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.

$\hat{y} = a + bx$, 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.

T-interval: Estimates the range of plausible values for the true slope. | T-test: Tests a hypothesis about the value of the slope (typically if it is zero).

Null Hypothesis ($H_0$): Assumes there is no linear relationship (slope = 0). | Alternative Hypothesis ($H_a$): Claims there is a linear relationship (slope ≠ 0, slope > 0, or slope < 0).

Correlation (r): Measures the strength and direction of a linear relationship, but is unitless and does not predict values. | Slope (b): Predicts the change in the response variable for a one-unit change in the explanatory variable and has units.

Deterministic language: Implies a certain outcome (e.g., 'studying one more hour will increase the score by 5 points'). | Predictive language: Acknowledges variability and makes predictions (e.g., 'studying one more hour is predicted to increase the score by 5 points').

Standard deviation of residuals (s): Measures the typical distance of the observed values from the regression line. | Standard error of the slope ($SE_b$): Measures the variability of the sample slopes around the true population slope.