Correlation & Regression (Multiple T/F)

The following is TRUE of the assumptions of linear regression:

• The relationship between the independent and dependent variables must be linear.

• The Y values are statistically dependent to X values.

• The error terms (residuals) should be normally distributed.

• Homoscedasticity is not important in regression analysis.

• The dependent variable must be categorical.

Answer explanation

The correct assumptions of linear regression include that the relationship between independent and dependent variables must be linear and that the error terms should be normally distributed. The other statements are incorrect.

The following is TRUE of scatterplots:

• A scatterplot with points closely following a straight line indicates a strong linear relationship.

• Scatterplots can only be used for simple linear regression, not multiple linear regression.

• If the points in a scatterplot are randomly dispersed, the correlation coefficient is likely close to 1 or -1.

• Scatterplots allow for the visualization of the relationship between the independent and dependent variables.

• A scatterplot can show the strength of a relationship but not its direction.

Answer explanation

A scatterplot with points closely following a straight line indicates a strong linear relationship, showing a clear correlation. Additionally, scatterplots effectively visualize the relationship between independent and dependent variables.

For a simple linear regression model,

• the dependent variable is always on the y-axis.

• the slope of the regression line is unaffected by the units of the variables.

• the relationship between variables is always positive.

• the intercept represents the expected value of Y when X is zero.

• all data points lie exactly on the regression line.

Answer explanation

In simple linear regression, the dependent variable (Y) is plotted on the y-axis. The intercept indicates the expected value of Y when X is zero, making these statements correct.

The following is TRUE of correlation:

• Correlation measures the degree of a linear relationship between two variables.

• A correlation of 1 or -1 indicates a perfect linear relationship.

• Correlation can be used to infer causation.

• A correlation of 0 means there is no relationship at all between the two variables.

• Correlation coefficients are sensitive to outliers.

Answer explanation

Correlation measures the linear relationship between two variables, with 1 or -1 indicating perfect relationships. A correlation of 0 suggests no relationship, and correlation coefficients can be affected by outliers.

The following is TRUE of simple linear regression:

• It can handle multiple independent variables.

• It requires the relationship between the independent and dependent variable to be linear.

• It can be used even if the residuals are not normally distributed.

• The slope of the line is the correlation coefficient.

• It is used to predict the value of one variable based on the value of another.

Answer explanation

Simple linear regression assumes a linear relationship between the independent and dependent variables and is used to predict one variable based on another. The other statements are incorrect.

For the equation of a regression line, Y=a+bX

• a is the slope of the line.

• b is the y-intercept.

• a represents the expected value of Y when X is zero.

• the equation can be used to make predictions for values of Y based on X.

• Changing the units of X does not affect the value of b.

Answer explanation

In the regression equation Y=a+bX, 'a' represents the expected value of Y when X is zero, and the equation allows predictions of Y based on X.

When interpreting scatterplots:

• A scatterplot with points closely following a straight ascending line indicates a strong positive linear correlation.

• Scatterplots can only depict linear relationships between variables.

• Scatterplots are not useful for identifying outliers in the data.

• The direction of the correlation cannot be determined from a scatterplot.

• A scatterplot with points randomly distributed with no apparent pattern indicates no or weak linear correlation.

Answer explanation

A scatterplot with points closely following a straight ascending line shows a strong positive linear correlation, while random distribution indicates no or weak correlation, confirming the correct choices.

The following is TRUE of scatterplots:

• The steeper the slope of the line of best fit in a scatterplot, the stronger the correlation.

• It cannot show negative correlations.

• It provides a visual representation of the relationship between two variables.

• The distance of the points from the line of best fit in a scatterplot is not related to the strength of the correlation.

• A scatterplot where points form a perfect straight line (either ascending or descending) indicates a correlation coefficient of 1 or -1.

Answer explanation

A scatterplot visually represents the relationship between two variables, making it easy to see correlations. A perfect straight line indicates a correlation coefficient of 1 (positive) or -1 (negative), showing strong relationships.

In the context of scatterplots:

• A scatterplot with a horizontal arrangement of points indicates a strong linear correlation.

• The correlation coefficient can be visually estimated from a scatterplot.

• Scatterplots are only useful for comparing two variables at a time.

• Scatterplots are not useful in regression analysis for checking the assumptions of linearity and homoscedasticity.

• Widely dispersed points indicate a high degree of variance and potentially a weaker linear correlation.

Answer explanation

The correlation coefficient can indeed be visually estimated from a scatterplot, as the pattern of points indicates the strength and direction of the relationship. Widely dispersed points suggest weaker correlation.

The correlation coefficient r has a value of -0.85. This indicates:

• A strong negative linear relationship between the two variables.

• A strong positive linear relationship between the two variables.

• That 72.25% of the variability in the dependent variable can be explained by the independent variable.

• That increases in the value of one variable are associated with decreases in the value of the other variable.

• That the slope of the regression line is positive.

Answer explanation

The correlation coefficient r = -0.85 indicates a strong negative linear relationship, meaning as one variable increases, the other decreases. Additionally, r^2 = 0.7225 shows that 72.25% of the variability in the dependent variable is explained.