Question
An economist wants to estimate the relationship between a family’s annual income and the amount that the family saves. The family data for nine families are obtained
| Annual income (thousands of Rs.) | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| Annual saving (thousands of Rs.) | 0.0 | 0.1 | 0.2 | 0.2 | 0.5 | 0.5 | 0.6 | 0.7 | 0.8 |
Calculate the least square regression line where annual savings is the dependent variable and annual income is the independent variable.
Answer :
Word Count : 508
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Skip in 10
To calculate the least squares regression line, we need to find the equation of the form: \[ y = \beta_0 + \beta_1 x \] where: - \( y \) is the dependent variable (annual savings), - \( x \) is the independent variable (annual income), - \( \beta_0 \) is the intercept, and - \( \beta_1 \) is the slope of the regression line. ### Step 1: Calculate the necessary sums The formulas for the slope \( \beta_1 \) and the intercept \( \beta_0 \) are: \[ \beta_1 = \frac{n \sum{x y} - \sum{x} \sum{y}}{n \sum{x^2} - (\sum{x})^2} \] \[ \beta_0 = \frac{\sum{y} - \beta_1 \sum{x}}{n} \] Where \( n \) is the number of data points. We will calculate the following sums: - \( \sum{x} \) = sum of all \( x \) values (annual income), - \( \sum{y} \) = sum of all \( y \) values (annual savings), - \( \sum{x^2} \) = sum of the squares of \( x \), - \( \sum{xy} \) = sum of the product of corresponding \( x \) and \( y \). ###
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