Final answer:
Regression inference requires conditions of linearity, independence, normality of residuals, and equal variance to be met, which can be checked through scatter plots, examining data collection methods, histograms or Q-Q plots of residuals, and residual plots, respectively. The y-intercept and slope have specific interpretations in the context of the regression model, and large residuals may signal outliers or influential points.
Step-by-step explanation:
The conditions for regression inference include linearity, independence, normality of residuals, and equal variance (homoscedasticity). To check for linearity, we inspect a scatter plot to ensure that the relationship between the independent and dependent variables appears to be linear. If a linear pattern is not evident, regression inference may not be appropriate.
Independence of observations can be checked by considering the data collection process. Observations should be collected randomly and independently from one another. Without this, the inference may not hold. The normality condition assumes that for any given value of the independent variable, the distribution of the residuals should be roughly normal. Inspect a histogram or a Q-Q plot of the residuals to assess this.
The condition of equal variance means that the residuals should have constant variance across all levels of the independent variable(s). One can visualize this by plotting residuals against the predicted values; looking for a 'funnel' shape can indicate violation of this condition. Diagnosis tools for regression also include calculating the correlation coefficient and the coefficient of determination (R-squared) to measure the strength and proportion of the variance in the dependent variable that's explained by the model. Interpretation of the y-intercept refers to the value of the dependent variable when all independent variables are zero. The slope, on the other hand, indicates the change in the dependent variable for a one-unit change in the independent variable.
The largest residual points to the data point with the greatest distance from the regression line. If a residual is noticeably larger than others, it may indicate that the point is an outlier or an influential point, which can disproportionally affect the slope of the regression line.