1. The problem is to understand the concepts of regression and correlation in statistics. 2. Regression analysis is used to model the relationship between a dependent variable $y$ and one or more independent variables $x$. The simplest form is linear regression, which uses the formula: $$y = mx + b$$ where $m$ is the slope and $b$ is the intercept. 3. Correlation measures the strength and direction of a linear relationship between two variables. The correlation coefficient $r$ ranges from $-1$ to $1$: - $r = 1$ means perfect positive correlation. - $r = -1$ means perfect negative correlation. - $r = 0$ means no linear correlation. 4. Important rules: - Correlation does not imply causation. - Regression predicts the value of $y$ given $x$. - The closer $|r|$ is to 1, the stronger the linear relationship. 5. Example: Given data points, calculate the regression line and correlation coefficient by: - Finding means of $x$ and $y$. - Calculating slope $m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$. - Calculating intercept $b = \bar{y} - m\bar{x}$. - Calculating correlation $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$. This process helps understand how variables relate and predict outcomes.