Formulas and Tables by Mario F. Triola

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Formulas and Tables by Mario F. Triola Copyright 010 Pearson Education, Inc. Ch. 3: Descriptive Statistics x Mean f # x x f Mean (frequency table) 1x - x s B n - 1 Standard deviation n 1 x - 1 x Standard deviation s B n 1n - 1 (shortcut) s B variance s Ch. 4: Probability P1A or B = P1A + P1B if A, B are mutually exclusive P1A or B = P1A + P1B - P1A and B if A, B are not mutually exclusive P1A and B = P1A # P1B if A, B are independent P1A and B = P1A # P1B ƒa if A, B are dependent P1A = 1 - P1A Rule of complements n! np r = 1n - r! n! Permutations (no elements alike) Permutations (n 1 alike, Á ) n 1! n!... n k! nc r = x n n 3 1f # x 4-3 1f # x4 n! 1n - r! r! n 1n - 1 Combinations Ch. 5: Probability Distributions m = x # P1x Mean (prob. dist.) s = 3x # P1x4 - m Standard deviation (prob. dist.) n! P 1x = # p x # q n - x Binomial probability 1n - x! x! m = n # p Mean (binomial) s = n # p # q Variance (binomial) s = n # p # q Standard deviation (binomial) # Poisson distribution P 1x = mx e -m where e.7188 x! Ch. 6: Normal Distribution z = x - x x - m or Standard score s s m x = m Central limit theorem Standard deviation (frequency table) Ch. 7: Confidence Intervals (one population) ˆp E p ˆp E Proportion pnqn where E = z a> B n x - E 6 m 6 x + E Mean s where E = z a> 1n (s known) s or E = t a> 1n (s unknown) 1n - 1s 1n - 6 s 1s 6 x R Ch. 7: Sample Size Determination 3z a>4. 0.5 n = 3z a>4 pnqn n = E E n = B z a>s E R Proportion Variance Proportion (ˆp and ˆq are known) Mean Ch. 9: Confidence Intervals (two populations) 1pN 1 - pn - E 6 1p 1 - p 6 1pN 1 - pn + E pn 1 qn 1 where E = z a> + pn qn B n 1 n 1x 1 - x - E 6 1m 1 - m 6 1x 1 - x + E s 1 where E = t a> Bn + s (df smaller of 1 n n 1 1, n 1) (s 1 and s unknown and not assumed equal) s p E = t a> + s p 1df = n Bn 1 n 1 + n - sp = 1n 1-1s 1 + 1n - 1s 1n 1-1 + 1n - 1 (s 1 and s unknown but assumed equal) s1 E = z a> + s B n 1 n (s 1, s known) d - E 6 m d 6 d + E x L (Matched pairs) (Indep.) s x = s n Central limit theorem (Standard error) s d where E = t a> (df n 1) 1n

Formulas and Tables by Mario F. Triola Copyright 010 Pearson Education, Inc. Ch. 8: Test Statistics (one population) z = pn - p pq B n z = x - m s> 1n t = x - m s> 1n 1n - x 1s = Proportion one population Ch. 9: Test Statistics (two populations) z = 1pN 1 - pn - 1p 1 - p Two proportions pq B n + pq 1 n p = x 1 + x n 1 + n t = 1x 1 - x - 1m 1 - m df smaller of s 1 + s Bn 1 n 1 1, n 1 F = s 1 s s Two means independent; s 1 and s unknown, and not assumed equal. t = 1x 1 - x - 1m 1 - m Two means independent; s 1 and s unknown, but assumed equal. z = 1x 1 - x - 1m 1 - m t = d - m d s d > 1n s p + s Bn 1 s 1 B n + s 1 n Standard deviation or variance two populations (where s 1 s ) Ch. 11: Goodness-of-Fit and Contingency Tables 1O - x E = g E n p n 1O - Contingency table x E = g [df (r 1)(c 1)] E 1row total1column total where E = 1grand total 1ƒb - c ƒ - x 1 = b + c Mean one population ( known) Mean one population ( unknown) Standard deviation or variance one population Two means matched pairs (df n 1) Goodness-of-fit (df k 1) (df n 1 n ) s p = 1n 1-1s 1 + 1n - 1s n 1 + n - Two means independent; 1, known. McNemar s test for matched pairs (df 1) Ch. 10: Linear Correlation/Regression n xy - 1 x1 y Correlation r = n1 x - 1 x n1 y - 1 y Slope: y-intercept: yn = b 0 + b 1 x or r = a Az x z y B n - 1 b 0 = y - b 1 x or b 0 = 1 y1 x - 1 x1 xy n 1 x - 1 x explained variation r = total variation s e = B 1y - yn n - yn - E 6 y 6 yn + E n xy - 1 x1 y b 1 = n 1 x - 1 x or b 1 = r Estimated eq. of regression line or B y - b 0 y - b 1 xy n - Prediction interval where E = t a> s e 1 + 1 B n + n1x 0 - x n1 x - 1 x Ch. 1: One-Way Analysis of Variance Procedure for testing H 0 : m 1 = m = m 3 = Á 1. Use software or calculator to obtain results.. Identify the P-value. 3. Form conclusion: If P-value a, reject the null hypothesis of equal means. If P-value a, fail to reject the null hypothesis of equal means. Ch. 1: Two-Way Analysis of Variance where z x = z score for x z y = z score for y Procedure: 1. Use software or a calculator to obtain results.. Test H 0 : There is no interaction between the row factor and column factor. 3. Stop if H 0 from Step is rejected. If H 0 from Step is not rejected (so there does not appear to be an interaction effect), proceed with these two tests: Test for effects from the row factor. Test for effects from the column factor. s y s x

