Formulas and Tables by Mario F. Triola

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1 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> 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 n - 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

2 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 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

3 Formulas and Tables by Mario F. Triola Copyright 010 Pearson Education, Inc. Ch. 13: Nonparametric Tests 1x n> 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 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 = 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 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

5 NEGATIVE z Scores z 0 TABLE A- Standard Normal (z) Distribution: Cumulative Area from the LEFT z and lower * * NOTE: For values of z below -3.49, use for the area. *Use these common values that result from interpolation: z score Area

6 0 z POSITIVE z Scores TABLE A- (continued ) Cumulative Area from the LEFT z * * and up NOTE: For values of z above 3.49, use for the area. *Use these common values that result from interpolation: Common Critical Values Confidence Critical z score Area Level Value

7 TABLE A-3 t Distribution: Critical t Values Area in One Tail Degrees of Area in Two Tails Freedom Large

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 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