Final answer:
The question pertains to statistical measures of relationship between two variables: the coefficient of determination, Pearson correlation coefficient, Spearman rank correlation coefficient, and regression coefficient. These values help in understanding the strength and direction of the association between the independent and dependent variables in a dataset.
Step-by-step explanation:
The student is asking about different statistical measures that assess the relationship between two variables. These measures include the coefficient of determination, the Pearson correlation coefficient, the Spearman rank correlation coefficient, and the regression coefficient. Understanding these concepts is crucial in fields such as statistics and data analysis.
The coefficient of determination (r²) represents the proportion of the variance in the dependent variable that is predictable from the independent variable. It is the square of the Pearson correlation coefficient. In the context provided, an r² value of 0.31 means that 31 percent of the variation in fuel efficiency can be explained by the variation in the bodyweight of the automobile.
The Pearson correlation coefficient (r) is a measure of linear correlation between two variables, giving a value between -1 and 1. A value of 1 implies a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 means no linear relationship. The formula for r can be complex, but it is readily calculable using various technological tools like spreadsheets and calculators.
The Spearman rank correlation coefficient is a non-parametric measure of rank correlation that assesses how well the relationship between two variables can be described by a monotonic function. It does not assume a linear relationship and is best used when evaluating ordinal data.
The regression coefficient (b in the equation ý = a + bx) represents the change in the dependent variable for each unit change in the independent variable and is a key part of the linear regression equation.