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 | 17 |
| 6 | 12 |
| 8 | 9 |
| 11 | 4 |
| 13 | 3 |
A. [tex]y=-22.1051 x+1.560[/tex]
B. [tex]y=1.560 x-22.105[/tex]
C. [tex]y=-1.560 x+22.105[/tex]
D. [tex]y=22.105 x-1.56[/tex]
Answer (2)
Calculate the mean of x-values: x ˉ = 8.4 .
Calculate the mean of y-values: y ˉ = 9 .
Calculate the slope: b ≈ − 1.560 .
Calculate the y-intercept: a ≈ 22.105 . The equation of the line of best fit is y = − 1.560 x + 22.105 .
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 = b x + a , where b is the slope and a is the y-intercept. We need to calculate these values and round them to three decimal places.
Calculating the Means First, we calculate the means of the x and y values.
The x-values are 4, 6, 8, 11, and 13. The y-values are 17, 12, 9, 4, and 3.
The mean of the x-values is: x ˉ = 5 4 + 6 + 8 + 11 + 13 = 5 42 = 8.4
The mean of the y-values is: y ˉ = 5 17 + 12 + 9 + 4 + 3 = 5 45 = 9
Calculating the Slope Next, we calculate the slope b using the formula: b = n ∑ i = 1 n x i 2 − ( ∑ i = 1 n x i ) 2 n ∑ i = 1 n x i y i − ∑ i = 1 n x i ∑ i = 1 n y i
We have n = 5 , ∑ i = 1 5 x i = 42 , ∑ i = 1 5 y i = 45 , and we need to calculate ∑ i = 1 5 x i y i and ∑ i = 1 5 x i 2 .
i = 1 ∑ 5 x i y i = ( 4 ) ( 17 ) + ( 6 ) ( 12 ) + ( 8 ) ( 9 ) + ( 11 ) ( 4 ) + ( 13 ) ( 3 ) = 68 + 72 + 72 + 44 + 39 = 295 i = 1 ∑ 5 x i 2 = 4 2 + 6 2 + 8 2 + 1 1 2 + 1 3 2 = 16 + 36 + 64 + 121 + 169 = 406
Plugging these values into the formula for b :
b = 5 ( 406 ) − ( 42 ) 2 5 ( 295 ) − ( 42 ) ( 45 ) = 2030 − 1764 1475 − 1890 = 266 − 415 ≈ − 1.560
Calculating the Y-Intercept Now, we calculate the y-intercept a using the formula: a = y ˉ − b x ˉ
We have x ˉ = 8.4 , y ˉ = 9 , and b ≈ − 1.560 . Plugging these values into the formula for a :
a = 9 − ( − 1.560 ) ( 8.4 ) = 9 + 13.104 = 22.104 Rounding to three decimal places, a ≈ 22.104 .
Final Equation Therefore, the equation of the line of best fit is approximately: y = − 1.560 x + 22.104 Rounding the y-intercept to three decimal places, we get 22.104 .
Final Answer The equation of the line of best fit, rounded to three decimal places, is: y = − 1.560 x + 22.105
Examples
In data analysis, the line of best fit helps predict future outcomes based on existing data. For example, if we are tracking the sales of a product over time, we can use the line of best fit to estimate future sales based on current trends. This can help businesses make informed decisions about inventory, marketing, and production. Understanding the relationship between variables through linear regression is a fundamental tool in business and science.
The equation of the line of best fit for the given data is approximately y = -1.560x + 22.105. The slope is -1.560 and the y-intercept is 22.105, rounded to three decimal places. Therefore, the correct choice is C. y = -1.560x + 22.105.
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