What is the equation for the line of best fit that matches the data shown in the table and in the graph?
x y
-7 9
-3 6
-1 3
2 1
6 -1
A. y = 0.79x - 3.12
B. y = -0.79x + 3.12
C. y = 3.12x - 0.79
D. y = -3.12x + 0.79
i can't show the graph here but it's basically a plot of the given points on a coordinate grid. i don't have the appropriate calculator to do this problem, so i can't get my answer.
You can use your calculator
B
To find the equation for the line of best fit, we can use the method of least squares regression. This method finds the line that minimizes the sum of the squared distances between the actual data points and the predicted values on the line.
To solve this manually without a calculator, we can follow these steps:
Step 1: Calculate the means of the x-values and y-values.
For the x-values:
mean(x) = (-7 - 3 - 1 + 2 + 6) / 5 = -3.6
For the y-values:
mean(y) = (9 + 6 + 3 + 1 - 1) / 5 = 3.6
Step 2: Calculate the differences for each x and y value from their respective means. Square these differences.
Differences for x-values:
(-7) - (-3.6) = -3.4
(-3) - (-3.6) = 0.6
(-1) - (-3.6) = 2.6
(2) - (-3.6) = 5.6
(6) - (-3.6) = 9.6
Differences for y-values:
9 - 3.6 = 5.4
6 - 3.6 = 2.4
3 - 3.6 = 0.6
1 - 3.6 = -2.6
-1 - 3.6 = -4.6
Squared differences for x-values:
(-3.4)^2 = 11.56
0.6^2 = 0.36
2.6^2 = 6.76
5.6^2 = 31.36
9.6^2 = 92.16
Squared differences for y-values:
5.4^2 = 29.16
2.4^2 = 5.76
0.6^2 = 0.36
(-2.6)^2 = 6.76
(-4.6)^2 = 21.16
Step 3: Calculate the sum of these squared differences.
Sum of squared differences for x-values = 11.56 + 0.36 + 6.76 + 31.36 + 92.16 = 142.2
Sum of squared differences for y-values = 29.16 + 5.76 + 0.36 + 6.76 + 21.16 = 63.20
Step 4: Calculate the sum of the products of the differences of x-values and y-values.
Product of differences for (-3.4) and 5.4 = -3.4 * 5.4 = -18.36
Product of differences for 0.6 and 2.4 = 0.6 * 2.4 = 1.44
Product of differences for 2.6 and 0.6 = 2.6 * 0.6 = 1.56
Product of differences for 5.6 and (-2.6) = 5.6 * (-2.6) = -14.56
Product of differences for 9.6 and (-4.6) = 9.6 * (-4.6) = -44.16
Sum of the products of differences = -18.36 + 1.44 + 1.56 + (-14.56) + (-44.16) = -74.08
Step 5: Calculate the slope of the line,
slope = (sum of the products of differences) / (sum of the squared differences for x-values)
slope = -74.08 / 142.2 ≈ -0.52
Step 6: Calculate the y-intercept of the line,
intercept = mean(y) - (slope * mean(x))
intercept = 3.6 - (-0.52 * -3.6) ≈ 1.87
Step 7: Write the equation in the form y = mx + b, where m is the slope and b is the y-intercept.
Therefore, the equation for the line of best fit is:
y = -0.52x + 1.87
Comparing this equation to the given answer choices, we observe that the closest match is option B:
y = -0.79x + 3.12
So, the correct answer is option B, y = -0.79x + 3.12.