Formulas and Tables by Mario F. Triola Copyright 010 Pearson Education, Inc. Ch. 13: Nonparametric Tests 1x + 0.5-1n> z = 1n> z = z = R - m R s R = H = B T - n 1n + 1>4 n 1n + 11n + 1 Sign test for n 5 1 N1N + 1 a R 1 + R +... + R k b - 31N + 1 n 1 n n k Kruskal-Wallis (chi-square df k 1) 6 d r s = 1 - n1n - 1 acritical value for n 7 30: z = G - m G s G = 4 Ch. 14: Control Charts R - n 11n 1 + n + 1 B R chart: Plot sample ranges UCL: D 4 R Centerline: R LCL: D 3 R x chart: Plot sample means UCL: xx + A R Centerline: xx LCL: xx - A R n 1 n 1n 1 + n + 1 1 Rank correlation p chart: Plot sample proportions pq UCL: p + 3 B n Centerline: p pq LCL: p - 3 B n Wilcoxon signed ranks (matched pairs and n 30) ; z 1n - 1 b G - a n 1n n 1 + n + 1b 1n 1 n 1n 1 n - n 1 - n B 1n 1 + n 1n 1 + n - 1 Wilcoxon rank-sum (two independent samples) Runs test for n 0 TABLE A-6 Critical Values of the Pearson Correlation Coefficient r n a =.05 a =.01 4.950.990 5.878.959 6.811.917 7.754.875 8.707.834 9.666.798 10.63.765 11.60.735 1.576.708 13.553.684 14.53.661 15.514.641 16.497.63 17.48.606 18.468.590 19.456.575 0.444.561 5.396.505 30.361.463 35.335.430 40.31.40 45.94.378 50.79.361 60.54.330 70.36.305 80.0.86 90.07.69 100.196.56 NOTE: To test H 0 : r = 0 against H 1 : r Z 0, reject H 0 if the absolute value of r is greater than the critical value in the table. Control Chart Constants Subgroup Size n A D 3 D 4 1.880 0.000 3.67 3 1.03 0.000.574 4 0.79 0.000.8 5 0.577 0.000.114 6 0.483 0.000.004 7 0.419 0.076 1.94

General considerations Context of the data Source of the data Sampling method Measures of center Measures of variation Nature of distribution Outliers Changes over time Conclusions Practical implications FINDING P-VALUES HYPOTHESIS TEST: WORDING OF FINAL CONCLUSION Inferences about M: choosing between t and normal distributions t distribution: s not known and normally distributed population or s not known and n 30 Normal distribution: s known and normally distributed population or s known and n 30 Nonparametric method or bootstrapping: Population not normally distributed and n 30

NEGATIVE z Scores z 0 TABLE A- Standard Normal (z) Distribution: Cumulative Area from the LEFT z.00.01.0.03.04.05.06.07.08.09-3.50 and lower.0001-3.4.0003.0003.0003.0003.0003.0003.0003.0003.0003.000-3.3.0005.0005.0005.0004.0004.0004.0004.0004.0004.0003-3..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 NOTE: For values of z below -3.49, use 0.0001 for the area. *Use these common values that result from interpolation: z score Area -1.645 0.0500 -.575 0.0050

