Formulas and Tables for Gerstman

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Formulas ad Tables for Gerstma Measuremet ad Study Desig Biostatistics is more tha a compilatio of computatioal techiques! Measuremet scales: quatitative, ordial, categorical Iformatio quality is primary (GIGO) Data table: observatios, variables, values Use Table A or radom umber geerator to choose simple radom sample or radomize a treatmet Comparative studies must strive for group comparability to make valid ifereces. Comparative studies may be either experimetal or observatioal i desig: Beware lurkig variables, esp. i o-experimetal studies: cofoudig! Explorig ad Describig Data Explore distributioal shape, locatio, ad spread; check for outliers Frequecy, relative frequecy, cumulative frequecy Sample mea: x xi Media: Form a ordered array. The media is the value with a depth of + ; whe is odd, average the two middle values. Quartiles (Tukey s higes): Divide the ordered array at the media; whe is odd, the media belogs to both the low group ad the high group. Q is media of the low group. Q3 is the media of the high group. IQR Q3 Q Boxplot: plot media ad quartiles (box); determie feces: F L Q.5 IQR, F U Q3 +.5 IQR; plot outside values (if ay); draw whiskers from higes to iside values Five-poit summary: miimum, Q, media, Q3, maximum Sample stadard deviatio: s ( xi x) C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page of

Probability Basic properties: () 0 Pr(A) ; () Pr(S) ; (3) Pr(Ā) Pr(A) (4) Pr(A or B) Pr(A) + Pr(B) for disjoit evets x Biomial: X b(, p), Pr( X x) C p q where µ p ad σ pq where q p x x C x! x!( x)! To fid probabilities o X N(μ, σ): () State () Stadardize z x μ (3) Sketch (4) Use Table B σ To fid percetile values o X N(μ, σ): () State () Sketch (3) Table B (4) Ustadardize: x μ + z p σ Samplig Distributios ad Itroductio to Iferece Samplig distributio of mea: x N(μ, σ ) whe populatio Normal or sample large Hypothesis testig procedure: (A.) H 0 ad H a (B.) Test statistic (C.) P-value (D.) Optioal: Sigificace level x μ0 σ Oe-sample test of mea (σ kow): H 0 : μ μ 0 ; z where SE x SE x σ Cofidece iterval for μ (σ kow): x ± z α / SE x where SE x Coditios for z procedures: SRS, Normal populatio or large sample, σ kow Iferece about Meas Sigle samples ad match-pairs (matched-pairs, aalyze the delta variable ) x μ0 s To test H 0 : μ μ 0 use tstat where SE x ad df SE ( α)00% Cofidece iterval for μ: α x x x ±, t SE Sample size ad power to limit margi of error m, use σ z α m σ Sample size to detect a differece Δ with stated power ad α, use Power of a test to detect Δ at give α: β Φ z Two idepedet samples α Δ + σ C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page of ( z β + z α ) x x s s To test H 0 : μ μ use tstat where SE x SE x x x + df coservative smaller of ( ) or ( ) [use df Welch whe workig with a computer] ( α)00% CI for μ μ : ( x x) ± ( t df α )( SE x x ) σ z To estimate μ μ with margi of error m, use α i each group m, σ z β + z α Sample size test H 0 : μ μ at give β ad α: use i each group Δ ( ) Δ

