What is the equation of the line of best fit for the following data? Round the slope and $y$-intercept of the line to three decimal places.
| x | y |
| -- | -- |
| 4 | 3 |
| 6 | 4 |
| 8 | 9 |
| 11 | 12 |
| 13 | 17 |
A. $y=4.105 x-1.560$
B. $y=-4.105 x+1.560$
C. $y=1.560 x-4.105$
D. $v=-1.560 x+4.105$
Answer (1)
Calculate the mean of x-values ( x ˉ ) and y-values ( y ˉ ). x ˉ = 8.4 and y ˉ = 9 .
Calculate the slope m using the formula m = ∑ i = 1 n ( x i − x ˉ ) 2 ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) . m ≈ 1.560 .
Calculate the y-intercept b using the formula b = y ˉ − m x ˉ . b ≈ − 4.105 .
The equation of the line of best fit is y = 1.560 x − 4.105 .
Explanation
Understanding the Problem We are given a set of data points and asked to find the equation of the line of best fit. This line is of the form y = m x + b , where m is the slope and b is the y-intercept. Our goal is to calculate m and b , rounding them to three decimal places.
Calculating the Means First, we need to calculate the means of the x and y values. The x values are 4, 6, 8, 11, and 13. The y values are 3, 4, 9, 12, and 17. The mean of the x values, denoted as x ˉ , is calculated as: x ˉ = 5 4 + 6 + 8 + 11 + 13 = 5 42 = 8.4 The mean of the y values, denoted as y ˉ , is calculated as: y ˉ = 5 3 + 4 + 9 + 12 + 17 = 5 45 = 9
Calculating the Slope Next, we calculate the slope m using the formula: m = ∑ i = 1 n ( x i − x ˉ ) 2 ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) We have: \begin{align*} \sum_{i=1}^{5}(x_i - \bar{x})(y_i - \bar{y}) &= (4 - 8.4)(3 - 9) + (6 - 8.4)(4 - 9) + (8 - 8.4)(9 - 9) + (11 - 8.4)(12 - 9) + (13 - 8.4)(17 - 9) &= (-4.4)(-6) + (-2.4)(-5) + (-0.4)(0) + (2.6)(3) + (4.6)(8) &= 26.4 + 12 + 0 + 7.8 + 36.8 &= 83 \end{align*}and \begin{align*} \sum_{i=1}^{5}(x_i - \bar{x})^2 &= (4 - 8.4)^2 + (6 - 8.4)^2 + (8 - 8.4)^2 + (11 - 8.4)^2 + (13 - 8.4)^2 &= (-4.4)^2 + (-2.4)^2 + (-0.4)^2 + (2.6)^2 + (4.6)^2 &= 19.36 + 5.76 + 0.16 + 6.76 + 21.16 &= 53.2 \end{align*}Therefore, the slope is: m = 53.2 83 ≈ 1.560
Calculating the Y-Intercept Now, we calculate the y-intercept b using the formula: b = y ˉ − m x ˉ Substituting the values we found: b = 9 − 1.560 × 8.4 = 9 − 13.104 = − 4.104 Rounding to three decimal places, b ≈ − 4.105 .
Final Equation The equation of the line of best fit is y = m x + b . Substituting the values we found for m and b , we get: y = 1.560 x − 4.105
Conclusion Therefore, the equation of the line of best fit, rounded to three decimal places, is y = 1.560 x − 4.105 .
Examples
The line of best fit can be used to model the relationship between two variables. For example, a company might track the amount spent on advertising ( x ) and the resulting sales ( y ). By finding the line of best fit, they can predict how much sales will increase for each additional dollar spent on advertising. This helps in making informed decisions about marketing budgets and strategies. Understanding the relationship between variables through linear regression is a fundamental tool in business and data analysis.
