Final answer:
The correlation coefficient, denoted as r, measures the strength and direction of the linear relationship between two variables, x and y. It is calculated using a specific formula, and its significance helps determine if there's a linear relationship between the variables.
Step-by-step explanation:
The Correlation Coefficient r
The coefficient of correlation, often denoted as r, is a statistical measure that calculates the strength and direction of the linear relationship between two variables, denoted as x and y. To find the correlation coefficient between x and y, you would use the formula:
nΣxy - (Σx)(Σy)
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√[(nΣx² - (Σx)²][nΣy² - (Σy)²]
This formula incorporates the sum of the product of the x and y variables, the sum of the x variables, the sum of the y variables, and the number of data points, or sample size (n). The value of r ranges from -1 to +1, indicating the strength and direction of the relationship. A positive r means that as x increases, y also tends to increase and vice versa. A negative r indicates an inverse relationship. Moreover, the coefficient of determination, which is r², provides the proportion of the variance for the dependent variable that's predictable from the independent variable.
When assessing the significance of the correlation coefficient, if it is significantly different from zero, it provides evidence of a linear relationship. To determine if the model is a good predictor, we need to look at the value of r alongside the sample size (n).