If ot equal group sizes are ot possible, calculate, fix, ad use k idepedet samples MSB To test H 0 : μ μ μ k : F stat with df B ad df W via ANOVA table MSW Variace Sum of Squares df Mea Square k Betwee SS B SSB ( ) groups i xi x df B k MSB df Withi groups SS W i k i i ( ) s df W N k Total SS T SS B + SS W df df B + df W xi x j Post hoc least square differece; tstat where SE SE i x i x j Cofidece iterval for μ i μ j ( x x ) ± ( t df α )( SE x x ), xi x j Post hoc Boferroi: multiply P by umber of comparisos ad use MSW i t α N k, c Correlatio ad Regressio (use calculator or computer) Correlatio coefficiet r z X zy To test H 0 : ρ 0, use t stat Cofidece iterval for ρ: Regressio lie: y ˆ a + bx s Slope estimate: b r s Itercept estimate: a y bx Y X r r where SE r ad df SE r r ϖ r + ϖ LCL ad UCL where ϖ rϖ + rϖ B SS MSW df W W + ; df N k j i the CI formula t α df, t α df, Stadard error of the regressio s Y x residuals with df sy x ( α)00% cofidece iterval for β b ± (t -,-α/ )(SE b ) where SEb s b To test H 0 : β 0, use t stat with df SE b Multiple regressio model: y ˆ a + b x + bx + + b k xk (determie regressio coefficiets with computer program) X + df C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 3 of

Iferece about Proportios Sigle sample umber of successes Sample proportio: pˆ ( α)00% CI for p p ± z pq α where x + p ad q p + 4 pˆ p0 To test H 0 : p p 0 use (use exact biomial procedure i small samples) p q z stat 0 0 z Sample size requiremet to limit the margi of error (m): use * * p q α m Two idepedet samples Notatio (-by- cross-tabulatio) + Total Exposed a b No-exposed a b Total m m N Sample proportios (cohort ad aturalistic samples) a a p ˆ ad p ˆ To test H 0 : p p, (z stat ad X stat are equivalet for -by-) pˆ pˆ umber of successes, both samples combied z stat where p total observatios, both samples combied pq + ( O E ) i i X stat where all Ei row total colum total E i with df (R ) (C ) table total I small samples, use Fisher s exact test or the Mid-P modificatio (computer) Risk differece pˆ pˆ (do ot use i case-cotrol sample) ( α)00% CI p p ( p p) ± z α SE p p where Risk ratio ai + p i ad SE + i p q p p + p q ˆ pˆ a / R R (do ot use i case-cotrol sample) pˆ a / ( α)00% CI for RR e l RR ˆ ± z SE α l RR ˆ where SE l R ˆ R + a a C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 4 of

Odds ratio ˆ a OR a b b ( α)00% CI for the OR e lor ˆ ± z SE α l OR ˆ where SE l O ˆ R + + + a b a b Matched-pairs Notatio (for case-cotrol data) Case E+ Case E Cotrol E+ a b Cotrol E c d c O ˆ R b ( α)00% CI for the OR e To test H 0 : OR, use lor ˆ ± z SE α l OR ˆ ( c b) z stat c + b where SE + c b l O ˆ R C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 5 of

