For a data set {x_1, y_1}, the best-fit line y = mx + b can be determined by the

Question

For a data set {x_1, y_1}, the best-fit line y = mx + b can be determined by the formula m = sigma_i(x_i - x)(y_1 - 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. If we assume a power-law relation T = b tims a^m, the linear regression between which tow 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. Using the data in Manual, make a table of the 2 quantities {x_i, y_i} to be analyzed. Make a table to compute the slope of best-fit line: m = sigma(x_i - x)(y_i - y)/sigma (x_i - x)^2 = ()/() = ______. Compute the y-intercept of best-fit line: b = y - mx = _____. SS_res = sigma (y_i - mx_i - b)^2 = _____.

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

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