The correlation between temperature in ◦F and age in weeks was r = −0.70. If we converted the temperature to ◦C and age to months, what would the 29 correlation be?

Question

The correlation between temperature in ◦F and age in weeks was r = −0.70. If we converted the temperature to ◦C and age to months, what would the 29 correlation be?

BlackTom AI · Accepted Answer

When both variables are transformed by linear, positive-slope conversions, the correlation coefficient r remains the same in magnitude and preserves its sign.

Option 1: 'the absolute value of r will increase.' This would imply a stronger linear relationship after converting units, which is not the case for standard linear unit conversions (Fahrenheit to Celsius and weeks to months) since these are simply scaling and shifting with positive slopes. The strength of the linear relationship (|r|) stays the same.

Option 2: '-0.70' This matches the principle that r is invariant under positive linear transformations of the variables. Converting temperature from F to C uses a = 5/9 > 0 and b = -32, and converting age from weeks to months also uses a positive scaling. Therefore the correlation coefficient keeps its magnitude and sign.

Option 3: 'the absolute value of r will decrease.' This would suggest a weakening of the linear association after unit changes, which contradicts the mathematical property that positive linear transformations do not change |r|.

Option 4: '+0.70' This would indicate a reversal of the direction of the relationship (a positive association) after the conversions, but since all transformations are with positive slopes, the sign of r should remain negative rather than flip to positive.

In summary, because the unit conversions are linear and with positive slopes, the correlation coefficient remains exactly -0.70.

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The correlation between temperature in ◦F and age in weeks was r = −0.70. If we converted the temperature to ◦C and age to months, what would the 29 correlation be?

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