Understanding the Concepts and Skills Suppose that, for a sample of pairs of observations from two variables, the linear correlation coefficient, r , is negative. Does this result necessarily imply that the variables are negatively linearly correlated? Explain.

Question
Bivariate numerical data
Understanding the Concepts and Skills
Suppose that, for a sample of pairs of observations from two variables, the linear correlation coefficient, r , is negative. Does this result necessarily imply that the variables are negatively linearly correlated? Explain.

2020-12-08
Step 1
Correlation coefficient, r:
The Karl Pearson’s product-moment correlation coefficient or simply, the Pearson’s correlation coefficient is a measure of the strength of a linear association between two variables and is denoted by r or $$\displaystyle{r}_{{{x}{y}}}$$.
The coefficient of correlation $$\displaystyle{r}_{{{x}{y}}}$$ between two variables x and y for the bivariate data set $$\displaystyle{\left({x}_{{i}},{y}_{{i}}\right)}{f}{\quad\text{or}\quad}{i}={1},{2},{3}…{N}$$ is given below:
$$\displaystyle{r}_{{{x}{y}}}=\frac{{{n}{\left(\sum{x}{y}\right)}-{\left(\sum{x}\right)}{\left(\sum{y}\right)}}}{{\sqrt{{{\left[{n}{\left(\sum{x}^{{2}}\right)}-{\left(\sum{x}^{{2}}\right)}\right]}\times{\left[{n}{\left(\sum{y}^{{2}}\right)}-{\left(\sum{y}^{{2}}\right)}\right]}}}}}$$
Step 2
The population linear correlation coefficient is $$\displaystyle\pi$$, which measures the linear correlation of all possible pair of variable in same way r measures.
The population linear correlation coefficient is $$\displaystyle\pi$$, which measures the linear correlation of all possible pair of variables in population that represent the numerical summary of the population.
The sample linear correlation coefficient is r, which measures the linear correlation of all possible pair of variables in sample that represent the numerical summary of the sample.
The linear correlation coefficient value could be negative due to the presence of error, where pi takes some other value, might be zero or positive. The statistic r is used to estimate the population linear coefficient $$\displaystyle\pi$$.
Thus, the given result not necessarily implies that, the variables are negatively linearly correlated.

Relevant Questions

Suppose that, for a sample of pairs of observations from two variables, the linear correlation coefficient, r , is negative. Does this result necessarily imply that the variables are negatively linearly correlated?
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