0 z POSITIVE z Scores TABLE A- (continued ) Cumulative Area from the LEFT 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 3.50.9999 and up NOTE: For values of z above 3.49, use 0.9999 for the area. *Use these common values that result from interpolation: Common Critical Values Confidence Critical z score Area Level Value 1.645 0.9500 0.90 1.645.575 0.9950 0.95 1.96 0.99.575

TABLE A-3 t Distribution: Critical t Values Area in One Tail 0.005 0.01 0.05 0.05 0.10 Degrees of Area in Two Tails Freedom 0.01 0.0 0.05 0.10 0.0 1 63.657 31.81 1.706 6.314 3.078 9.95 6.965 4.303.90 1.886 3 5.841 4.541 3.18.353 1.638 4 4.604 3.747.776.13 1.533 5 4.03 3.365.571.015 1.476 6 3.707 3.143.447 1.943 1.440 7 3.499.998.365 1.895 1.415 8 3.355.896.306 1.860 1.397 9 3.50.81.6 1.833 1.383 10 3.169.764.8 1.81 1.37 11 3.106.718.01 1.796 1.363 1 3.055.681.179 1.78 1.356 13 3.01.650.160 1.771 1.350 14.977.64.145 1.761 1.345 15.947.60.131 1.753 1.341 16.91.583.10 1.746 1.337 17.898.567.110 1.740 1.333 18.878.55.101 1.734 1.330 19.861.539.093 1.79 1.38 0.845.58.086 1.75 1.35 1.831.518.080 1.71 1.33.819.508.074 1.717 1.31 3.807.500.069 1.714 1.319 4.797.49.064 1.711 1.318 5.787.485.060 1.708 1.316 6.779.479.056 1.706 1.315 7.771.473.05 1.703 1.314 8.763.467.048 1.701 1.313 9.756.46.045 1.699 1.311 30.750.457.04 1.697 1.310 31.744.453.040 1.696 1.309 3.738.449.037 1.694 1.309 33.733.445.035 1.69 1.308 34.78.441.03 1.691 1.307 35.74.438.030 1.690 1.306 36.719.434.08 1.688 1.306 37.715.431.06 1.687 1.305 38.71.49.04 1.686 1.304 39.708.46.03 1.685 1.304 40.704.43.01 1.684 1.303 45.690.41.014 1.679 1.301 50.678.403.009 1.676 1.99 60.660.390.000 1.671 1.96 70.648.381 1.994 1.667 1.94 80.639.374 1.990 1.664 1.9 90.63.368 1.987 1.66 1.91 100.66.364 1.984 1.660 1.90 00.601.345 1.97 1.653 1.86 300.59.339 1.968 1.650 1.84 400.588.336 1.966 1.649 1.84 500.586.334 1.965 1.648 1.83 1000.581.330 1.96 1.646 1.8 000.578.38 1.961 1.646 1.8 Large.576.36 1.960 1.645 1.8

Formulas and Tables by Mario F. Triola Copyright 010 Pearson Education, Inc. x TABLE A-4 Chi-Square ( ) Distribution Area to the Right of the Critical Value 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 9.60 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 40 0.707.164 4.433 6.509 9.051 51.805 55.758 59.34 63.691 66.766 50 7.991 9.707 3.357 34.764 37.689 63.167 67.505 71.40 76.154 79.490 60 35.534 37.485 40.48 43.188 46.459 74.397 79.08 83.98 88.379 91.95 70 43.75 45.44 48.758 51.739 55.39 85.57 90.531 95.03 100.45 104.15 80 51.17 53.540 57.153 60.391 64.78 96.578 101.879 106.69 11.39 116.31 90 59.196 61.754 65.647 69.16 73.91 107.565 113.145 118.136 14.116 18.99 100 67.38 70.065 74. 77.99 8.358 118.498 14.34 19.561 135.807 140.169 From Donald B. Owen, Handbook of Statistical Tables, 196 Addison-Wesley Publishing Co., Reading, MA. Reprinted with permission of the publisher. Degrees of Freedom n - 1 for confidence intervals or hypothesis tests with a standard deviation or variance k - 1 for goodness-of-fit with k categories (r - 1)(c - 1) for contingency tables with r rows and c columns k - 1 for Kruskal-Wallis test with k samples

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