Formulas and Tables
Formulas and Tables Formulas and Tables
Formulas and Tables by Mario F. TriolaCopyright 2010 Pearson Education, Inc.Ch. 8: Test Statistics (one population)F = s2 1s 2 2Standard deviation or variance—two populations (where s 2 1 s 2 2 )Ch. 11: Goodness-of-Fit and Contingency Tables1O -x 2 E22= gEGoodness-of-fit(df k 1)1O - Contingency tablex 2 E22= g [df (r 1)(c 1)]E1row total21column total2where E =1grand total21ƒb - c ƒ - McNemar’s test for matchedx 2 122= pairs (df 1)b + cCh. 10: Linear Correlation/Regressionz = pN - pProportion—one populationn©xy - 1©x21©y2pqCorrelation r =2n1©x 2 2 - 1©x2 2 2n1©y 2 2 - 1©y2 2B nMean—one populationz = x - ms> 1n ( known)or r = a where zAz x z y Bx = z score for xz y = z score for yn - 1Mean—one populationt = x - mn©xy - 1©x21©y2( unknown)Slope:bs> 1n1 =n 1©x 2 2 - 1©x2 21n - Standard deviation or variance—x 2 12s2s= ys one populationor b 1 = rs xCh. 9: Test Statistics (two populations)z = 1pN y-Intercept:1 - pN 2 2 - 1p 1 - p 2 2Two proportionspqB n+ pqb 0 = y - b 1 x or b 0 = 1©y21©x 2 2 - 1©x21©xy21 n 2‹ p = x 1 + x 2n 1©x 2 2 - 1©x2 2n 1 + n 2t = 1x 1 - x 2 2 - 1m 1 - m 2 2yN = b 0 + b 1 x Estimated eq. of regression linedf smaller ofs 2 1n 1 1, n 2 1+ s2 2explained variationBn 1 nr 2 =2total variationTwo means—independent; s 1and s 2unknown, and not©1y - yN2 2 ©y 2 - b 0 ©y - b 1 ©xyassumed equal.s e = or Bt = 1x n - 2 B n - 21 - x 2 2 - 1m 1 - m 2 2(df n 1 n 2 2)yN - E 6 y 6 yN + E Prediction intervalsp2 + s2 pBn sp 2 = 1n 1 - 12s 2 1 + 1n 2 - 12s 2 21 n 2where E = t a>2 s e 1 + 1 n 1 + n 2 - 2B n + n1x 0 - x2 2n1©x 2 2 - 1©x2 2Two means—independent; s 1and s 2unknown, butassumed equal.Ch. 12: One-Way Analysis of VarianceTwo means—independent;z = 1x 1 - x 2 2 - 1m 1 - m 2 2Procedure for testing H 0 : m 1 = m 2 = m 3 = Á 1, 2known.s12B n+ s 22 1. Use software or calculator to obtain results.1 n 2 2. Identify the P-value.Two means—matched pairst = d - m 3. Form conclusion:dIf P-value a, reject the null hypothesiss (df n 1)d > 1nof equal means.If P-value a, fail to reject the null hypothesisof equal means.‹ ‹‹Ch. 12: Two-Way Analysis of VarianceProcedure:1. Use software or a calculator to obtain results.2. Test H 0: There is no interaction between the row factor andcolumn factor.3. Stop if H 0from Step 2 is rejected.If H 0from Step 2 is not rejected (so there does not appear tobe an interaction effect), proceed with these two tests:Test for effects from the row factor.Test for effects from the column factor.
