# 2 df se b b t 005 2 18 se b 10139 2100041

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Unformatted text preview: df SE b b ! &quot;0 t= SEb Example: 95% confidence interval for slope of smoking / excise tax relationship b ± t! [ 2 ],df SE b = b ± t 0.05[ 2 ],18 SE b = 1.0139 ± 2.10(0.041) = 1.0139 ± 0.0866 Hypothesis tests can use t: Confidence bands: confidence intervals for predictions of mean Y Prediction intervals: confidence intervals for predictions of individual Y Hypothesis tests on slopes H 0: HA: =0 !0 t= Non-linear relationships Transformations Quadratic regression Splines t= b ! &quot;0 SEb 1.0139 ! 0 = 24.58 0.041 t0.05(2),18= ±2.10 So we can reject H0 Transformations If Y = aX b then ln Y = ln a + b ln X . If Y = ab X then ln Y = ln a + X ln b . If Y = a + Non-linear relationship: Number of fish species vs. Size of desert pool b 1 then set X ! = , and calculate Y = a + bX ! . X X All of the equations on the right have the form Y=a+bX. Residual plots help assess assumptions Original: Residual plot Transformed data Logs: Residual plot Polynomial regression Do not fit a polynomial with too many terms (the sample size should be at least 7 times the number of terms) Number of species = 0.046 + 0.185 Biomass - 0.00044 Biomass2 Comparing two slopes Example: Comparing species-area curves for islands to those of mainland populations L og10(Number o f s pecies) B y L og 1 0(Area o f &quot; island&quot;) 2.5 Hypotheses H 0: HA: M = # I. I. M Log10(Number of species) 2.0 1.5 1.0 0.5 -1 0 1 2 3 4 5 Log 10(Area of &quot;island&quot;) Linear Fit Type of island=I Linear Fit Type of island=M 6 7 Linear F it T ype o f i sland=I Log10(Number of species) = 0.24537 + 0.27554 Log 10(Area of &quot;island&quot;) Summary o f F it RSquare RSquare Adj Root Mean Square Error Mean of Response Obs...
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