For a data set {x_i, y_i}, the best-fit line y = mx + b can be determined by the

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Question

For a data set {x_i, y_i}, the best-fit line y = mx + b can be determined by the formula m = sigma_i(x_i - x) (y_i - y)/sigma_i(x_i - x)^2, and b = y - mx Here x and y are the average of {x_i} and {y_i}, respectively. Let's apply the regression analysis to several solar planets and find a power-law relation between their semi-major axes a and orbital periods T, as listed below. * 1 astronomical unit is 149.6 million km (the distance from Earth to the Sun). If we assume a power-law relation T = b times a^m, the linear regression between which two quantities do we need to analyze? (A) T vs a; (B) log T vs a; (C) T vs log a; (D) log T vs log a.

Explanation / Answer

For Mercury

xi = log0.387 = - 0.412

yi = log0.241 = - 0.618

For Venus

xi = log0.723 = - 0.141

yi = log0.615 = - 0.211

For Earth

xi = log1 = 0

yi = log1 = 0

For Mars

xi = log1.52 = 0.182

yi = log1.88 = 0.274

For Jupiter

xi = log5.2 = 0.716

yi = log11.9 = 1.075

For Saturn

xi = log9.55 = 0.98

yi = log29.5 = 1.47

Mean of xi = 0.2208

Mean of yi = 0.3316

Thus, we need to analyze logT vs log a .

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For a data set {x_i, y_i}, the best-fit line y = mx + b can be determined by the

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