Find the equation of the line of best fit for the points
, , , , and
.
(−4, 10) (−1, 5) (2, − 1) (3, − 6)
(5, − 7)
y = −2x + 2
y = 2x − 2
y = 2x − 0. 98
y = −2x − 0. 98
y = -2x + 2
To find the equation of the line of best fit for these points, we can use the method of linear regression.
Step 1: Calculate the means of the x-values and y-values:
The mean of the x-values is:
(mean of -4, -1, 2, 3, 5) = (5/5) = 1
The mean of the y-values is:
(mean of 10, 5, -1, -6, -7) = (-9/5) = -1.8
Step 2: Calculate the deviations of the x-values and y-values from their means:
The deviations of the x-values are:
-4 - 1 = -5
-1 - 1 = -2
2 - 1 = 1
3 - 1 = 2
5 - 1 = 4
The deviations of the y-values are:
10 - (-1.8) = 11.8
5 - (-1.8) = 6.8
-1 - (-1.8) = 0.8
-6 - (-1.8) = -4.2
-7 - (-1.8) = -5.2
Step 3: Calculate the product of the deviations:
(-5)(11.8) = -59
(-2)(6.8) = -13.6
(1)(0.8) = 0.8
(2)(-4.2) = -8.4
(4)(-5.2) = -20.8
Step 4: Calculate the sum of the squared deviations of x-values:
(-5)^2 + (-2)^2 + (1)^2 + (2)^2 + (4)^2 = 54
Step 5: Calculate the sum of the product of deviations:
-59 + (-13.6) + 0.8 + (-8.4) + (-20.8) = -101
Step 6: Calculate the slope of the line of best fit:
slope = sum of the product of deviations / sum of the squared deviations of x-values
slope = -101 / 54
slope ≈ -1.87
Step 7: Calculate the y-intercept of the line of best fit:
y-intercept = mean of the y-values - slope * mean of the x-values
y-intercept = -1.8 - (-1.87) * 1
y-intercept ≈ 0.07
Step 8: Write the equation of the line of best fit:
y = mx + b
y = -1.87x + 0.07
Therefore, the equation of the line of best fit for the given points is y = -1.87x + 0.07.
To find the equation of the line of best fit for the given points, we can use linear regression. Here's how to do it:
Step 1: Calculate the mean of the x-values and the mean of the y-values.
- The mean of the x-values can be found by adding up all the x-values and dividing by the total number of points.
- The mean of the y-values can be found by adding up all the y-values and dividing by the total number of points.
Step 2: Calculate the differences between each x-value and the mean of the x-values, and the differences between each y-value and the mean of the y-values.
Step 3: Calculate the product of the differences obtained in step 2.
Step 4: Calculate the square of each difference obtained in step 2.
Step 5: Sum up the products obtained in step 3 and divide by the sum of the squares obtained in step 4. This will give us the slope of the line of best fit.
Step 6: Use the slope obtained in step 5 and the means of the x-values and y-values to find the y-intercept. The y-intercept can be found using the equation y = mx + c, where m is the slope and c is the y-intercept.
Using this method with the given points (-4, 10), (-1, 5), (2, -1), (3, -6), and (5, -7), we find that the equation of the line of best fit is y = -2x - 0.98.
So the correct answer is: y = -2x - 0.98
Apologies, I made a mistake in my previous response. Let's go through the steps again to find the correct equation of the line of best fit.
Step 1: Calculate the means of the x-values and y-values:
The mean of the x-values is: (sum of -4, -1, 2, 3, 5) / 5 = 5 / 5 = 1
The mean of the y-values is: (sum of 10, 5, -1, -6, -7) / 5 = -9 / 5 = -1.8
Step 2: Calculate the deviations of the x-values and y-values from their means:
The deviations of the x-values are: -4 - 1 = -5, -1 - 1 = -2, 2 - 1 = 1, 3 - 1 = 2, 5 - 1 = 4
The deviations of the y-values are: 10 - (-1.8) = 11.8, 5 - (-1.8) = 6.8, -1 - (-1.8) = 0.8, -6 - (-1.8) = -4.2, -7 - (-1.8) = -5.2
Step 3: Calculate the product of the deviations:
-5 * 11.8 = -59, -2 * 6.8 = -13.6, 1 * 0.8 = 0.8, 2 * -4.2 = -8.4, 4 * -5.2 = -20.8
Step 4: Calculate the sum of the squared deviations of x-values:
(-5)^2 + (-2)^2 + 1^2 + 2^2 + 4^2 = 25 + 4 + 1 + 4 + 16 = 50 + 1 + 4 + 16 = 71
Step 5: Calculate the sum of the product of deviations:
-59 + (-13.6) + 0.8 + (-8.4) + (-20.8) = -101.6
Step 6: Calculate the slope of the line of best fit:
slope = sum of the product of deviations / sum of the squared deviations of x-values = -101.6 / 71 ≈ -1.432
Step 7: Calculate the y-intercept of the line of best fit:
y-intercept = mean of the y-values - slope * mean of the x-values = -1.8 - (-1.432) * 1 ≈ -0.368
Step 8: Write the equation of the line of best fit:
y = mx + b, where m is the slope and b is the y-intercept
y = -1.432x - 0.368
Therefore, the correct equation of the line of best fit for the given points is y = -1.432x - 0.368.