Exploring Two–Variable Data

The independent variable explains most of the variability in the dependent variable.

There is almost no correlation between independent and dependent variables.

The slope of the regression line is approaching zero, suggesting weak association.

The variance around the regression line increases significantly with each unit increase in x-variable.

Total sum of squares in a dataset

Strength and direction of a linear relationship

Proportion of variation explained by the model

Slope of the least squares regression line

Time series plot displaying trends over time for residual errors after fitting model.

Histogram showing frequency distribution of independent variable values.

Normal probability plot comparing standardized residuals against theoretical quantiles.

Scatterplot with Cook's distance superimposed on each point.

To connect all the data points on a scatterplot with a straight line.

To maximize the sum of the squared differences between observed and predicted values.

To model the linear relationship between a dependent variable and one or more independent variables.

To find the average of the dependent variable values.

$H_{a} : b_{1} = 0$

$H_{a} : b_{1} > 0$

$H_{a} : b_{1} < 0$

$H_{a} : b_{1} \neq 0$

The correlation coefficient ($r$).

The y-intercept of the least squares regression line.

The slope of the least squares regression line.

The sum of squared residuals from each plot.

Using computer-generated random numbers to assign individuals to experimental conditions before starting the collection process.

Ensuring equal gender distribution across different treatment levels studying their effect on outcome.

Randomly assigning subjects into different treatment groups during experiment design.

Collecting multiple observations from each participant within your study.

Principal component analysis (PCA) which transforms correlated variables into a set of uncorrelated components.

Stepwise selection by gradually adding or removing predictors based on their statistical significance.

Lasso regression which performs variable selection and regularization simultaneously to enhance prediction accuracy.

Ridge regression which adds bias but reduces variance through shrinkage parameters.

Inverse transformation to reduce right skewness.

Cubic transformation to capture polynomial trends.

Logarithmic transformation to stabilize variance.

Square root transformation to linearize the relationship.

It will remain unchanged.

It will decrease.

It will become zero.

It will increase.