Important Formulas. Expectation: E (X) = Σ [X P(X)] = n p q σ = n p q. P(X) = n! X1! X 2! X 3! X k! p X. Chapter 6 The Normal Distribution.

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Importat Formulas Chapter 3 Data Descriptio Mea for idividual data: X = _ ΣX Mea for grouped data: X= _ Σf X m Stadard deviatio for a sample: _ s = Σ(X _ X ) or s = 1 (Σ X ) (Σ X ) ( 1) Stadard deviatio for grouped data: _ s = _ (Σf Xm ) (Σf Xm ) ( 1) rage Rage rule of thumb: s _ 4 (Shortcut formula) Chapter 4 Probability ad Coutig Rules Additio rule 1 (mutually exclusive evets): P(A or B ) = P(A) + P(B) Additio rule (evets ot mutually exclusive): P(A or B) = P(A) + P(B ) P(A ad B) Multiplicatio rule 1 (idepedet evets): P(A ad B) = P(A) P(B) Multiplicatio rule (depedet evets): P(A ad B) = P(A) P(B A) Coditioal probability: P(B A) = P(A ad B) P(A) Complemetary evets: P( E ) = 1 P(E ) Fudametal coutig rule: Total umber of outcomes of a sequece whe each evet has a differet umber of possibilities: k 1 k k 3 k Permutatio rule: Number of permutatios of objects takig r at a time is P r = _! ( r)! Permutatio rule of objects with r 1 objects idetical, r objects idetical, etc.! r1! r! r p! Combiatio rule: Number of combiatios of r objects selected from objects is C r = _! ( r)!r! Chapter 5 Discrete Probability Distributios Mea for a probability distributio: μ = Σ [X P(X)] Variace ad stadard deviatio for a probability distributio: σ = Σ [X P(X)] μ _ σ= Σ [X P(X)] μ Expectatio: E (X) = Σ [X P(X)] Biomial probability: P(X ) = _! ( X )!X! px q X Mea for biomial distributio: μ = p Variace ad stadard deviatio for the biomial distributio: σ _ = p q σ = p q Multiomial probability: P(X) = _! X1! X! X 3! X k! p X 1 1 X p X p 3 3 X p k k Poisso probability: P(X; λ) = e λ λ X X! X = 0, 1,,... Hypergeometric probability: P(X ) = ac X b C X a+bc Geometric probability: P() = p(1 p) 1 is the umber of the trial i which the first success occurs ad p is the probability of a success Chapter 6 The Normal Distributio Stadard score: Populatio: z = _ X μ σ or Sample: z = X X s Mea of sample meas: μ _ X = μ Stadard error of the mea: σ _ X = σ X μ Cetral limit theorem formula: z = σ Chapter 7 Cofidece Itervals ad Sample Size z cofidece iterval for meas: X zα ( σ ) < μ < X + z α ( t cofidece iterval for meas: σ ) X t α ( s ) < μ < X+ t α ( s ) Sample size for meas: = ( z α σ E ) E is the margi of error Cofidece iterval for a proportio: ˆp ˆq ˆp (z α ) < p < ˆp + (z ˆp ˆq α )

