A data set is shown below.
x y
1 6.00
2 5.04
3 4.23
4 3.56
5 2.99
6 2.51
7 2.11
Using an exponential best-fit model, what is the residual when x = 1. Round your answer to the nearest hundredth.
This is the problem i dont understand line of best fit i dont get at all
To find the exponential best-fit model for a given data set, we need to perform exponential regression. This involves finding the equation of an exponential curve that best represents the relationship between the variables x and y.
To explain how to find the exponential best-fit model, follow these steps:
Step 1: Plot the data points on a graph. The x-values should represent the independent variable, and the y-values should represent the dependent variable.
Step 2: Determine the form of the exponential function. The general form of an exponential function is y = a * e^(bx), where a and b are constants that we need to find.
Step 3: Take the natural logarithm of both sides of the equation to simplify the equation: ln(y) = ln(a) + bx.
Step 4: Convert the problem into a linear regression problem by letting Y = ln(y), A = ln(a), and B = b. The equation becomes Y = A + BX.
Step 5: Calculate the values ΣX, ΣY, ΣXY, ΣX^2, ΣY^2 using the given data points, where Σ denotes the summation.
Step 6: Use the formulas for finding the slope (B) and the y-intercept (A) of the line that best represents the data:
B = (n∑XY - ∑X∑Y) / (n∑X^2 - (∑X)^2)
A = (∑Y - B∑X) / n
where n is the number of data points.
Step 7: Substitute the values of A and B back into the equation Y = A + BX to get the equation of the exponential best-fit model.
Step 8: Once you have the equation of the exponential best-fit model, you can find the residual for a specific value of x by substituting it into the equation and finding the difference between the predicted y-value and the actual y-value.
Now, let's apply these steps to find the exponential best-fit model for the given data and calculate the residual when x = 1.