Table A. Two thousad radom digits Lie 0 79587 9407 4985 58687 99639 8670 73457 53546 309 7574 0 03 54407 97 6739 5745 5334 7445 6658 9597 38440 03 77633 43390 63003 5585 6374 4043 9576 9098 7540 04987 04 097 3996 38879 9749 543 658 75594 3430 648 4885 05 7673 0860 9904 537 8560 03805 738 8333 8080 65863 06 5330 098 097 50444 3447 783 954 55366 8300 9865 07 4337 8867 59836 05054 5874 59309 7740 5805 60603 5596 08 67893 0573 37080 6409 75438 3959 644 50847 3344 99647 09 5453 47973 68704 47487 9668 3437 068 38 4304 563 0 34867 89777 96947 4409 49866 9483 7694 78305 3354 306 956 43739 4836 85438 7033 878 56655 4865 53784 36693 7030 9840 05955 30586 3850 48 88039 66 03304 800 3 955 6306 59709 55085 893 50503 7570 440 64 9765 4 337 8396 46680 94704 6987 496 3849 68858 646 003 5 7693 9589 55809 938 56686 37898 3675 588 985 5568 6 88630 595 7694 53000 8909 690 5597 9669 34893 97543 7 5078 87768 693 9054 5804 64398 849 96407 97303 93459 8 5677 8748 65 04353 74 4304 4940 59906 696 36837 9 0447 38798 7386 99890 09907 760 0469 4785 7470 9887 0 760 83064 66743 580 4954 5685 585 837 06368 68488 77403 6093 6895 6903 0578 08934 89067 96693 07387 94489 70045 45404 8065 60568 9438 0857 34838 60958 94947 98568 3 7444 09905 65366 695 68 87077 965 979 4537 6760 4 39950 05637 4388 0366 6793 997 7973 55083 83840 4579 5 350 3969 807 86599 5738 5330 38380 8773 976 756 6 3008 0696 5945 0073 7760 8647 04865 5333 83736 546 7 0098 9303 773 7499 6938 78075 74684 98037 885 754 8 47948 4765 54 65500 86080 47438 404 56085 0446 30 9 54985 64 5648 433 466 844 74549 6900 8983 6596 30 48786 657 65 54949 5774 05975 87 05667 3 3879 3 855 66957 596 547 50576 4745 87903 8030 7690 8060 3 9955 007 073 0563 060 63964 5949 860 7494 87038 33 6788 88556 638 85038 67970 3366 6743 78854 63456 6789 34 066 4056 69688 98904 839 890 34 5743 3805 840 35 6376 643 64 94569 8043 637 99795 9047 958 95604 36 46558 56764 3508 863 43490 98 38375 9905 37766 59 37 40 65556 483 6588 37766 54388 80069 78335 79539 605 38 5365 5355 393 834 95995 4878 448 60663 0307 9508 39 585 75 3388 93730 36 94 933 905 64565 874 40 5649 4400 9678 94 08990 0549 4065 5756 6466 96748 C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 6 of

Table B: Cumulative probabilities for a Stadard Normal Z variable; traditioal z table. z.00.0.0.03.04.05.06.07.08.09 3.4.0003.0003.0003.0003.0003.0003.0003.0003.0003.000 3.3.0005.0005.0005.0004.0004.0004.0004.0004.0004.0003 3..0007.0007.0006.0006.0006.0006.0006.0005.0005.0005 3..000.0009.0009.0009.0008.0008.0008.0008.0007.0007 