Length Time 1 1 10.2 2 2 14.2 3 3 16.4 4 4 19.2 5 5 21.5 6 6 23.3 7 7 25.2 8 8 27.0 9 9 29.0 10 10 30.5 11 11 31.9 12 12 33.6 13 13 34.7 14 14 35.9 15 15 37.4 16 16 38.5 17 17 39.2 18 18 40.4 19 19 41.9 20 20 43.0 len per lenlen sqrtlen loglen logper 1 0.0254 0.3400000 0.00064516 0.1593738 -3.6730061 -1.07880966 2 0.0508 0.4733333 0.00258064 0.2253886 -2.9798589 -0.74795542 3 0.0762 0.5466667 0.00580644 0.2760435 -2.5743938 -0.60391605 4 0.1016 0.6400000 0.01032256 0.3187475 -2.2867117 -0.44628710 5 0.1270 0.7166667 0.01612900 0.3563706 -2.0635682 -0.33314445 6 0.1524 0.7766667 0.02322576 0.3903844 -1.8812466 -0.25274402 7 0.1778 0.8400000 0.03161284 0.4216634 -1.7270960 -0.17435339 8 0.2032 0.9000000 0.04129024 0.4507771 -1.5935646 -0.10536052 9 0.2286 0.9666667 0.05225796 0.4781213 -1.4757815 -0.03390155 10 0.2540 1.0166667 0.06451600 0.5039841 -1.3704210 0.01652930 11 0.2794 1.0633333 0.07806436 0.5285830 -1.2751108 0.06140863 12 0.3048 1.1200000 0.09290304 0.5520869 -1.1880995 0.11332869 13 0.3302 1.1566667 0.10903204 0.5746303 -1.1080567 0.14554231 14 0.3556 1.1966667 0.12645136 0.5963221 -1.0339488 0.17953991 15 0.3810 1.2466667 0.14516100 0.6172520 -0.9649559 0.22047332 16 0.4064 1.2833333 0.16516096 0.6374951 -0.9004174 0.24946086 17 0.4318 1.3066667 0.18645124 0.6571149 -0.8397928 0.26747937 18 0.4572 1.3466667 0.20903184 0.6761657 -0.7826343 0.29763240 19 0.4826 1.3966667 0.23290276 0.6946942 -0.7285671 0.33408845 20 0.5080 1.4333333 0.25806400 0.7127412 -0.6772738 0.36000273 #################################### Model 1: Simple Linear Regression # #################################### Call: lm(formula = per ~ len, dataset = pendul) Residuals: Min 1Q Median 3Q Max -0.13486 -0.03898 0.01713 0.04233 0.05941 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.42081 0.02494 16.88 1.77e-12 *** len 2.12796 0.08195 25.96 1.02e-15 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.05368 on 18 degrees of freedom Multiple R-squared: 0.974, Adjusted R-squared: 0.9726 F-statistic: 674.2 on 1 and 18 DF, p-value: 1.021e-15 #################################### Model 2: Quadratic Regression # #################################### Call: lm(formula = per ~ len + lenlen, dataset = pendul) Residuals: Min 1Q Median 3Q Max -0.041177 -0.007715 0.001316 0.008940 0.025203 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.29426 0.01204 24.45 1.1e-14 *** len 3.48676 0.10392 33.55 < 2e-16 *** lenlen -2.54743 0.18924 -13.46 1.7e-10 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.01618 on 17 degrees of freedom Multiple R-squared: 0.9978, Adjusted R-squared: 0.9975 F-statistic: 3802 on 2 and 17 DF, p-value: < 2.2e-16 #################################### Model 3: Using Sqrt Transform # #################################### Call: lm(formula = per ~ sqrtlen, dataset = pendul) Residuals: Min 1Q Median 3Q Max -0.013638 -0.007926 0.001996 0.004806 0.013708 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.011651 0.006215 1.875 0.0772 . sqrtlen 1.987563 0.012034 165.157 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.008548 on 18 degrees of freedom Multiple R-squared: 0.9993, Adjusted R-squared: 0.9993 F-statistic: 2.728e+04 on 1 and 18 DF, p-value: < 2.2e-16 #################################### Model 4: Using Log-log transform # #################################### Call: lm(formula = logper ~ loglen, dataset = pendul) Residuals: Min 1Q Median 3Q Max -0.032357 -0.010117 0.002458 0.008493 0.026870 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.680052 0.007022 96.85 <2e-16 *** loglen 0.486177 0.004021 120.91 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.01424 on 18 degrees of freedom Multiple R-squared: 0.9988, Adjusted R-squared: 0.9987 F-statistic: 1.462e+04 on 1 and 18 DF, p-value: < 2.2e-16