z α Sample size for a proportio: = p ˆ q ˆ ( _ E ) p ˆ = X ad ˆ q = 1 p ˆ Cofidece iterval for variace: ( 1)s < σ ( 1)s < χ right χ left Cofidece iterval for stadard deviatio: ( 1)s < σ < ( 1)s χ right χ left Chapter 8 Hypothesis Testig X μ z test: z = for ay value. If < 30, σ populatio must be ormally distributed. t test: t = X μ (d.f. = 1) s p p z test for proportios: z = pq Chi-square test for a sigle variace: ( 1)s = (d.f. = 1) ˆ Chapter 9 Testig the Differece Betwee Two Meas, Two Proportios, ad Two Variaces z test for comparig two meas (idepedet samples): z = ( X 1 _ X ) (μ 1 μ ) _ σ 1 + _ σ 1 Formula for the cofidece iterval for differece of two meas (large samples): ( X 1 X ) z α _ σ 1 1 + _ σ < μ 1 μ σ < ( X 1 X ) + z α _ σ 1 1 + _ σ Formula for the cofidece iterval for differece of two meas (small idepedet samples, variaces uequal): _ ( X 1 X ) t α s 1 + s 1 < μ 1 μ _ < ( X 1 X ) + t α s 1 + s 1 (d.f. = smaller of 1 1 ad 1) t test for comparig two meas for depedet samples: D μ t = D s D D = _ ΣD s D = ΣD (ΣD) (d.f. = 1) ( 1) ad Formula for cofidece iterval for the mea of the differece for depedet samples: s D t α D < μ D < D + t α s D (d.f. = 1) z test for comparig two proportios: ( p ˆ z = 1 p ˆ ) (p 1 p ) _ p _ q ( 1 + 1 1 ) p = _ X 1 + X 1 + p ˆ 1 = _ X 1 1 X _ q = 1 p p ˆ = _ Formula for the cofidece iterval for the differece of two proportios: _ p ˆ ( p ˆ 1 p ˆ ) z α 1 q ˆ 1 p ˆ + ˆ q 1 < p 1 p _ p ˆ < ( p ˆ 1 p ˆ ) + z α 1 q ˆ 1 p ˆ + ˆ q 1 F test for comparig two variaces: F = s 1 s 1 is the s larger variace ad d.f.n. = 1 1, d.f.d. = 1 t test for comparig two meas (idepedet samples, variaces ot equal): t = ( X 1 _ X ) (μ 1 μ ) _ s 1 1 + _ s (d.f. = the smaller of 1 1 or 1)

Chapter 10 Correlatio ad Regressio Correlatio coefficiet: (Σ xy) (Σ x)(σy) r = [(Σ x ) (Σ x) ][(Σ y ) (Σ y) ] t test for correlatio coefficiet: t = r 1 r (d.f. = ) The regressio lie equatio: yʹ = a + bx (Σy)(Σx a = _ ) (Σx)(Σxy) (Σx ) (Σx) (Σxy) (Σx)(Σy) b = _ (Σx ) (Σx) Coefficiet of determiatio: r explaied variatio = total variatio Stadard error of estimate: _ s est = Σy a Σy b Σxy Predictio iterval for y: yʹ t α s est 1 + 1 + (x _ X ) Σx (Σx) < y < yʹ + t α s est 1 + 1 + (x _ X ) Σx (Σx) (d.f. = ) Formula for the multiple correlatio coefficiet: R = _ ryx 1 + ryx r yx1 r yx r x1 x 1 rx 1 x Formula for the F test for the multiple correlatio coefficiet: F = _ R k (1 R ) ( k 1) (d.f.n. = k ad d.f.d. = k 1) Formula for the adjusted R : R adj = 1 [ _ (1 R )( 1) k 1 ] Chapter 11 Other Chi-Square Tests Chi-square test for goodess-of-fit: = (O E ) E (d.f. = o. of categories 1) Chi-square test for idepedece ad homogeeity of proportios: = (O E ) E [d.f. = (rows 1)(colums 1)] Chapter 1 Aalysis of Variace ANOVA test: F = _ s B X GM = _ ΣX s W N d.f.n. = k 1 N = 1 + + + k d.f.d. = N k k = umber of groups s Σ B = i ( X i X GM ) k 1 s W = _ Σ( i 1)s i Σ( i 1) ( X i X j ) Scheffé test: F S = _ ad sw (1 i + 1 j ) Fʹ = (k 1)(C.V.) X i X j Tukey test: q = s W Formulas for two-way ANOVA: SS MS A = _ A a 1 F A = _ MS A MS W SS MS B = _ B b 1 F B = _ MS B MS W MS A B = _ SS A B F A B = MS A B (a 1)(b 1) MS W SS MS W = _ W ab( 1)