3.0.003.003.003.00.00.00.00.00.000.000.9.009.008.008.007.006.006.005.005.004.004.8.006.005.004.003.003.00.00.00.000.009.7.0035.0034.0033.003.003.0030.009.008.007.006.6.0047.0045.0044.0043.004.0040.0039.0038.0037.0036.5.006.0060.0059.0057.0055.0054.005.005.0049.0048.4.008.0080.0078.0075.0073.007.0069.0068.0066.0064.3.007.004.00.0099.0096.0094.009.0089.0087.0084..039.036.03.09.05.0.09.06.03.00..079.074.070.066.06.058.054.050.046.043.0.08.0.07.0.007.00.097.09.088.083.9.087.08.074.068.06.056.050.044.039.033.8.0359.035.0344.0336.039.03.034.0307.030.094.7.0446.0436.047.048.0409.040.039.0384.0375.0367.6.0548.0537.056.056.0505.0495.0485.0475.0465.0455.5.0668.0655.0643.0630.068.0606.0594.058.057.0559.4.0808.0793.0778.0764.0749.0735.07.0708.0694.068.3.0968.095.0934.098.090.0885.0869.0853.0838.083..5.3..093.075.056.038.00.003.0985..357.335.34.9.7.5.30.0.90.70.0.587.56.539.55.49.469.446.43.40.379 0.9.84.84.788.76.736.7.685.660.635.6 0.8.9.090.06.033.005.977.949.9.894.867 0.7.40.389.358.37.96.66.36.06.77.48 0.6.743.709.676.643.6.578.546.54.483.45 0.5.3085.3050.305.98.946.9.877.843.80.776 0.4.3446.3409.337.3336.3300.364.38.39.356.3 0.3.38.3783.3745.3707.3669.363.3594.3557.350.3483 0..407.468.49.4090.405.403.3974.3936.3897.3859 0..460.456.45.4483.4443.4404.4364.435.486.447 0.0.5000.4960.490.4880.4840.480.476.47.468.464 0.0.5000.5040.5080.50.560.599.539.579.539.5359 0..5398.5438.5478.557.5557.5596.5636.5675.574.5753 0..5793.583.587.590.5948.5987.606.6064.603.64 0.3.679.67.655.693.633.6368.6406.6443.6480.657 0.4.6554.659.668.6664.6700.6736.677.6808.6844.6879 0.5.695.6950.6985.709.7054.7088.73.757.790.74 0.6.757.79.734.7357.7389.74.7454.7486.757.7549 0.7.7580.76.764.7673.7704.7734.7764.7794.783.785 0.8.788.790.7939.7967.7995.803.805.8078.806.833 0.9.859.886.8.838.864.889.835.8340.8365.8389.0.843.8438.846.8485.8508.853.8554.8577.8599.86..8643.8665.8686.8708.879.8749.8770.8790.880.8830..8849.8869.8888.8907.895.8944.896.8980.8997.905.3.903.9049.9066.908.9099.95.93.947.96.977.4.99.907.9.936.95.965.979.99.9306.939.5.933.9345.9357.9370.938.9394.9406.948.949.944.6.945.9463.9474.9484.9495.9505.955.955.9535.9545.7.9554.9564.9573.958.959.9599.9608.966.965.9633.8.964.9649.9656.9664.967.9678.9686.9693.9699.9706.9.973.979.976.973.9738.9744.9750.9756.976.9767.0.977.9778.9783.9788.9793.9798.9803.9808.98.987..98.986.9830.9834.9838.984.9846.9850.9854.9857..986.9864.9868.987.9875.9878.988.9884.9887.9890.3.9893.9896.9898.990.9904.9906.9909.99.993.996.4.998.990.99.995.997.999.993.993.9934.9936.5.9938.9940.994.9943.9945.9946.9948.9949.995.995.6.9953.9955.9956.9957.9959.9960.996.996.9963.9964.7.9965.9966.9967.9968.9969.9970.997.997.9973.9974.8.9974.9975.9976.9977.9977.9978.9979.9979.9980.998.9.998.998.998.9983.9984.9984.9985.9985.9986.9986 3.0.9987.9987.9987.9988.9988.9989.9989.9989.9990.9990 3..9990.999.999.999.999.999.999.999.9993.9993 3..9993.9993.9994.9994.9994.9994.9994.9995.9995.9995 3.3.9995.9995.9995.9996.9996.9996.9996.9996.9996.9997 3.4.9997.9997.9997.9997.9997.9997.9997.9997.9997.9998 C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 7 of

