Formulas and Tables by Mario F. Triola

Transcription

1 Copyright 010 Pearson Education, Inc. Ch. 3: Descriptive Statistics x f # x x f Mean 1x - x s - 1 n 1 x - 1 x s 1n - 1 s B variance s Ch. 4: Probability Mean (frequency table) Standard deviation 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 np r 1n - r! Permutations (no elements alike) Permutations (n 1 alike, Á ) n 1! n!... n k! nc r x n Combinations Ch. 5: Probability Distributions m x # P1x m n # p n 3 1f # x 4-3 1f # x4 1n - r! r! s n # p # q s n # p # q P 1x mx # e -m x! n 1n - 1 Mean (prob. dist.) s 3x # P1x4 - m P 1x 1n - x! x! # p x # q n - x Ch. 6: Normal Distribution Standard deviation (shortcut) Standard deviation (prob. dist.) Binomial probability Mean (binomial) Variance (binomial) z x - x x - m or Standard score s s m x m Central limit theorem Standard deviation (binomial) Poisson distribution where e.7188 Standard deviation (frequency table) Ch. 7: Confidence Intervals (one population) ˆp E p ˆp E Proportion pnqn where E z a> 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 1 n 1x 1 - x - E 6 1m 1 - m 6 1x 1 - x + E where E t a> Bn + s 1 n (s 1 and s unknown and not assumed equal) s p E t a> + s p 1df n Bn 1 n 1 + n - s p 1n 1-1s 1 + 1n - 1s 1n n - 1 (s 1 and s unknown but assumed equal) s1 E z a> + s 1 n (s 1, s known) s 1 d - E 6 m d 6 d + E x L (df smaller of n 1 1, n 1) (Matched pairs) (Indep.) s x s n Central limit theorem (Standard error) s d where E t a> (df n 1) 1n

2 Copyright 010 Pearson Education, Inc. Ch. 8: Test Statistics (one population) z pn - p 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 + 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 + 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 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 Copyright 010 Pearson Education, Inc. Ch. 13: Nonparametric Tests 1x n> z 1n> z H B T - n 1n + 1>4 n 1n + 11n + 1 z R - m R s R 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 Ch. 14: Control Charts 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 Rank correlation p chart: Plot sample proportions UCL: p + 3 Centerline: p 4 LCL: p - 3 B R - n 11n 1 + n + 1 B n 1 n 1n 1 + n + 1 n 1 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 Table entry for p and C is the point t* with probability p lying above it and probability C lying between t * and t*. Probability p t* Table B t distribution critical values Tail probability p df % 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% Confidence level C $ 6

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: z score Area Common Critical Values Confidence Critical Level Value

7 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