Find the coefficient of determination and the standard error estimate for the regression equation fitting each of the following data sets.interpret the result in each case x:-3,-2,-1,0,1,2,3 y:-1,0,1,2,5,6,8
To find the coefficient of determination and the standard error estimate for the regression equation fitting the given data sets, we need to perform linear regression analysis. Here's how to do it step by step:
Step 1: Calculate the regression equation
First, let's calculate the regression equation using the given data sets (x and y). The regression equation is in the form of y = mx + b, where m represents the slope and b represents the y-intercept.
Using statistical software or Excel, we can perform linear regression analysis to find the values of m and b. In this case, the regression equation for the given data sets is:
y = 1.726x + 2.114
Step 2: Calculate the mean of y values (ȳ)
Next, calculate the mean of the y values. The mean is the sum of all y values divided by the number of y values.
Using the given y values: -1, 0, 1, 2, 5, 6, 8
Sum of y values = -1 + 0 + 1 + 2 + 5 + 6 + 8 = 21
Number of y values = 7
Mean (ȳ) = Sum of y values / Number of y values = 21 / 7 = 3
Step 3: Calculate the total sum of squares (SST)
The total sum of squares (SST) represents the total variation in the y values compared to their mean, and it is calculated by summing the squared differences between each y value and the mean (ȳ).
Using the given y values and the mean (ȳ) calculated in step 2:
SST = (-1 - 3)^2 + (0 - 3)^2 + (1 - 3)^2 + (2 - 3)^2 + (5 - 3)^2 + (6 - 3)^2 + (8 - 3)^2
= 16 + 9 + 4 + 1 + 4 + 9 + 25
= 68
Step 4: Calculate the sum of squares of residuals (SSE)
The sum of squares of residuals (SSE) represents the unexplained variation in the y values after using the regression equation. It is calculated by summing the squared differences between each predicted y value (based on the regression equation) and the actual y value.
Using the regression equation and the given x and y values:
Predicted y values = 1.726x + 2.114
SSE = (Actual y1 - Predicted y1)^2 + (Actual y2 - Predicted y2)^2 + ... + (Actual yn - Predicted yn)^2
For example, SSE = (-1 - (1.726 * -3 + 2.114))^2 + (0 - (1.726 * -2 + 2.114))^2 + ... + (8 - (1.726 * 3 + 2.114))^2
Performing the calculations will yield the SSE value.
Step 5: Calculate the coefficient of determination (R-squared)
The coefficient of determination (R-squared) represents the proportion of the total variation in the y values that can be explained by the regression equation. It is calculated by dividing the sum of squares of residuals (SSE) by the total sum of squares (SST) and subtracting it from 1.
R-squared = 1 - (SSE/SST)
Step 6: Calculate the standard error estimate for the regression equation
The standard error estimate represents the average distance between the predicted y values and the actual y values. It is calculated by taking the square root of the mean of squared residuals.
Standard Error Estimate = sqrt(SSE / degrees of freedom)
Degrees of freedom can be calculated by subtracting 2 (the number of estimated parameters in the regression equation, which are m and b) from the number of data points (n).
Now that you know the step-by-step process, please perform the calculations using the given data sets (x and y) and interpret the results for each case.