Table C. Traditioal t table; tables etries represet t values. Cumulative 0.75 0.80 0.85 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995 Upper tail 0.5 0.0 0.5 0.0 0.05 0.05 0.0 0.005 0.005 0.00 0.0005 df.000.376.963 3.078 6.34.7 3.8 63.66 7.3 38.3 636.6 0.86.06.386.886.90 4.303 6.965 9.95 4.09.33 3.60 3 0.765 0.978.50.638.353 3.8 4.54 5.84 7.453 0..9 4 0.74 0.94.90.533.3.776 3.747 4.604 5.598 7.73 8.60 5 0.77 0.90.56.476.05.57 3.365 4.03 4.773 5.893 6.869 6 0.78 0.906.34.440.943.447 3.43 3.707 4.37 5.08 5.959 7 0.7 0.896.9.45.895.365.998 3.499 4.09 4.785 5.408 8 0.706 0.889.08.397.860.306.896 3.355 3.833 4.50 5.04 9 0.703 0.883.00.383.833.6.8 3.50 3.690 4.97 4.78 0 0.700 0.879.093.37.8.8.764 3.69 3.58 4.44 4.587 0.697 0.876.088.363.796.0.78 3.06 3.497 4.05 4.437 0.695 0.873.083.356.78.79.68 3.055 3.48 3.930 4.38 3 0.694 0.870.079.350.77.60.650 3.0 3.37 3.85 4. 4 0.69 0.868.076.345.76.45.64.977 3.36 3.787 4.40 5 0.69 0.866.074.34.753.3.60.947 3.86 3.733 4.073 6 0.690 0.865.07.337.746.0.583.9 3.5 3.686 4.05 7 0.689 0.863.069.333.740.0.567.898 3. 3.646 3.965 8 0.688 0.86.067.330.734.0.55.878 3.97 3.60 3.9 9 0.688 0.86.066.38.79.093.539.86 3.74 3.579 3.883 0 0.687 0.860.064.35.75.086.58.845 3.53 3.55 3.850 0.686 0.859.063.33.7.080.58.83 3.35 3.57 3.89 0.686 0.858.06.3.77.074.508.89 3.9 3.505 3.79 3 0.685 0.858.060.39.74.069.500.807 3.04 3.485 3.768 4 0.685 0.857.059.38.7.064.49.797 3.09 3.467 3.745 5 0.684 0.856.058.36.708.060.485.787 3.078 3.450 3.75 6 0.684 0.856.058.35.706.056.479.779 3.067 3.435 3.707 7 0.684 0.855.057.34.703.05.473.77 3.057 3.4 3.690 8 0.683 0.855.056.33.70.048.467.763 3.047 3.408 3.674 9 0.683 0.854.055.3.699.045.46.756 3.038 3.396 3.659 30 0.683 0.854.055.30.697.04.457.750 3.030 3.385 3.646 40 0.68 0.85.050.303.684.0.43.704.97 3.307 3.55 60 0.679 0.848.045.96.67.000.390.660.95 3.3 3.460 80 0.678 0.846.043.9.664.990.374.639.887 3.95 3.46 00 0.677 0.845.04.90.660.984.364.66.87 3.74 3.390 000 0.675 0.84.037.8.646.96.330.58.83 3.098 3.300 (z) 0.674 0.84.036.8.645.960.36.576.807 3.090 3.9 50% 60% 70% 80% 90% 95% 98% 99% 99.5% 99.8% 99.9% Cofidece level: ( α/)00% C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 8 of

Table E. Chi-square table Upper tail P df.975.5.0.5.0.05.05.0.0.00.0005 0.00.3.64.07.7 3.84 5.0 6.63 7.88 0.83. 0.05.77 3. 3.79 4.6 5.99 7.38 9. 0.60 3.8 5.0 3 0.6 4. 4.64 5.3 6.5 7.8 9.35.34.84 6.7 7.73 4 0.48 5.39 5.99 6.74 7.78 9.49.4 3.8 4.86 8.47 0.00 5 0.83 6.63 7.9 8. 9.4.07.83 5.09 6.75 0.5. 6.4 7.84 8.56 9.45 0.64.59 4.45 6.8 8.55.46 4.0 7.69 9.04 9.80 0.75.0 4.07 6.0 8.48 0.8 4.3 6.0 8.8 0..03.03 3.36 5.5 7.53 0.09.95 6. 7.87 9.70.39.4 3.9 4.68 6.9 9.0.67 3.59 7.88 9.67 0 3.5.55 3.44 4.53 5.99 8.3 0.48 3. 5.9 9.59 3.4 3.8 3.70 4.63 5.77 7.8 9.68.9 4.7 6.76 3.6 33.4 4.40 4.85 5.8 6.99 8.55.03 3.34 6. 8.30 3.9 34.8 3 5.0 5.98 6.98 8.0 9.8.36 4.74 7.69 9.8 34.53 36.48 4 5.63 7. 8.5 9.4.06 3.68 6. 9.4 3.3 36. 38. 