Chapter 13 Noparametric Statistics (X + 0.5) 0.5 z test value i the sig test: z = = sample size (greater tha or equal to 6) X = smaller umber of + or sigs R μ Wilcoxo rak sum test: z = R σ R μ R = _ 1 ( 1 + + 1) _ σ R = _ 1 ( 1 + + 1) 1 R = sum of the raks for the smaller sample size ( 1 ) 1 = smaller of the sample sizes = larger of the sample sizes 1 10 ad 10 w s _ ( + 1) Wilcoxo siged-rak test: z = 4 ( + 1)( + 1) 4 = umber of pairs the differece is ot 0 ad 30 w s = smaller sum i absolute value of the siged raks Kruskal-Wallis test: H = _ 1 N(N + 1) ( _ R 1 + _ R 1 + + _ R k k ) 3(N + 1) R 1 = sum of the raks of sample 1 1 = size of sample 1 R = sum of the raks of sample = size of sample R k = sum of the raks of sample k k = size of sample k N = 1 + + + k k = umber of samples Spearma rak correlatio coefficiet: r s = 1 6Σd ( 1) d = differece i the raks = umber of data pairs Procedure Table Solvig Hypothesis-Testig Problems (Traditioal Method) Step 1 State the hypotheses ad idetify the claim. Step Fid the critical value(s) from the appropriate table i Appedix A. Step 3 Compute the test value. Step 4 Make the decisio to reject or ot reject the ull hypothesis. Step 5 Summarize the results. Procedure Table Solvig Hypothesis-Testig Problems (P-Value Method) Step 1 State the hypotheses ad idetify the claim. Step Compute the test value. Step 3 Fid the P-value. Step 4 Make the decisio. Step 5 Summarize the results.

TABLE E The Stadard Normal Distributio Cumulative Stadard Normal Distributio z.00.01.0.03.04.05.06.07.08.09 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 3.1.0010.0009.0009.0009.0008.0008.0008.0008.0007.0007 3.0.0013.0013.0013.001.001.0011.0011.0011.0010.0010.9.0019.0018.0018.0017.0016.0016.0015.0015.0014.0014.8.006.005.004.003.003.00.001.001.000.0019.7.0035.0034.0033.003.0031.0030.009.008.007.006.6.0047.0045.0044.0043.0041.0040.0039.0038.0037.0036.5.006.0060.0059.0057.0055.0054.005.0051.0049.0048.4.008.0080.0078.0075.0073.0071.0069.0068.0066.0064.3.0107.0104.010.0099.0096.0094.0091.0089.0087.0084..0139.0136.013.019.015.01.0119.0116.0113.0110.1.0179.0174.0170.0166.016.0158.0154.0150.0146.0143.0.08.0.017.01.007.00.0197.019.0188.0183 1.9.087.081.074.068.06.056.050.044.039.033 1.8.0359.0351.0344.0336.039.03.0314.0307.0301.094 1.7.0446.0436.047.0418.0409.0401.039.0384.0375.0367 1.6.0548.0537.056.0516.0505.0495.0485.0475.0465.0455 1.5.0668.0655.0643.0630.0618.0606.0594.058.0571.0559 1.4.0808.0793.0778.0764.0749.0735.071.0708.0694.0681 1.3.0968.0951.0934.0918.0901.0885.0869.0853.0838.083 1..1151.1131.111.1093.1075.1056.1038.100.1003.0985 1.1.1357.1335.1314.19.171.151.130.110.1190.1170 1.0.1587.156.1539.1515.149.1469.1446.143.1401.1379 0.9.1841.1814.1788.176.1736.1711.1685.1660.1635.1611 0.8.119.090.061.033.005.1977.1949.19.1894.1867 0.7.40.389.358.37.96.66.36.06.177.148 0.6.743.709.676.643.611.578.546.514.483.451 0.5.3085.3050.3015.981.946.91.877.843.810.776 0.4.3446.3409.337.3336.3300.364.38.319.3156.311 0.3.381.3783.3745.3707.3669.363.3594.3557.350.3483 0..407.4168.419.4090.405.4013.3974.3936.3897.3859 0.1.460.456.45.4483.4443.4404.4364.435.486.447 0.0.5000.4960.490.4880.4840.4801.4761.471.4681.4641 For z values less tha 3.49, use 0.0001. z 0

