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Predict from Correlation

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Paired measurements have sample means $10$ for $x$ and $50$ for $y$ , sample standard deviations $4$ for $x$ and $10$ for $y$ , and sample correlation $\frac{3}{5}$ . Using the least-squares line of $y$ on $x$ , what is the predicted value at $x = 12$ ?

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Hint

The slope equals the correlation times the ratio of standard deviations, and the line passes through the point of means.

Solution

The slope is $\frac{3}{5} \times \frac{10}{4} = \frac{3}{2}$ .
The line passes through $(10, 50)$ , so at $x = 12$ it rises by $\frac{3}{2} \times 2 = 3$ .
The predicted value is $50 + 3 = 53$ .

Answer: $53$

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