5 6.6 8.5 9.3 0.60.3 5.00 7.49 30.58 3.80 37.70 39.7 6 6.9 9.37 0.47.79 3.54 6.30 8.85 3.00 34.7 39.5 4.3 7 7.56 0.49.6.98 4.77 7.59 30.9 33.4 35.7 40.79 4.88 8 8.3.60.76 4.6 5.99 8.87 3.53 34.8 37.6 4.3 44.43 9 8.9.7 3.90 5.33 7.0 30.4 3.85 36.9 38.58 43.8 45.97 0 9.59 3.83 5.04 6.50 8.4 3.4 34.7 37.57 40.00 45.3 47.50 0.8 4.93 6.7 7.66 9.6 3.67 35.48 38.93 4.40 46.80 49.0 0.98 6.04 7.30 8.8 30.8 33.9 36.78 40.9 4.80 48.7 50.5 3.69 7.4 8.43 9.98 3.0 35.7 38.08 4.64 44.8 49.73 5.00 4.40 8.4 9.55 3.3 33.0 36.4 39.36 4.98 45.56 5.8 53.48 5 3. 9.34 30.68 3.8 34.38 37.65 40.65 44.3 46.93 5.6 54.95 6 3.84 30.43 3.79 33.43 35.56 38.89 4.9 45.64 48.9 54.05 56.4 7 4.57 3.53 3.9 34.57 36.74 40. 43.9 46.96 49.64 55.48 57.86 8 5.3 3.6 34.03 35.7 37.9 4.34 44.46 48.8 50.99 56.89 59.30 9 6.05 33.7 35.4 36.85 39.09 4.56 45.7 49.59 5.34 58.30 60.7 30 6.79 34.80 36.5 37.99 40.6 43.8 47.0 50.9 53.7 59.7 6. 40 4.43 45.6 47.7 49.4 5.8 55.76 59.34 63.69 66.77 73.40 76.09 50 3.36 56.33 58.6 60.35 63.7 67.50 7.4 76.5 79.49 86.66 89.56 60 40.48 66.98 68.97 7.34 74.40 79.08 83.30 88.38 9.95 99.6 0.69 80 57.5 88.3 90.4 93. 96.58 0.88 06.63.33 6.3 4.84 8.6 00 74. 09..7 4.7 8.5 4.3 9.6 35.8 40. 49.4 53. C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 9 of

Table F: Two tails of z; table etries are two-sided P-values for z. z.00.0.0.03.04.05.06.07.08.09 0.0.0000.990.98404.97607.96809.960.956.9449.9364.989 0..9034.94.90448.89657.88866.88076.8788.8650.8575.8493 0..8448.83367.8587.8809.8033.8059.79486.7876.77948.778 0.3.7648.75656.74897.7440.73386.7634.7885.738.70395.69654 0.4.6896.688.67449.6670.65994.657.6455.63836.633.643 0.5.6708.6005.60306.596.5890.583.57548.56868.569.5559 0.6.5485.5486.5356.5869.57.5569.5095.5086.49650.4909 0.7.48393.47770.475.46539.45930.4535.4475.4430.43539.4953 0.8.437.4794.4.40654.4009.39533.38979.38430.37886.37347 0.9.368.368.35757.3537.347.34.33706.3305.3709.37.0.373.350.30773.3030.9834.937.894.846.804.757..733.6700.67.5848.549.504.4605.400.3800.3405..304.68.46.870.498.30.0767.0408.0055.9705.3.9360.900.8684.835.805.770.7383.7069.6759.6453.4.65.5854.556.57.4987.4706.449.456.3887.36.5.336.304.85.60.356.4.876.64.4.83.6.0960.0740.053.030.00.09894.0969.0949.0996.0903.7.0893.0877.08543.08363.0886.080.0784.07673.07508.07345.8.0786.07030.06876.0675.06577.0643.0689.0648.060.05876.9.05743.0563.05486.0536.0538.058.05000.04884.04770.04659.0.04550.04443.04338.0436.0435.04036.03940.03845.03753.0366..03573.03486.0340.0337.0335.0356.03077.0300.096.085..078.07.064.0575.0509.0445.038.03.06.00.3.045.0089.0034.098.098.0877.087.0779.073.0685.4.0640.0595.055.050.0469.049.0389.035.034.077.5.04.007.074.04.009.0077.0047.007.00988.00960.6.0093.00905.00879.00854.0089.00805.0078.00759.00736.0075.7.00693.00673.00653.00633.0064.00596.00578.0056.00544.0057.8.005.00495.00480.00465.0045.00437.0044.0040.00398.00385.9.00373.0036.00350.00339.0038.0038.00308.0098.0088.0079 3.0.0070.006.0053.0045.0037.009.00.004.0007.0000 3..0094.0087.008.0075.0069.0063.0058.005.0047.004 