To create a linear model \( y = mx + b \) from the provided data points, we first need to calculate the slope \( m \) and the y-intercept \( b \).
-
Data Points: You provided the following pairs:
- (3, 5)
- (7, 10)
- (10, 15)
- (11, 17)
- (12, 18)
- (14, 21)
- (18, 26)
-
Calculating the Slope \( m \): The formula for the slope \( m \) 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 pair, \( \sum x \) is the sum of the x values, \( \sum y \) is the sum of the y values, and \( \sum x^2 \) is the sum of the squares of the x values.
Let's calculate these values:
- \( n = 7 \)
- \( \sum x = 3 + 7 + 10 + 11 + 12 + 14 + 18 = 85 \)
- \( \sum y = 5 + 10 + 15 + 17 + 18 + 21 + 26 = 112 \)
- \( \sum xy = (35) + (710) + (1015) + (1117) + (1218) + (1421) + (18*26) \) \[ = 15 + 70 + 150 + 187 + 216 + 294 + 468 = 1400 \]
- \( \sum x^2 = 3^2 + 7^2 + 10^2 + 11^2 + 12^2 + 14^2 + 18^2 \) \[ = 9 + 49 + 100 + 121 + 144 + 196 + 324 = 943 \]
Now plugging these values into the slope formula: \[ m = \frac{7(1400) - (85)(112)}{7(943) - (85)^2} \] \[ = \frac{9800 - 9520}{6601 - 7225} = \frac{280}{-624} = -0.449 \]
-
Calculating the y-Intercept \( b \): The formula to find \( b \) is: \[ b = \frac{\sum y - m \sum x}{n} \] Plugging the values into this formula: \[ b = \frac{112 - (-0.449)(85)}{7} \] \[ = \frac{112 + 38.165}{7} = \frac{150.165}{7} = 21.38 \]
-
Final Model: The linear model can be summarized as: \[ y = -0.449x + 21.38 \]
Final Answer:
Thus, the linear model in the required format is: \[ y = -0.449x + 21.38 \]
Feel free to make adjustments or round the numbers to your required precision!
1 answer