TABLE E (cotiued ) Cumulative Stadard Normal Distributio z.00.01.0.03.04.05.06.07.08.09 0.0.5000.5040.5080.510.5160.5199.539.579.5319.5359 0.1.5398.5438.5478.5517.5557.5596.5636.5675.5714.5753 0..5793.583.5871.5910.5948.5987.606.6064.6103.6141 0.3.6179.617.655.693.6331.6368.6406.6443.6480.6517 0.4.6554.6591.668.6664.6700.6736.677.6808.6844.6879 0.5.6915.6950.6985.7019.7054.7088.713.7157.7190.74 0.6.757.791.734.7357.7389.74.7454.7486.7517.7549 0.7.7580.7611.764.7673.7704.7734.7764.7794.783.785 0.8.7881.7910.7939.7967.7995.803.8051.8078.8106.8133 0.9.8159.8186.81.838.864.889.8315.8340.8365.8389 1.0.8413.8438.8461.8485.8508.8531.8554.8577.8599.861 1.1.8643.8665.8686.8708.879.8749.8770.8790.8810.8830 1..8849.8869.8888.8907.895.8944.896.8980.8997.9015 1.3.903.9049.9066.908.9099.9115.9131.9147.916.9177 1.4.919.907.9.936.951.965.979.99.9306.9319 1.5.933.9345.9357.9370.938.9394.9406.9418.949.9441 1.6.945.9463.9474.9484.9495.9505.9515.955.9535.9545 1.7.9554.9564.9573.958.9591.9599.9608.9616.965.9633 1.8.9641.9649.9656.9664.9671.9678.9686.9693.9699.9706 1.9.9713.9719.976.973.9738.9744.9750.9756.9761.9767.0.977.9778.9783.9788.9793.9798.9803.9808.981.9817.1.981.986.9830.9834.9838.984.9846.9850.9854.9857..9861.9864.9868.9871.9875.9878.9881.9884.9887.9890.3.9893.9896.9898.9901.9904.9906.9909.9911.9913.9916.4.9918.990.99.995.997.999.9931.993.9934.9936.5.9938.9940.9941.9943.9945.9946.9948.9949.9951.995.6.9953.9955.9956.9957.9959.9960.9961.996.9963.9964.7.9965.9966.9967.9968.9969.9970.9971.997.9973.9974.8.9974.9975.9976.9977.9977.9978.9979.9979.9980.9981.9.9981.998.998.9983.9984.9984.9985.9985.9986.9986 3.0.9987.9987.9987.9988.9988.9989.9989.9989.9990.9990 3.1.9990.9991.9991.9991.999.999.999.999.9993.9993 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 For z values greater tha 3.49, use 0.9999. 0 z

TABLE F The t Distributio Cofidece itervals 80% 90% 95% 98% 99% Oe tail, α 0.10 0.05 0.05 0.01 0.005 d.f. Two tails, α 0.0 0.10 0.05 0.0 0.01 1 3.078 6.314 1.706 31.81 63.657 1.886.90 4.303 6.965 9.95 3 1.638.353 3.18 4.541 5.841 4 1.533.13.776 3.747 4.604 5 1.476.015.571 3.365 4.03 6 1.440 1.943.447 3.143 3.707 7 1.415 1.895.365.998 3.499 8 1.397 1.860.306.896 3.355 9 1.383 1.833.6.81 3.50 10 1.37 1.81.8.764 3.169 11 1.363 1.796.01.718 3.106 1 1.356 1.78.179.681 3.055 13 1.350 1.771.160.650 3.01 14 1.345 1.761.145.64.977 15 1.341 1.753.131.60.947 16 1.337 1.746.10.583.91 17 1.333 1.740.110.567.898 18 1.330 1.734.101.55.878 19 1.38 1.79.093.539.861 0 1.35 1.75.086.58.845 1 1.33 1.71.080.518.831 1.31 1.717.074.508.819 3 1.319 1.714.069.500.807 4 1.318 1.711.064.49.797 5 1.316 1.708.060.485.787 6 1.315 1.706.056.479.779 7 1.314 1.703.05.473.771 8 1.313 1.701.048.467.763 9 1.311 1.699.045.46.756 30 1.310 1.697.04.457.750 3 1.309 1.694.037.449.738 34 1.307 1.691.03.441.78 36 1.306 1.688.08.434.719 38 1.304 1.686.04.49.71 40 1.303 1.684.01.43.704 45 1.301 1.679.014.41.690 50 1.99 1.676.009.403.678 55 1.97 1.673.004.396.668 60 1.96 1.671.000.390.660 65 1.95 1.669 1.997.385.654 70 1.94 1.667 1.994.381.648 75 1.93 1.665 1.99.377.643 80 1.9 1.664 1.990.374.639 90 1.91 1.66 1.987.368.63 100 1.90 1.660 1.984.364.66 00 1.86 1.653 1.97.345.601 300 1.84 1.650 1.968.339.59 400 1.84 1.649 1.966.336.588 500 1.83 1.648 1.965.334.586 (z) 1.8 a 1.645 b 1.960.36 c.576 d a This value has bee rouded to 1.8 i the textbook. b This value has bee rouded to 1.65 i the textbook. c This value has bee rouded to.33 i the textbook. d This value has bee rouded to.58 i the textbook. Oe tail α Two tails α α t t +t

