To create a linear model from the given data, we can determine the slope (m) and y-intercept (b) of the line in the form \( y = mx + b \).
Given the data:
\[ \begin{array}{|c|c|} \hline x & y \ \hline 3 & 7 \ 6 & 10 \ 9 & 15 \ 10 & 16 \ 11 & 18 \ 13 & 21 \ 17 & 26 \ \hline \end{array} \]
Step 1: Calculate the slope (m)
The slope formula is:
\[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]
Where:
- \( N \) is the number of data points
- \( \sum xy \) is the sum of the product of each \( x \) and \( y \)
- \( \sum x \) is the sum of \( x \) values
- \( \sum y \) is the sum of \( y \) values
- \( \sum x^2 \) is the sum of the squares of \( x \) values
Step 2: Sum the values
- \( \sum x = 3 + 6 + 9 + 10 + 11 + 13 + 17 = 69 \)
- \( \sum y = 7 + 10 + 15 + 16 + 18 + 21 + 26 = 113 \)
- \( N = 7 \)
- \( \sum xy = (37) + (610) + (915) + (1016) + (1118) + (1321) + (17*26) = 21 + 60 + 135 + 160 + 198 + 273 + 442 = 1290 \)
- \( \sum x^2 = 3^2 + 6^2 + 9^2 + 10^2 + 11^2 + 13^2 + 17^2 = 9 + 36 + 81 + 100 + 121 + 169 + 289 = 805 \)
Step 3: Plug values into the slope formula
\[ m = \frac{7(1290) - (69)(113)}{7(805) - (69)^2} \]
Calculating the numerator:
\[ = \frac{9030 - 7797}{5635 - 4761} = \frac{1233}{874} ≈ 1.41 \]
Calculating the denominator:
\[ = 5635 - 4761 = 874 \]
So,
\[ m ≈ 1.41 \]
Step 4: Calculate y-intercept (b)
Once we have the slope \( m \), we can use the formula \( b = \frac{\sum y - m \sum x}{N} \)
\[ b = \frac{113 - 1.41 \times 69}{7} \] Calculating \( m \sum x \):
\[ 1.41 \times 69 ≈ 97.29 \]
Now,
\[ b = \frac{113 - 97.29}{7} ≈ \frac{15.71}{7} ≈ 2.24 \]
Step 5: Write the final linear model
Thus, the linear model in the form \( y = mx + b \) is:
\[ y = 1.41x + 2.24 \]
Final response:
So the complete linear model is:
\( y = 1.41x + 2.24 \)
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