$r$

Definition

$r$ is the correlation coefficient, which is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It is a key concept in the analysis of bivariate data and is central to the topic of testing the significance of the correlation coefficient.

5 Must Know Facts For Your Next Test

1. $r$ can take values between -1 and 1, with -1 indicating a perfect negative linear relationship, 0 indicating no linear relationship, and 1 indicating a perfect positive linear relationship.
2. The sign of $r$ (positive or negative) indicates the direction of the linear relationship, while the magnitude of $r$ (closer to 1 or -1) indicates the strength of the linear relationship.
3. The coefficient of determination, $r^2$, is the square of the correlation coefficient and represents the proportion of the variance in one variable that can be explained by the linear relationship with the other variable.
4. Hypothesis testing for the significance of the correlation coefficient is used to determine whether the observed correlation in the sample data is likely to have occurred by chance or if it reflects a true linear relationship in the population.
5. The test statistic used to assess the significance of the correlation coefficient is the t-statistic, which follows a t-distribution with $n-2$ degrees of freedom, where $n$ is the sample size.

Review Questions

• Explain the meaning and interpretation of the correlation coefficient, $r$.
  • The correlation coefficient, $r$, is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It can take values between -1 and 1, with -1 indicating a perfect negative linear relationship, 0 indicating no linear relationship, and 1 indicating a perfect positive linear relationship. The sign of $r$ (positive or negative) indicates the direction of the linear relationship, while the magnitude of $r$ (closer to 1 or -1) indicates the strength of the linear relationship. The coefficient of determination, $r^2$, represents the proportion of the variance in one variable that can be explained by the linear relationship with the other variable.
• Describe the purpose and process of testing the significance of the correlation coefficient, $r$.
  • The purpose of testing the significance of the correlation coefficient, $r$, is to determine whether the observed correlation in the sample data is likely to have occurred by chance or if it reflects a true linear relationship in the population. The process involves formulating a null hypothesis (H0: $r = 0$, indicating no linear relationship) and an alternative hypothesis (H1: $r \neq 0$, indicating a linear relationship). The test statistic used is the t-statistic, which follows a t-distribution with $n-2$ degrees of freedom, where $n$ is the sample size. The p-value obtained from the test is then compared to the chosen significance level to determine whether to reject or fail to reject the null hypothesis, and thus conclude whether the observed correlation is statistically significant.
• Explain how the correlation coefficient, $r$, is related to the coefficient of determination, $r^2$, and discuss the implications of this relationship.
  • The correlation coefficient, $r$, is directly related to the coefficient of determination, $r^2$, through the equation $r^2 = (r)^2$. The coefficient of determination, $r^2$, represents the proportion of the variance in one variable that can be explained by the linear relationship with the other variable. This means that the magnitude of $r$ (closer to 1 or -1) indicates the strength of the linear relationship, while $r^2$ indicates the proportion of the variance in one variable that is accounted for by the linear relationship with the other variable. For example, if $r = 0.8$, then $r^2 = 0.64$, which means that 64% of the variance in one variable can be explained by the linear relationship with the other variable.

Related terms

Correlation: Correlation is a statistical measure that describes the strength and direction of the linear relationship between two variables.

Bivariate Data: Bivariate data refers to a dataset that contains measurements on two variables for each observation or data point.

Hypothesis Testing: Hypothesis testing is a statistical method used to determine whether a particular claim or hypothesis about a population parameter is supported by the sample data.