Master Regression Analysis for CBSE Class 11 Applied Mathematics. Learn linear regression, coefficients, and prediction with step-by-step examples.
Regression analysis is a statistical method used to model the relationship between a dependent variable (y) and one or more independent variables (x). In simple linear regression, we assume a linear relationship represented by the line y = a + bx. This line minimizes the sum of the squares of the vertical deviations between each data point and the fitted line.
Step 1: Given the equation y = 2.5 + 0.8x.
Step 2: Compare with y = a + bx.
Step 3: The intercept a = 2.5 and the slope b = 0.8.
Answer: The slope is 0.8.
To find the best-fit line, we calculate the slope (b) and the intercept (a) using the method of least squares. The slope b represents the change in y for a unit change in x, while the intercept a is the value of y when x is zero. These parameters are derived from the means and variances of the datasets.
Step 1: Given Σ(x-x̄)(y-ȳ) = 40 and Σ(x-x̄)² = 50.
Step 2: Apply b = 40 / 50.
Step 3: b = 0.8.
Answer: The regression coefficient b is 0.8.
Once the regression equation is established, it can be used to predict the value of y for any given value of x within the observed range. This predictive power makes regression a vital tool in economics and business forecasting. Accuracy depends on the strength of the correlation between the variables.
Step 1: Given y = 2 + 0.5x and x = 10.
Step 2: Substitute x = 10 into the equation: y = 2 + 0.5(10).
Step 3: y = 2 + 5 = 7.
Answer: The predicted value is 7.