Exploring Two–Variable Data

Observational studies do not control for lurking variables which could influence both explanatory and response variables, potentially inflating $r^2$ without causal linkages.

Correlation coefficients derived from observational data typically underestimate true cause-and-effect relationships leading to deflated $r^2$ values even when causation exists.

The computation method for $r^2$ inherently includes confounding factors that always imply direct causation when high values are observed in observational studies.

Controlled experiments do not utilize $r^2$ because it's only applicable in non-experimental contexts like observational studies where causation can be inferred directly from it.

A completely random pattern showing no identifiable trend or structure.

Clear clustering around certain values on the x-axis without any apparent curve.

A consistent horizontal banding suggesting perfectly equal variance among errors.

A curved pattern indicating that relationship between variables may be non-linear.

Assuming direct cause-and-effect based solely on model fit.

Looking for confounding variables that could explain any observed association.

Accepting causation because high coefficient determination ($r^2$).

Relying purely on p-values without considering practical significance.

A clear curved pattern.

Points equally distributed above and below the x-axis.

No distinct patterns or trends in residuals across different values of x.

A random scatter around zero.

An r-squared comparison alone determines necessity cubic inclusion without further testing.

Linear correlation coefficient increases sufficiently when adding cubic term compared prior version.

A partial F-test comparing full cubic model against reduced quadratic model could show improvement in fit.

Simple t-tests on coefficients for individual cubic terms would suffice independent assessment context.

Stratified sampling

Cluster sampling

Simple random sampling

Multi-stage sampling

High-leverage point

Low-leverage point

Outlier point

Influential point

The number of data points

The y-intercept of the model

Residual plots and $R^2$ value

Magnitude of influential points

To confirm that all outliers have been removed from the dataset.

To calculate new values for x-variable coefficients.

To ensure that linear models are appropriate for making predictions.

To increase the correlation coefficient to improve model accuracy.

The distribution shape changes.

The width increases.

Variability increases.

The width decreases.