Slope from correlation, sleep and alertness

Question

A sleep study relates nightly hours of sleep ( $x$ ) to a next-day alertness score ( $y$ ). The summary numbers are

$r = 0.8, \qquad s_x = 2, \qquad s_y = 10.$

What slope would the least-squares line use to predict alertness from hours of sleep?

Accepted Answer

Use the relationship between slope, correlation, and the two standard deviations: b = r \cdot \frac{sy}{sx}. The correlation gives the strength and sign; the ratio of spreads rescales it into real units (alertness points per hour of sleep). Substitute: b = 0.8 \cdot \frac{10}{2} = 0.8 	imes 5 = 4. Each extra hour of sleep is associated with about 4 more alertness points. The strong correlation (0.8) and the fact that y varies five times as widely as x combine to give a fairly steep slope. Takeaway. Correlation times the spread ratio gives the slope: 0.8 	imes (10/2) = 4 points per hour of s…

Slope from correlation, sleep and alertness

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