Math Problem Statement

Find the line of best fit for the given data points and predict y when x = 5.

Solution

The problem asks to find the line of best fit y′=a+bxy' = a + bx for the given data points and to predict yy when x=5x = 5. Let’s solve this step by step:

1. Extracting Data Points

From the graph, the points appear to be: (0,1), (1,3), (2,2), (3,4), (4,3)(0, 1),\ (1, 3),\ (2, 2),\ (3, 4),\ (4, 3)

2. Line of Best Fit Formula

The line of best fit is: y′=a+bxy' = a + bx Where:

  • b=n∑(xiyi)−∑xi∑yin∑xi2−(∑xi)2b = \frac{n \sum (x_i y_i) - \sum x_i \sum y_i}{n \sum x_i^2 - (\sum x_i)^2}
  • a=∑yi−b∑xina = \frac{\sum y_i - b \sum x_i}{n}

3. Calculations

Compute the necessary sums:

n &= 5 \quad (\text{number of data points}), \\ \sum x_i &= 0 + 1 + 2 + 3 + 4 = 10, \\ \sum y_i &= 1 + 3 + 2 + 4 + 3 = 13, \\ \sum x_i^2 &= 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 30, \\ \sum x_i y_i &= (0)(1) + (1)(3) + (2)(2) + (3)(4) + (4)(3) = 0 + 3 + 4 + 12 + 12 = 31. \end{aligned}$$ #### Calculate slope ($$b$$): $$b = \frac{n \sum (x_i y_i) - \sum x_i \sum y_i}{n \sum x_i^2 - (\sum x_i)^2}$$ $$b = \frac{5(31) - (10)(13)}{5(30) - (10)^2} = \frac{155 - 130}{150 - 100} = \frac{25}{50} = 0.5$$ #### Calculate intercept ($$a$$): $$a = \frac{\sum y_i - b \sum x_i}{n}$$ $$a = \frac{13 - (0.5)(10)}{5} = \frac{13 - 5}{5} = \frac{8}{5} = 1.6$$ ### 4. Line of Best Fit Equation The equation of the line is: $$y' = 1.6 + 0.5x$$ ### 5. Predict $$y$$ When $$x = 5$$ $$y' = 1.6 + 0.5(5) = 1.6 + 2.5 = 4.1$$ ### Final Answer: When $$x = 5$$, $$y' = 4.1$$ (rounded to the nearest thousandth). --- Would you like me to elaborate on any step? Here are 5 questions for further exploration: 1. How is the formula for the slope $$b$$ derived in linear regression? 2. What are alternative methods for finding the line of best fit? 3. How would you calculate the residual for each data point? 4. What does the correlation coefficient tell us about the data's fit to the line? 5. Can the line of best fit always predict accurately outside the given data range? **Tip:** Always double-check the sums during regression calculations to avoid errors!

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Line of Best Fit
Slope-Intercept Form

Formulas

y' = a + bx
b = (n Σ(xi yi) - Σxi Σyi) / (n Σ(xi^2) - (Σxi)^2)
a = (Σyi - b Σxi) / n

Theorems

Method of Least Squares
Linear Regression

Suitable Grade Level

Grades 10-12

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