What is the natural logarithmic regression equation for the following data? Estimate the value for x = 15
x 2, 4, 7, 10
y 3, 8, 11, 12
Thank you!
See what you can find here
https://www.google.ca/search?rlz=1C5CHFA_enCA690CA690&ei=F-XRW5X6Done5gKwzrrwCw&q=natural+logarithm+regression+equation+calculator+youtube&oq=natural+logarithm+regression+equation+calculator+youtube&gs_l=psy-ab.3...19564.21538.0.22139.8.8.0.0.0.0.170.781.0j6.6.0....0...1c.1.64.psy-ab..2.5.669...0i22i30k1j33i160k1j33i21k1.0.G-xVjNY39F0
y = -0.498 + 5.688 lnx
To find the natural logarithmic regression equation for the given data, we need to first calculate the natural logarithm of the x-values and then perform a linear regression on the transformed data.
1. Calculate the natural logarithm of the x-values:
x ln(x)
----------------
2 0.693
4 1.386
7 1.946
10 2.303
2. Perform linear regression on the transformed data:
We need to find the equation of a line in the form of y = a + bx.
Using the provided data points:
ln(x) y
---------------
0.693 3
1.386 8
1.946 11
2.303 12
Let's calculate the mean of ln(x) and y:
mean(ln(x)) = (0.693 + 1.386 + 1.946 + 2.303) / 4 = 1.582
mean(y) = (3 + 8 + 11 + 12) / 4 = 8.5
Now, calculate the slope (b) using the formula:
b = Σ((ln(xi) - mean(ln(x))) * (yi - mean(y))) / Σ((ln(xi) - mean(ln(x)))^2)
Substituting the values:
b = ((0.693 - 1.582) * (3 - 8.5) + (1.386 - 1.582) * (8 - 8.5) + (1.946 - 1.582) * (11 - 8.5) + (2.303 - 1.582) * (12 - 8.5)) / ((0.693 - 1.582)^2 + (1.386 - 1.582)^2 + (1.946 - 1.582)^2 + (2.303 - 1.582)^2)
b ≈ -3.076
Next, calculate the y-intercept (a) using the formula:
a = mean(y) - b * mean(ln(x))
Substituting the values:
a = 8.5 - (-3.076) * 1.582
a ≈ 13.669
Therefore, the natural logarithmic regression equation is:
y = 13.669 - 3.076 * ln(x)
To estimate the value for x = 15, substitute x = 15 into the equation:
y ≈ 13.669 - 3.076 * ln(15)
y ≈ 13.669 - 3.076 * 2.708
y ≈ 13.669 - 8.357
y ≈ 5.312
Hence, the estimated value for x = 15 is approximately 5.312.
To find the natural logarithmic regression equation for the given data, we can use a statistical software or perform the calculations manually.
Step 1: Calculate the natural logarithm of each x-value.
- ln(2) = 0.693
- ln(4) = 1.386
- ln(7) = 1.946
- ln(10) = 2.303
Step 2: Using the transformed x-values and the original y-values, perform linear regression on the data (using a statistical software or manual calculations) to find the equation of the regression line in the form of y = a + b * ln(x).
- The linear regression equation for the transformed data is: y = -0.407 + 1.177 * ln(x)
Step 3: Substitute x = 15 into the equation to estimate the value for y.
- y = -0.407 + 1.177 * ln(15)
- y ≈ 12.326
Therefore, the estimated value for x = 15 is approximately 12.326.