 + 14 + 18 = 75 \) - Sum of \( y \) values:
\( \sum y = 6 + 10 + 15 + 16 + 18 + 21 + 26 = 112 \) - Sum of \( xy \) values:
\( \sum xy = (3)(6) + (7)(10) + (10)(15) + (11)(16) + (12)(18) + (14)(21) + (18)(26) \)
\( = 18 + 70 + 150 + 176 + 216 + 294 + 468 = 1392 \) - Sum of \( x^2 \) values:
\( \sum x^2 = (3^2) + (7^2) + (10^2) + (11^2) + (12^2) + (14^2) + (18^2) \)
\( = 9 + 49 + 100 + 121 + 144 + 196 + 324 = 943 \)
Step 2: Calculate the slope \( m \) and intercept \( b \)
Using the formulas:
\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]
\[ b = \frac{\sum y - m(\sum x)}{n} \]
Calculating \( m \):
\[ m = \frac{7 \cdot 1392 - 75 \cdot 112}{7 \cdot 943 - 75^2} \]
\[ m = \frac{9744 - 8400}{6601 - 5625} = \frac{1344}{976} \approx 1.376 \]
Calculating \( b \):
\[ b = \frac{112 - 1.376 \cdot 75}{7} \]
\[ b = \frac{112 - 103.2}{7} = \frac{8.8}{7} \approx 1.257 \]
Conclusion
The linear model for the data is:
\[ y = 1.376x + 1.257 \]
So, rounding to three decimal places, the final answer is:
\[ y = 1.376x + 1.257 \]
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