Exploring Two–Variable Data

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

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.

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.

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

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

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

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

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.

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.

Coefficient of determination and correlation coefficient.

Median and mode.

Slope and y-intercept.

Mean and standard deviation.

Skewness doesn't influence outcomes as long Central Limit Theorem applies regardlessly ensuring sampling distributions maintain their expected bell-shape form.

Inference precision improves automatically whenever skewness appears regardlessly provided condition known asymptotic properties retain their relevance across all circumstances still persisting here too.

It enhances precision inference since skewness adds directionality interpretation aiding hypothesis testing procedures regarding slope coefficients.

It can compromise validity due its effect on t-distribution approximation accuracy without large samples involved allowing Central Limit Theorem application fully counteracting it.

Snowball Sampling

Multi-Stage Sampling

Random Digit Dialing

Proportional Allocation