The standard error of the slope ($SE_b$) measures the variability of sample slopes around the true population slope. A smaller $SE_b$ indicates a more precise estimate of the slope.

Checking conditions (Linearity, Independence, Normality, Equal Variance) ensures the validity of the inference procedures. If conditions are not met, the results of the t-test or t-interval may be unreliable.

Correlation does not equal causation. A significant slope indicates a linear association, but other factors (lurking variables, confounding variables) may be influencing the response variable.

$R^2$ represents the proportion of the variance in the response variable that is explained by the explanatory variable. A higher $R^2$ indicates a better fit of the regression model.

Residuals are the differences between the observed values and the predicted values from the regression line. They represent the error in the model's predictions.

The explanatory variable (independent variable) is plotted on the x-axis and explains the patterns seen in a scatterplot; consider it the 'cause'.

The response variable (dependent variable) is plotted on the y-axis and responds to the explanatory variable; consider it the 'effect'.

Inference uses sample data to make predictions or test claims about a population parameter, moving from describing data to making informed decisions.

A confidence interval used to estimate the true slope of the population regression line, providing a range of plausible values.

A hypothesis test used to determine if there is a significant linear relationship between two variables by testing if the slope is significantly different from zero.

$\hat{y} = a + bx$, where $\hat{y}$ is the predicted value, a is the y-intercept, and b is the slope.

$r = \pm \sqrt{R^2}$. The sign of r matches the sign of the slope.

$t = \frac{b - 0}{SE_b}$, where b is the sample slope and $SE_b$ is the standard error of the slope.

$b \pm t^[object Object]$, where b is the sample slope, t^ is the critical t-value, and $SE_b$ is the standard error of the slope.

$df = n - 2$, where n is the number of data points.