answer:
To find the r-squared value for the equation y = 2x using the given set of data x = [0.5, 1, 1.5, 2, 2.5] and y = [1, 2, 3, 4, 5], you can use MATLAB to perform a linear regression analysis.
Here are the steps to calculate the r-squared value:
1. Open MATLAB and create two vectors to represent the data:
x = [0.5, 1, 1.5, 2, 2.5]
y = [1, 2, 3, 4, 5]
2. Fit a linear regression model to the data using the "polyfit" function in MATLAB:
p = polyfit(x, y, 1)
The "polyfit" function fits a polynomial curve of degree 1 (a straight line) to the data points and returns the coefficients of the line. In this case, the first coefficient represents the slope of the line (2) and the second coefficient represents the y-intercept.
3. Calculate the predicted values of y using the fitted line equation:
y_predicted = polyval(p, x)
The "polyval" function evaluates the polynomial equation using the coefficients obtained from the linear regression.
4. Calculate the residuals (the differences between the predicted and actual y-values):
residuals = y - y_predicted
5. Calculate the total sum of squares (SST):
SST = sum((y - mean(y)).^2)
The "mean" function calculates the mean value of the y-values.
6. Calculate the sum of squares of residuals (SSE):
SSE = sum(residuals.^2)
7. Calculate the r-squared value:
r_squared = 1 - SSE/SST
The r-squared value measures the proportion of the total variation in the dependent variable (y) that can be explained by the independent variable (x). It ranges from 0 to 1, where a value of 1 indicates a perfect fit.
By following these steps in MATLAB, you will be able to obtain the r-squared value for the equation y = 2x using the given set of data.
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