Formulas and Tables by Mario F. TriolaCopyright 2010 Pearson Education, Inc.Ch. 13: Nonparametric Tests1x + 0.52 - 1n>22z =1n>2z =z = R - m Rs R=H =BT - n 1n + 12>4n 1n + 1212n + 12Sign test for n 2512N1N + 12 a R 2 1+ R 2 2+ . . . + R 2 kb - 31N + 12n 1 n 2 n kKruskal-Wallis (chi-square df k 1)6©d 2r s = 1 -n1n 2 - 12acritical value for n 7 30:z = G - m Gs G=24Ch. 14: Control ChartsR - n 11n 1 + n 2 + 122BR chart: Plot sample rangesUCL: D 4 RCenterline: RLCL: D 3 Rx chart: Plot sample meansUCL: xx + A 2 RCenterline: xxLCL: xx - A 2 Rn 1 n 2 1n 1 + n 2 + 1212Rank correlationp chart: Plot sample proportionspqUCL: p + 3 B nCenterline: ppqLCL: p - 3 B nWilcoxon signed ranks(matched pairs and n 30); z1n - 1 bG - a 2n 1n 2n 1 + n 2+ 1b12n 1 n 2 212n 1 n 2 - n 1 - n 2 2B 1n 1 + n 2 2 2 1n 1 + n 2 - 12Wilcoxon rank-sum(two independentsamples)Runs testfor n 20TABLE A-6Critical Values of thePearson CorrelationCoefficient rn a = .05 a = .014 .950 .9905 .878 .9596 .811 .9177 .754 .8758 .707 .8349 .666 .79810 .632 .76511 .602 .73512 .576 .70813 .553 .68414 .532 .66115 .514 .64116 .497 .62317 .482 .60618 .468 .59019 .456 .57520 .444 .56125 .396 .50530 .361 .46335 .335 .43040 .312 .40245 .294 .37850 .279 .36160 .254 .33070 .236 .30580 .220 .28690 .207 .269100 .196 .256NOTE: To test H 0 : r = 0 against H 1 : r Z 0,reject H 0 if the absolute value of r isgreater than the critical value in the table.Control Chart ConstantsSubgroup Sizen A 2D 3D 42 1.880 0.000 3.2673 1.023 0.000 2.5744 0.729 0.000 2.2825 0.577 0.000 2.1146 0.483 0.000 2.0047 0.419 0.076 1.924
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<strong>Formulas</strong> <strong>and</strong> <strong>Tables</strong> by Mario F. TriolaCopyright 2010 Pearson Education, Inc.Ch. 8: Test Statistics (one population)F = s2 1s 2 2St<strong>and</strong>ard deviation or variance—two populations (where s 2 1 s 2 2 )Ch. 11: Goodness-of-Fit <strong>and</strong> Contingency <strong>Tables</strong>1O -x 2 E22= gEGoodness-of-fit(df k 1)1O - Contingency tablex 2 E22= g [df (r 1)(c 1)]E1row total21column total2where E =1gr<strong>and</strong> total21ƒb - c ƒ - McNemar’s test for matchedx 2 122= pairs (df 1)b + cCh. 10: Linear Correlation/Regressionz = pN - pProportion—one populationn©xy - 1©x21©y2pqCorrelation r =2n1©x 2 2 - 1©x2 2 2n1©y 2 2 - 1©y2 2B nMean—one populationz = x - ms> 1n ( known)or r = a where zAz x z y Bx = z score for xz y = z score for yn - 1Mean—one populationt = x - mn©xy - 1©x21©y2( unknown)Slope:bs> 1n1 =n 1©x 2 2 - 1©x2 21n - St<strong>and</strong>ard deviation or variance—x 2 12s2s= ys one populationor b 1 = rs xCh. 9: Test Statistics (two populations)z = 1pN y-Intercept:1 - pN 2 2 - 1p 1 - p 2 2Two proportionspqB n+ pqb 0 = y - b 1 x or b 0 = 1©y21©x 2 2 - 1©x21©xy21 n 2‹ p = x 1 + x 2n 1©x 2 2 - 1©x2 2n 1 + n 2t = 1x 1 - x 2 2 - 1m 1 - m 2 2yN = b 0 + b 1 x Estimated eq. of regression linedf smaller ofs 2 1n 1 1, n 2 1+ s2 2explained variationBn 1 nr 2 =2total variationTwo means—independent; s 1<strong>and</strong> s 2unknown, <strong>and</strong> not©1y - yN2 2 ©y 2 - b 0 ©y - b 1 ©xyassumed equal.s e = or Bt = 1x n - 2 B n - 21 - x 2 2 - 1m 1 - m 2 2(df n 1 n 2 2)yN - E 6 y 6 yN + E Prediction intervalsp2 + s2 pBn sp 2 = 1n 1 - 12s 2 1 + 1n 2 - 12s 2 21 n 2where E = t a>2 s e 1 + 1 n 1 + n 2 - 2B n + n1x 0 - x2 2n1©x 2 2 - 1©x2 2Two means—independent; s 1<strong>and</strong> s 2unknown, butassumed equal.Ch. 12: One-Way Analysis of VarianceTwo means—independent;z = 1x 1 - x 2 2 - 1m 1 - m 2 2Procedure for testing H 0 : m 1 = m 2 = m 3 = Á 1, 2known.s12B n+ s 22 1. Use software or calculator to obtain results.1 n 2 2. Identify the P-value.Two means—matched pairst = d - m 3. Form conclusion:dIf P-value a, reject the null hypothesiss (df n 1)d > 1nof equal means.If P-value a, fail to reject the null hypothesisof equal means.‹ ‹‹Ch. 12: Two-Way Analysis of VarianceProcedure:1. Use software or a calculator to obtain results.2. Test H 0: There is no interaction between the row factor <strong>and</strong>column factor.3. Stop if H 0from Step 2 is rejected.If H 0from Step 2 is not rejected (so there does not appear tobe an interaction effect), proceed with these two tests:Test for effects from the row factor.Test for effects from the column factor.
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