This page intentionally left blank Important Formulas Chapter 3 Data Description Chapter 5 Discrete Probability Distributions X Mean for a probability distribution: m [X P(X)] Mean for individual data: X n Variance and standard deviation for a probability distribution: f • Xm Mean for grouped data: X s2 [X 2 P(X)] m2 n Standard deviation for a sample: s [X 2 • PX ] m2 Expectation: E(X) [X P(X)] X X 2 n X 2 X 2 n! s or s Binomial probability: PX • pX • q nX n1 nn 1 n X !X! (Shortcut formula) Mean for binomial distribution: m n p Standard deviation for grouped data: Variance and standard deviation for the binomial distribution: s2 n p q s n • p • q n f • X m2 f • Xm 2 Multinomial probability: s nn 1 n! PX • p X 1 • p2X 2 • p3X 3 • • • pkX k range X1!X2!X3!. Xk! 1 Range rule of thumb: s 4 e X Poisson probability: P(X; l) where X! Chapter 4 Probability and Counting Rules X 0, 1, 2,. C • C Addition rule 1 (mutually exclusive events): Hypergeometric probability: PX a X b nX abCn P(A or B) P(A) P(B) Chapter 6 The Normal Distribution Addition rule 2 (events not mutually exclusive): P(A or B) P(A) P(B) P(A and B) X XX Standard score z or z s Multiplication rule 1 (independent events): Mean of sample means: mX m P(A and B) P(A) P(B) Standard error of the mean: sX Multiplication rule 2 (dependent events): n X P(A and B) P(A) P(B A) Central limit theorem formula: z n P A and B Conditional probability: PB A Chapter 7 Confidence Intervals and Sample P A Size Complementary events: P(E ) 1 P(E) z confidence interval for means: Fundamental counting rule: Total number of outcomes of a sequence when each event has a different X z 2 n X z 2 n number of possibilities: k 1 k 2 k 3 k n t confidence interval for means: Permutation rule: Number of permutations of n objects taking r at a time is n Pr n! X t 2 s n X t 2 s n n r ! z 2 • 2 Combination rule: Number of combinations of r objects Sample size for means: n E where E is the n! maximum error of estimate selected from n objects is n Cr n r !r! Confidence interval for a proportion: p̂ q̂ p̂ q̂ p̂ z 2 p p̂ z 2 n n z 2 2 Sample size for a proportion: n p̂ q̂ E Formula for the confidence interval for difference of two means (small independent samples, variance X unequal): where p̂ and q̂ 1 p̂ n s21 s22 X1 X2 t 2 1 Confidence interval for variance: n1 n2 2 n 1 s 2 n 1 s 2 2 s21 s22 2right 2left X1 X2 t 2 n1 n2 Confidence interval for standard deviation: (d. smaller of n1 1 and n2 1) n 1 s2 n 1 s2 2 2 t test for comparing two means for dependent samples: right left D D D t where D and sD n n Chapter 8 Hypothesis Testing X nD 2 D 2 z test: z for any value n.
n 1 n nn 1 population must be normally distributed. Formula for confidence interval for the mean of the X difference for dependent samples: t test: t (d. n 1) s n SD SD p̂ p D t 2 D D t 2 z test for proportions: z n n pq n (d. n 1) n 1 s 2 Chi-square test for a single variance: 2 2 z test for comparing two proportions: (d.
n 1) p̂1 p̂2 p1 p2 z 1 1 __ pq n1 n2 Chapter 9 Testing the Difference Between Two Means, Two Proportions, _ X1 X2 X1 and Two Variances where p p̂1 n1 n2 n1 z test for comparing two means (independent samples): _ _ X2 q1p p̂2 X1 X2 n2 1 2 z 2 1 2 2 Formula for the confidence interval for the difference of two proportions: n1 n2 Formula for the confidence interval for difference of two p̂1 q̂1 p̂2 q̂2 p̂1 p̂2 z 2 p1 p2 means (large samples): n1 n2 2 1 2 2 p̂1 q̂1 p̂2 q̂2 X1 X2 z 2 1 2 p̂1 p̂2 z 2 n1 n2 n1 n2 s2 2 2 X1 X2 z 2 1 2 F test for comparing two variances: F 12 where s 21 is the n1 n2 s2 larger variance and d. n2 1 t test for comparing two means (independent samples, variances not equal): X1 X2 1 2 t s21 s22 n1 n2 (d. the smaller of n 1 1 or n2 1) Chapter 10 Correlation and Regression Chapter 11 Other Chi-Square Tests Correlation coefficient: Chi-square test for goodness-of-fit: nxy xy O E 2 r x2 a [nx2 x 2][n y2 y 2] E (d. of categories 1) n2 t test for correlation coefficient: t r 1 r2 Chi-square test for independence and homogeneity of (d.
