Final answer:
The correlation coefficient (r) measures the strength and direction of the linear relationship between two variables, with values ranging from -1 to +1. The coefficient of determination (r²) represents the variance in the dependent variable explained by the independent variable. Correlation does not imply causation, and sample size affects the reliability of correlation estimates.
Step-by-step explanation:
The correlation coefficient (r) is a statistical measurement that describes the strength and direction of a linear relationship between two variables.
The value of r ranges from -1 to +1, where -1 indicates a perfect negative linear correlation, +1 indicates a perfect positive linear correlation, and 0 indicates no correlation. Calculating r provides insight into how two variables move in relation to one another.
When r is positive, an increase in one variable tends to be associated with an increase in the other variable, while a negative r indicates that as one variable increases, the other tends to decrease. The closer the value of r is to -1 or +1, the stronger the linear relationship between the variables.
To understand the coefficient of determination (r²), which is the square of the correlation coefficient, it's important to know that it represents the proportion of the variance in the dependent variable that is predictable from the independent variable.
For example, an r² value of 0.4397 could be interpreted as approximately 44 percent of the variance in the dependent variable being explained by the variance in the independent variable using the regression line.
It is also important to note that correlation should not be confused with causation. A high correlation does not imply that one variable causes changes in another.
Lastly, sample size is essential in validating the reliability of the correlation estimation for the larger population. Thus, both the sample correlation coefficient and the number of observations need to be considered.