Build and predict with a negative correlation

Question

A study relates hours of daily practice ( $x$ ) to the time in minutes to finish a task ( $y$ ), where more practice means less time. The summaries are

$\bar{x} = 8, \quad \bar{y} = 50, \quad s_x = 2, \quad s_y = 6, \quad r = -0.6.$

Predict the completion time when practice is $x = 10$ hours.

Accepted Answer

The negative correlation means signs matter at every step; carry them carefully. Slope: b = r \cdot \frac{sy}{sx} = -0.6 \cdot \frac{6}{2} = -0.6 	imes 3 = -1.8. Each extra hour of practice is associated with about 1.8 fewer minutes on the task (the minus sign encodes "less time"). Intercept, watching the double negative: a = \bar{y} - b \, \bar{x} = 50 - (-1.8)(8) = 50 + 14.4 = 64.4. The line is \hat{y} = 64.4 - 1.8x. Prediction at x = 10: \hat{y} = 64.4 - 1.8 	imes 10 = 64.4 - 18 = 46.4 	ext{ minutes.} The strength of the relationship is R^2 = (-0.6)^2 = 0.36, so about 36\% of the variat…

Build and predict with a negative correlation

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