3..0037.0033.008.004.000.005.00.0008.0004.0000 3.3.00097.00093.00090.00087.00084.0008.00078.00075.0007.00070 3.4.00067.00065.00063.00060.00058.00056.00054.0005.00050.00048 3.5.00047.00045.00043.0004.00040.00039.00037.00036.00034.00033 3.6.0003.0003.0009.0008.0007.0006.0005.0004.0003.000 3.7.000.000.0000.0009.0008.0008.0007.0006.0006.0005 3.8.0004.0004.0003.0003.000.000.000.000.0000.0000 3.9.0000.00009.00009.00008.00008.00008.00007.00007.00007.00007 C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 0 of

Table G: Two-sided P-values from t statistics ("o-traditioal t table") df ( to 5) t 3 4 5 6 7 8 9 0 3 4 5 0.0.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000 0. 0.937 0.99 0.97 0.95 0.94 0.94 0.93 0.93 0.93 0.9 0.9 0.9 0.9 0.9 0.9 0. 0.874 0.860 0.854 0.85 0.849 0.848 0.847 0.846 0.846 0.845 0.845 0.845 0.845 0.844 0.844 0.3 0.84 0.79 0.784 0.779 0.776 0.774 0.773 0.77 0.77 0.770 0.770 0.769 0.769 0.769 0.768 0.4 0.758 0.78 0.76 0.70 0.706 0.703 0.70 0.700 0.698 0.698 0.697 0.696 0.696 0.695 0.695 0.5 0.705 0.667 0.65 0.643 0.638 0.635 0.63 0.63 0.69 0.68 0.67 0.66 0.65 0.65 0.64 0.6 0.656 0.609 0.59 0.58 0.575 0.570 0.567 0.565 0.563 0.56 0.56 0.560 0.559 0.558 0.557 0.7 0.6 0.556 0.534 0.53 0.55 0.50 0.507 0.504 0.50 0.500 0.498 0.497 0.496 0.495 0.495 0.8 0.570 0.508 0.48 0.469 0.460 0.454 0.450 0.447 0.444 0.44 0.44 0.439 0.438 0.437 0.436 0.9 0.533 0.463 0.434 0.49 0.409 0.403 0.398 0.394 0.39 0.389 0.387 0.386 0.384 0.383 0.38.0 0.500 0.43 0.39 0.374 0.363 0.356 0.35 0.347 0.343 0.34 0.339 0.337 0.336 0.334 0.333. 0.470 0.386 0.35 0.333 0.3 0.33 0.308 0.303 0.300 0.97 0.95 0.93 0.9 0.90 0.89. 0.44 0.353 0.36 0.96 0.84 0.75 0.69 0.64 0.6 0.58 0.55 0.53 0.5 0.50 0.49.3 0.47 0.33 0.84 0.63 0.50 0.4 0.35 0.30 0.6 0.3 0.0 0.8 0.6 0.5 0.3.4 0.395 0.96 0.56 0.34 0.0 0. 0.04 0.99 0.95 0.9 0.89 0.87 0.85 0.83 0.8.5 0.374 0.7 0.3 0.08 0.94 0.84 0.77 0.7 0.68 0.65 0.6 0.59 0.58 0.56 0.54.6 0.356 0.5 0.08 0.85 0.70 0.6 0.54 0.48 0.44 0.4 0.38 0.36 0.34 0.3 0.30.7 0.339 0.3 0.88 0.64 0.50 0.40 0.33 0.8 0.3 0.0 0.7 0.5 0.3 0. 0.0.8 0.33 0.4 0.70 0.46 0.3 0. 0.5 0.0 0.05 0.0 0.099 0.097 0.095 0.093 0.09.9 0.308 0.98 0.54 0.30 0.6 0.06 0.099 0.094 0.090 0.087 0.084 0.08 0.080 0.078 0.077.0 0.95 0.84 0.39 0.6 0.0 0.09 0.086 0.08 0.077 0.073 0.07 0.069 0.067 0.065 0.064. 0.83 0.7 0.7 0.04 0.090 0.080 0.074 0.069 0.065 0.06 0.060 0.058 0.056 0.054 0.053. 0.7 0.59 0.5 0.093 0.079 0.070 0.064 0.059 0.055 0.05 0.050 0.048 0.046 0.045 0.044.3 0.6 0.48 0.05 0.083 0.070 0.06 0.055 0.050 0.047 0.044 0.04 0.040 0.039 0.037 0.036.4 0.5 0.38 0.096 0.074 0.06 0.053 0.047 0.043 0.040 0.037 0.035 0.034 0.03 0.03 0.030.5 0.4 0.30 0.088 0.067 0.054 0.047 0.04 0.037 0.034 0.03 0.030 0.08 0.07 0.05 0.05.6 0.34 0. 0.080 0.060 0.048 0.04 0.035 0.03 0.09 0.06 0.05 0.03 0.0 0.0 0.00.7 0.6 0.4 0.074 0.054 0.043 0.036 0.03 0.07 0.04 0.0 0.0 0.09 0.08 0.07 0.06.8 0.8 0.07 0.068 0.049 0.038 0.03 0.07 0.03 0.0 0.09 0.07 0.06 0.05 0.04 0.03.9 0. 