Lines of Best Fit, Linear Regression and Predictions

m = (55 − 15) ÷ (10 − 2) = 40 ÷ 8 = 5. 15 = 5(2) + b, so b = 5. Equation: y = 5x + 5.

y = 0.074(70) + 4.98 = 5.18 + 4.98 = 10.16, about 10 free throws.

Predicted: 0.074(150) + 4.98 = 11.1 + 4.98 = 16.08. Residual = 17 − 16.08 = 0.92 (positive: the player did better than predicted).

x = 25 is interpolation (inside 5 to 40). x = 60 is extrapolation (outside).

Each extra centimetre of diameter is associated with about $0.55 more in price.

y = −0.43(50) + 100 = −21.5 + 100 = 78.5%.

It makes the sum of the squared residuals (vertical distances from the points to the line) as small as possible, so it is the line that fits the data most closely by that measure, and everyone using the method gets the same line.

The slope (rate of change).

Residual = actual y − predicted y.

Negative.

y = 12 − 2 = 10.

False.

y = 0.91(160) + 15 = 145.6 + 15 = 160.6 cm. At x = 0, the model gives 15 cm, which has no sensible meaning (no person has an arm span of 0); the intercept is outside the useful range of the model.

m = (47 − 32) ÷ (2015 − 1995) = 15 ÷ 20 = 0.75. The club grew by about 0.75 members per year (3 members every 4 years).

Negative snowfall is impossible, so the prediction is outside the range where the model works (probably an extrapolation). Linear models can give impossible values beyond the data.

Both lines could be reasonable lines drawn by eye. Neither is "the" answer; linear regression with technology gives one agreed best line by minimizing the squared residuals.

0 = −0.43x + 100, so x = 100 ÷ 0.43 ≈ 233 minutes. It is an extrapolation far beyond the data; many phones also switch off or go into power-saving mode at low battery, changing the rate.

Predicted: 3.1, 4.7, 6.3, 7.9. Residuals: 3 − 3.1 = −0.1; 5 − 4.7 = 0.3; 6 − 6.3 = −0.3; 8 − 7.9 = 0.1. Sum = 0. For a least-squares line, the residuals always add to 0: the positive and negative errors balance.

A negative slope means the predicted time keeps decreasing forever. Eventually the model would predict a time of 0 seconds or even negative times, which is impossible. Human performance has limits, so the improvement must slow down: the true relationship is not linear over the long term.

r = 0.35 is a weak to moderate correlation, so the points are widely scattered around the line. Individual predictions could be far off. The line shows a general trend, but predictions should be described as rough estimates.