Which one of the following statements is true?

Question

BlackTom AI · Accepted Answer

Let's evaluate each statement one by one and unpack why they are true or false, keeping an eye on what correlation and scatter relationships actually imply.

Option 1: 'If you correlate two quantitative variables, called “x” and “y,” changing the units of measurement for “x” and “y” will change the correlation.' In fact, the correlation coefficient is unitless and is invariant under linear changes of scale (including changing units). Scaling or shifting x or y does not alter the strength or direction of the linear relationship as measured by the correlation r. Therefore this statement is incorrect.

Option 2: 'When a relationship between two quantitative variables is strong, linear, and negative, low values of one variable tend to correspond with high values of the other variable.' A strong negative linear relationship means that as one variable increases, the other tends to decrease in a predictable, linear fashion. Practically, low values of one variable tend to align with high values of the other variable, which is exactly what a negative slope conveys. Thus this statement correctly describes a strong negative linear association and is true.

Option 3: 'If r is positive and we want to predict one variable based on the other, the intercept in the regression equation must also be positive.' The sign of the intercept is not constrained to be positive merely because r is positive. The intercept depends on the means of the variables and the slope, and can be negative, zero, or positive depending on the data sample. Therefore this statement is incorrect.

Option 4: 'If you correlate two quantitative variables, called “x” and “y,” the correlation will change if you switch which variable is “x” and which is “y.”' The correlation is symmetric: r between x and y is the same as between y and x. Switching the labels does not change the correlation value. Hence this statement is false.

Option 5: 'A correlation of r = –0.49 is weaker than a correlation of r = 0.15.' When comparing correlation magnitudes, we consider the absolute value. |–0.49| = 0.49 is larger than |0.15| = 0.15, so –0.49 represents a stronger (more linear) relationship in magnitude than 0.15. Therefore this statement is incorrect.

In summary, the only statement that accurately captures the true nature of the relationship between two quantitative variables in this context is Option 2, while all other options contain a misconception or a false claim about correlation and regression properties.

Which one of the following statements is true?

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