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221FinalEquations

# 221FinalEquations - Constants R = 8.314 J-K-1-mol-1 N0 =...

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Constants R = 8 . 314 J-K - 1 -mol - 1 k B = 1 . 381 × 10 - 23 J-K - 1 e = 1 . 1602 × 10 - 19 C m e = 9 . 11 × 10 - 31 kg N 0 = 6 . 022 × 10 23 mol - 1 h = 6 . 626 × 10 - 34 J-s c = 2 . 998 × 10 8 m-s - 1 0 = 8 . 854 × 10 - 12 C 2 N - 1 m - 2 Related Information Quadratic Equation: if ax 2 + bx + c = 0 , then x = - b ± b 2 - 4 ac 2 a Standard Deviation: s = ( x i - x ) 2 N - 1 Error Propagation Equation: σ 2 y ( u, v, . . . ) = σ 2 u dy du 2 + σ 2 v dy dv 2 + · · · t-tests: True Mean Comparison: x 1 - x 2 = ± ts/ N Measured Means Comparison (when σ 2 1 = σ 2 2 ): x 1 - x 2 = ± ts pooled N 1 + N 2 N 1 N 2 s pooled = s 2 1 ( N 1 - 1) + s 2 2 ( N 2 - 1) N 1 + N 2 - 2 Measured Means Comparison (when σ 2 1 = σ 2 2 ): | x 1 - x 2 | = ± t s 2 1 /N 1 + s 2 2 /N 2 ν = ( s 2 1 /N 1 + s 2 2 /N 2 ) 2 ( s 2 1 /N 1 ) 2 N 1 + 1 + ( s 2 2 /N 2 ) 2 N 2 + 1 - 2 Degrees of Freedom, ν t(90%) t(95%) t(99%) 1 6.314 12.706 63.657 2 2.920 4.303 9.925 3 2.353 3.182 5.841 4 2.132 2.776 4.604 5 2.015 2.571 4.032 6 1.943 2.447 3.707 7 1.895 2.365 3.500 8 1.860 2.306 3.355 9 1.833 2.262 3.250 10 1.812 2.228 3.169 1.64 1.96 2.576 Table 1: Values of t for ν degrees of freedom for 95% confidence levels. ν = N - 1 = degrees of freedom, and confidence limit = x ± ts N . 12

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F Distribution F = s 2 A s 2 B . (1) Note that critical F values in the two-tailed F -test table with probability p/ 2 can be found in the one-tailed table F -test with probability p . ν 1 =1 2 3 4 5 6 7 8 9 10 12 15 20 ν 2 1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 243.9 245.9 248.0 254.3 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.50 3 10.13 9.552 9.277 9.117 9.013 8.941 8.887 8.845 8.812 8.786 8.745 8.703 8.660 8.53 4 7.709 6.944 6.591 6.388 6.256 6.163 6.094 6.041 5.999 5.964 5.912 5.858 5.803 5.63 5 6.608 5.786 5.409 5.192 5.050 4.950 4.876 4.818 4.772 4.735 4.678 4.619 4.558 4.36 6 5.987 5.143 4.757 4.535 4.387 4.284 4.207 4.147 4.099 4.060 4.000 3.938 3.874 3.67 7 5.591 4.737 4.347 4.120 3.972 3.866 3.787 3.726 3.677 3.637 3.575 3.511 3.445 3.23 8 5.318 4.459 4.066 3.838 3.687 3.581 3.500 3.438 3.388 3.347 3.284 3.218 3.150 2.93 9 5.117 4.256 3.863 3.633 3.482 3.374 3.293 3.230 3.179 3.137 3.073 3.006 2.936 2.71 10 4.965 4.103 3.708 3.478 3.326 3.217 3.135 3.072 3.020 2.978 2.913 2.845 2.774 2.54 12 4.747 3.885 3.490 3.259 3.106 2.996 2.913 2.849 2.796 2.753 2.687 2.617 2.544 2.30 15 4.543 3.682 3.287 3.056 2.901 2.790 2.707 2.641 2.588 2.544 2.475 2.403 2.328 2.07 20 4.351 3.493 3.098 2.866 2.711 2.599 2.514 2.447 2.393 2.348 2.278 2.203 2.124 1.84 Table 2: Critical values of F
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