0.0 0.063 0.044 0.034 0.07 0.03 0.00 0.08 0.06 0.04 0.03 0.0 0.0 0.0 3.0 0.05 0.095 0.058 0.040 0.030 0.04 0.00 0.07 0.05 0.03 0.0 0.0 0.00 0.00 0.009 df (6 to 30) t 6 7 8 9 0 3 4 5 6 7 8 9 30 0.0.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000 0. 0.937 0.99 0.97 0.95 0.94 0.94 0.93 0.93 0.93 0.9 0.9 0.9 0.9 0.9 0.9 0. 0.874 0.860 0.854 0.85 0.849 0.848 0.847 0.846 0.846 0.845 0.845 0.845 0.845 0.844 0.844 0.3 0.84 0.79 0.784 0.779 0.776 0.774 0.773 0.77 0.77 0.770 0.770 0.769 0.769 0.769 0.768 0.4 0.758 0.78 0.76 0.70 0.706 0.703 0.70 0.700 0.698 0.698 0.697 0.696 0.696 0.695 0.695 0.5 0.705 0.667 0.65 0.643 0.638 0.635 0.63 0.63 0.69 0.68 0.67 0.66 0.65 0.65 0.64 0.6 0.656 0.609 0.59 0.58 0.575 0.570 0.567 0.565 0.563 0.56 0.56 0.560 0.559 0.558 0.557 0.7 0.6 0.556 0.534 0.53 0.55 0.50 0.507 0.504 0.50 0.500 0.498 0.497 0.496 0.495 0.495 0.8 0.570 0.508 0.48 0.469 0.460 0.454 0.450 0.447 0.444 0.44 0.44 0.439 0.438 0.437 0.436 0.9 0.533 0.463 0.434 0.49 0.409 0.403 0.398 0.394 0.39 0.389 0.387 0.386 0.384 0.383 0.38.0 0.500 0.43 0.39 0.374 0.363 0.356 0.35 0.347 0.343 0.34 0.339 0.337 0.336 0.334 0.333. 0.470 0.386 0.35 0.333 0.3 0.33 0.308 0.303 0.300 0.97 0.95 0.93 0.9 0.90 0.89. 0.44 0.353 0.36 0.96 0.84 0.75 0.69 0.64 0.6 0.58 0.55 0.53 0.5 0.50 0.49.3 0.47 0.33 0.84 0.63 0.50 0.4 0.35 0.30 0.6 0.3 0.0 0.8 0.6 0.5 0.3.4 0.395 0.96 0.56 0.34 0.0 0. 0.04 0.99 0.95 0.9 0.89 0.87 0.85 0.83 0.8.5 0.374 0.7 0.3 0.08 0.94 0.84 0.77 0.7 0.68 0.65 0.6 0.59 0.58 0.56 0.54.6 0.356 0.5 0.08 0.85 0.70 0.6 0.54 0.48 0.44 0.4 0.38 0.36 0.34 0.3 0.30.7 0.339 0.3 0.88 0.64 0.50 0.40 0.33 0.8 0.3 0.0 0.7 0.5 0.3 0. 0.0.8 0.33 0.4 0.70 0.46 0.3 0. 0.5 0.0 0.05 0.0 0.099 0.097 0.095 0.093 0.09.9 0.308 0.98 0.54 0.30 0.6 0.06 0.099 0.094 0.090 0.087 0.084 0.08 0.080 0.078 0.077.0 0.95 0.84 0.39 0.6 0.0 0.09 0.086 0.08 0.077 0.073 0.07 0.069 0.067 0.065 0.064. 0.83 0.7 0.7 0.04 0.090 0.080 0.074 0.069 0.065 0.06 0.060 0.058 0.056 0.054 0.053. 0.7 0.59 0.5 0.093 0.079 0.070 0.064 0.059 0.055 0.05 0.050 0.048 0.046 0.045 0.044.3 0.6 0.48 0.05 0.083 0.070 0.06 0.055 0.050 0.047 0.044 0.04 0.040 0.039 0.037 0.036.4 0.5 0.38 0.096 0.074 0.06 0.053 0.047 0.043 0.040 0.037 0.035 0.034 0.03 0.03 0.030.5 0.4 0.30 0.088 0.067 0.054 0.047 0.04 0.037 0.034 0.03 0.030 0.08 0.07 0.05 0.05.6 0.34 0. 0.080 0.060 0.048 0.04 0.035 0.03 0.09 0.06 0.05 0.03 0.0 0.0 0.00.7 0.6 0.4 0.074 0.054 0.043 0.036 0.03 0.07 0.04 0.0 0.0 0.09 0.08 0.07 0.06.8 0.8 0.07 0.068 0.049 0.038 0.03 0.07 0.03 0.0 0.09 0.07 0.06 0.05 0.04 0.03.9 0. 0.0 0.063 0.044 0.034 0.07 0.03 0.00 0.08 0.06 0.04 0.03 0.0 0.0 0.0 3.0 0.05 0.095 0.058 0.040 0.030 0.04 0.00 0.07 0.05 0.03 0.0 0.0 0.00 0.00 0.009 C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page of

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