Rebuild and forecast, factory output from machine hours

Question

A plant relates machine hours (x) to units produced (y). The summaries are \bar{x} = 100, \quad \bar{y} = 500, \quad sx = 25, \quad sy = 100, \quad r = 0.75. Predict output when machine hours are x = 120.

Accepted Answer

The recipe does not care about the size of the numbers. Build the line, then predict. Slope: b = r \cdot \frac{sy}{sx} = 0.75 \cdot \frac{100}{25} = 0.75 	imes 4 = 3. Each extra machine hour is associated with about 3 more units. Intercept: a = \bar{y} - b \, \bar{x} = 500 - 3 	imes 100 = 500 - 300 = 200. The line is \hat{y} = 200 + 3x. Prediction at x = 120: \hat{y} = 200 + 3 	imes 120 = 200 + 360 = 560 	ext{ units.} This is a strong fit (R^2 = 0.75^2 = 0.5625, about 56\% explained), and 120 hours is well within a standard deviation of the average 100, so 560 units is a confident interpo…

Rebuild and forecast, factory output from machine hours

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