Find an equation of the least squares line, then use it to complete the table.
X- 111 169 230 287 335 ?
Y- 2 3 4 5 6 7
Extrapolation
I missed the lesson on this and as far as I can tell I think the least squares line is y= .01764x+ .oo693
If anyone could verify my line and and help me with the rest? Find the "?" in the table. I also don't know what it means by extrapolation? Is that just finding the value in the table?
Your least squares line looks okay.
Now, to find the ?, use 7 for y in the equation and solve for x. That will be the value you seek.
Here's how the equation would look:
7 = .01764x + .00693
I hope this helps.
352.4285714286
To find the least squares line, we need to use the method of linear regression. This involves finding the equation of a line that minimizes the sum of the squared differences between the actual data points and the predicted values on the line.
To obtain the equation, we need to calculate the slope and the y-intercept.
Step 1: Calculate the mean of X and Y.
X-mean (X) = (111 + 169 + 230 + 287 + 335) / 5
= 226.4
Y-mean (Y) = (2 + 3 + 4 + 5 + 6) / 5
= 4
Step 2: Calculate the sum of the products of the differences between X and the mean of X, and the differences between Y and the mean of Y.
Sum of ((X - mean (X)) * (Y - mean (Y))) = ((111 - 226.4) * (2 - 4)) + ((169 - 226.4) * (3 - 4)) + ((230 - 226.4) * (4 - 4)) + ((287 - 226.4) * (5 - 4)) + ((335 - 226.4) * (6 - 4))
= (-115.4 * -2) + (-57.4 * -1) + (3.6 * 0) + (60.6 * 1) + (108.6 * 2)
= 230.8 + 57.4 + 0 + 60.6 + 217.2
= 566
Step 3: Calculate the sum of the squared differences between X and the mean of X.
Sum of ((X - mean (X))^2) = ((111 - 226.4)^2) + ((169 - 226.4)^2) + ((230 -226.4)^2) + ((287 - 226.4)^2) + ((335 - 226.4)^2)
= (115.4^2) + (57.4^2) + (3.6^2) + (60.6^2) + (108.6^2)
= 13333.6 + 3305.6 + 12.96 + 3672.36 + 11808.96
= 42133.48
Step 4: Calculate the slope (m) of the least squares line.
m = Sum of ((X - mean (X)) * (Y - mean (Y))) / Sum of ((X - mean (X))^2)
= 566 / 42133.48
≈ 0.01342
Step 5: Calculate the y-intercept (b) of the least squares line.
b = mean (Y) - (m * mean (X))
= 4 - (0.01342 * 226.4)
≈ 1.97948
Therefore, the equation of the least squares line is:
y = 0.01342x + 1.97948
To complete the table, substitute the given values of X into the equation to find the corresponding values of Y.
For example, for X = 111:
Y = 0.01342 * 111 + 1.97948
≈ 3.44182
Upon completing the table, we can find the value for X = 495 by substituting it into the equation:
Y = 0.01342 * 495 + 1.97948
≈ 8.62690
Extrapolation refers to estimating or predicting values beyond the given data range. In this case, we are estimating the Y value for X = 495 using the equation of the least squares line.