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 |
| -- | -- |
| 5 | 4 |
| 6 | 6 |
| 9 | 9 |
| 10 | 11 |
| 14 | 12 |

A. $y=0.535 x+0.894$
B. $y=0.894 x+0.535$
C. $y=-0.894 x+0.535$
D. $y=-0.535 x+0.894

Calculate the mean of x-values: x ˉ = 8.8 .
Calculate the mean of y-values: y ˉ ​ = 8.4 .
Calculate the slope: m ≈ 0.894 .
Calculate the y-intercept: b ≈ 0.535 . The equation of the line of best fit is y = 0.894 x + 0.535 ​ .

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. We need to calculate m and b and round 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 5, 6, 9, 10, and 14. The y values are 4, 6, 9, 11, and 12. The number of data points is n = 5 .

Calculating the Means (Continued) The mean of the x values is: x ˉ = 5 5 + 6 + 9 + 10 + 14 ​ = 5 44 ​ = 8.8 The mean of the y values is: y ˉ ​ = 5 4 + 6 + 9 + 11 + 12 ​ = 5 42 ​ = 8.4

Calculating the Slope Next, we calculate the slope m using the formula: m = ∑ ( x i ​ − x ˉ ) 2 ∑ ( x i ​ − x ˉ ) ( y i ​ − y ˉ ​ ) ​ We have: ∑ ( x i ​ − x ˉ ) ( y i ​ − y ˉ ​ ) = ( 5 − 8.8 ) ( 4 − 8.4 ) + ( 6 − 8.8 ) ( 6 − 8.4 ) + ( 9 − 8.8 ) ( 9 − 8.4 ) + ( 10 − 8.8 ) ( 11 − 8.4 ) + ( 14 − 8.8 ) ( 12 − 8.4 ) = ( − 3.8 ) ( − 4.4 ) + ( − 2.8 ) ( − 2.4 ) + ( 0.2 ) ( 0.6 ) + ( 1.2 ) ( 2.6 ) + ( 5.2 ) ( 3.6 ) = 16.72 + 6.72 + 0.12 + 3.12 + 18.72 = 45.4 And: ∑ ( x i ​ − x ˉ ) 2 = ( 5 − 8.8 ) 2 + ( 6 − 8.8 ) 2 + ( 9 − 8.8 ) 2 + ( 10 − 8.8 ) 2 + ( 14 − 8.8 ) 2 = ( − 3.8 ) 2 + ( − 2.8 ) 2 + ( 0.2 ) 2 + ( 1.2 ) 2 + ( 5.2 ) 2 = 14.44 + 7.84 + 0.04 + 1.44 + 27.04 = 50.8 Therefore: m = 50.8 45.4 ​ ≈ 0.8937 ≈ 0.894

Calculating the Y-Intercept Now, we calculate the y-intercept b using the formula: b = y ˉ ​ − m x ˉ = 8.4 − 0.894 × 8.8 = 8.4 − 7.8672 = 0.5328 ≈ 0.533

Final Equation Therefore, the equation of the line of best fit is approximately y = 0.894 x + 0.533 . However, looking at the options, the closest one is y = 0.894 x + 0.535 .

Final Answer Confirmed Using the python calculation tool, we find that the slope is approximately 0.894 and the y-intercept is approximately 0.535. Therefore, the equation of the line of best fit is: y = 0.894 x + 0.535


Examples
The line of best fit can be used to model the relationship between two variables. For example, if we have data on the number of hours students study and their exam scores, we can use the line of best fit to predict the exam score of a student based on the number of hours they study. This can help students understand how much they need to study to achieve a certain score. Similarly, businesses can use lines of best fit to predict sales based on advertising expenditure, or to model the relationship between production costs and output.

Answered by GinnyAnswer | 2025-07-07