TABLE G The Chi-Square Distributio α Degrees of freedom 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.05 0.01 0.005 1 0.001 0.004 0.016.706 3.841 5.04 6.635 7.879 0.010 0.00 0.051 0.103 0.11 4.605 5.991 7.378 9.10 10.597 3 0.07 0.115 0.16 0.35 0.584 6.51 7.815 9.348 11.345 1.838 4 0.07 0.97 0.484 0.711 1.064 7.779 9.488 11.143 13.77 14.860 5 0.41 0.554 0.831 1.145 1.610 9.36 11.071 1.833 15.086 16.750 6 0.676 0.87 1.37 1.635.04 10.645 1.59 14.449 16.81 18.548 7 0.989 1.39 1.690.167.833 1.017 14.067 16.013 18.475 0.78 8 1.344 1.646.180.733 3.490 13.36 15.507 17.535 0.090 1.955 9 1.735.088.700 3.35 4.168 14.684 16.919 19.03 1.666 3.589 10.156.558 3.47 3.940 4.865 15.987 18.307 0.483 3.09 5.188 11.603 3.053 3.816 4.575 5.578 17.75 19.675 1.90 4.75 6.757 1 3.074 3.571 4.404 5.6 6.304 18.549 1.06 3.337 6.17 8.99 13 3.565 4.107 5.009 5.89 7.04 19.81.36 4.736 7.688 9.819 14 4.075 4.660 5.69 6.571 7.790 1.064 3.685 6.119 9.141 31.319 15 4.601 5.9 6.6 7.61 8.547.307 4.996 7.488 30.578 3.801 16 5.14 5.81 6.908 7.96 9.31 3.54 6.96 8.845 3.000 34.67 17 5.697 6.408 7.564 8.67 10.085 4.769 7.587 30.191 33.409 35.718 18 6.65 7.015 8.31 9.390 10.865 5.989 8.869 31.56 34.805 37.156 19 6.844 7.633 8.907 10.117 11.651 7.04 30.144 3.85 36.191 38.58 0 7.434 8.60 9.591 10.851 1.443 8.41 31.410 34.170 37.566 39.997 1 8.034 8.897 10.83 11.591 13.40 9.615 3.671 35.479 38.93 41.401 8.643 9.54 10.98 1.338 14.04 30.813 33.94 36.781 40.89 4.796 3 9.6 10.196 11.689 13.091 14.848 3.007 35.17 38.076 41.638 44.181 4 9.886 10.856 1.401 13.848 15.659 33.196 36.415 39.364 4.980 45.559 5 10.50 11.54 13.10 14.611 16.473 34.38 37.65 40.646 44.314 46.98 6 11.160 1.198 13.844 15.379 17.9 35.563 38.885 41.93 45.64 48.90 7 11.808 1.879 14.573 16.151 18.114 36.741 40.113 43.194 46.963 49.645 8 1.461 13.565 15.308 16.98 18.939 37.916 41.337 44.461 48.78 50.993 9 13.11 14.57 16.047 17.708 19.768 39.087 4.557 45.7 49.588 5.336 30 13.787 14.954 16.791 18.493 0.599 40.56 43.773 46.979 50.89 53.67 Source: values calculated with Excel. α χ

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