n 2) proportions: The regression line equation: y a bx O E 2 x2 a E y x2 x xy where a [d. 1)] nx x 2 2 nxy xy b n x 2 x 2 Chapter 12 Analysis of Variance explained variation s2 X Coefficient of determination: r 2 ANOVA test: F B2 where XGM total variation sW N Standard error of estimate: d. N k where k number of groups y2 a y b xy sest n2 niXi XGM 2 sB2 k1 Prediction interval for y: ni 1 s2i sW2 1 n x X 2 ni 1 y t 2 sest 1 n n x 2 x 2 2 Xi Xj Scheffé test: FS and 1 n x X 2 sW 1 ni 1 nj 2 y y t 2s est 1 n n x2 x 2 F (k 1)(C. n 2) Xi Xj Formula for the multiple correlation coefficient: Tukey test: q sW2 n Formulas for two-way ANOVA: 2 r yx r yx 2 2ryx 1 • ryx 2 • rx 1x2 R 1 2 1 r 2x 1 x 2 SSA MSA MSA FA Formula for the F test for the multiple correlation a1 MSW coefficient: SSB MSB MSB FB R2 k b1 MSW F 1 R n k 1 2 SSAB MSAB MSAB FAB a 1 b 1 MSW (d.
n k 1) SSW Formula for the adjusted R2: MSW ab n 1 1 R2 n 1 R 2adj 1 nk1 Chapter 13 Nonparametric Statistics Kruskal-Wallis test: X 0.5 n 2 12 R21 R22 R2 z test value in the sign test: z n 2 H NN 1 n1 n2 nk • • • k 3N 1 where n sample size (greater than or equal to 26) X smaller number of or signs where R1 sum of the ranks of sample 1 R mR Wilcoxon rank sum test: z n1 size of sample 1 sR where R2 sum of the ranks of sample 2 n2 size of sample 2 n1n1 n2 1 R 2 n 1 n 2n1 n 2 1 Rk sum of the ranks of sample k R 12 nk size of sample k R sum of the ranks for the smaller sample N n1 n2 nk size (n1) k number of samples n1 smaller of the sample sizes Spearman rank correlation coefficient: n2 larger of the sample sizes n1 10 and n2 10 6 d 2 rS 1 nn2 1 nn 1 ws where 4 Wilcoxon signed-rank test: z d difference in the ranks nn 12n 1 n number of data pairs A 24 where n number of pairs where the difference is not 0 ws smaller sum in absolute value of the signed ranks Procedure Table Solving Hypothesis-Testing Problems (Traditional Method) Step 1 State the hypotheses and identify the claim. Step 2 Find the critical value(s) from the appropriate table in Appendix C. ISBN-13: 978–0–07–743861–6 Step 3 Compute the test value. ISBN-10: 0–07–743861–2 Step 4 Make the decision to reject or not reject the null hypothesis.
Step 5 Summarize the results. Procedure Table Solving Hypothesis-Testing Problems (P-value Method) Step 1 State the hypotheses and identify the claim. Step 2 Compute the test value. Step 3 Find the P-value.
Step 4 Make the decision. Step 5 Summarize the results. Table E The Standard Normal Distribution Cumulative Standard Normal Distribution z .4641 For z values less than 3. Area z 0 Table E (continued ) Cumulative Standard Normal Distribution z .9998 For z values greater than 3.
Area 0 z Table F The t Distribution Confidence intervals 80% 90% 95% 98% 99% One tail, A 0.576d a This value has been rounded to 1.28 in the textbook. One tail Two tails b This value has been rounded to 1.65 in the textbook. c This value has been rounded to 2.33 in the textbook. Area d Area Area This value has been rounded to 2.58 in the textbook.
␣ ␣ ␣ 2 2 Source: Adapted from W. Beyer, Handbook of Tables for Probability and Statistics, 2nd ed., CRC Press, Boca Raton, Fla. Reprinted with permission. t ⫺t ⫹t Table G The Chi-Square Distribution Degrees of A freedom 0.169 Source: Owen, Handbook of Statistical Tables, Table A–4 “Chi-Square Distribution Table,” © 1962 by Addison-Wesley Publishing Company, Inc.
Reproduced by permission of Pearson Education, Inc. Area ␣ 2 blu38582_IFC.