Build the whole regression line from summaries, then predict

Question

A study relates weekly advertising spend ( $x$ ) to units sold ( $y$ ). You never see the raw data, only these five summaries:

$\bar{x} = 50, \quad \bar{y} = 140, \quad s_x = 10, \quad s_y = 30, \quad r = 0.8.$

Using only these, predict the units sold when spend is $x = 60$ .

Accepted Answer

Five summary numbers are enough to rebuild the entire least-squares line and then use it. Do it in three steps. Step 1, the slope. The slope is the correlation rescaled by the ratio of spreads: b = r \cdot \frac{sy}{sx} = 0.8 \cdot \frac{30}{10} = 0.8 	imes 3 = 2.4. Each extra unit of spend is associated with about 2.4 more units sold. Step 2, the intercept. The line passes through the point of averages, so a = \bar{y} - b \, \bar{x} = 140 - 2.4 	imes 50 = 140 - 120 = 20. The line is therefore \hat{y} = 20 + 2.4x. Step 3, the prediction. Plug in x = 60: \hat{y} = 20 + 2.4 	imes 60 = 20 + 1…

Build the whole regression line from summaries, then predict

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