Important Formulas. Discrete Probability Distributions. Probability and Counting Rules. The Normal Distribution. Confidence Intervals and Sample Size

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1 blu38582_if_1-8.qxd 9/27/10 9:19 PM Page 1 Important Formulas Chapter 3 Data Description Mean for individual data: Mean for grouped data: Standard deviation for a sample: X2 s X n 1 or Standard deviation for grouped data: s nf X 2 m f X m 2 nn 1 Range rule of thumb: Chapter 4 s 2 X 2 nx nn 1 (Shortcut formula) Probability and Counting Rules Addition rule 1 (mutually exclusive events): P(A or B) P(A) P(B) Addition rule 2 (events not mutually exclusive): P(A or B) P(A) P(B) P(A and B) Multiplication rule 1 (independent events): P(A and B) P(A) P(B) Multiplication rule 2 (dependent events): P(A and B) P(A) P(B A) Conditional probability: s range 4 PA and B PB A PA Complementary events: P( ) 1 P(E) E X X n X f X m n Fundamental counting rule: Total number of outcomes of a sequence when each event has a different number of possibilities: k 1 k 2 k 3 k n Permutation rule: Number of permutations of n objects taking r at a time is np r n! n r! Combination rule: Number of combinations of r objects n! selected from n objects is n C r n r!r! Chapter 5 Discrete Probability Distributions Mean for a probability distribution: m [X P(X)] Variance and standard deviation for a probability distribution: s 2 [X 2 P(X)] m 2 s [X 2 PX] m 2 Expectation: E(X) [X P(X)] n! Binomial probability: PX n X!X! px q nx Mean for binomial distribution: m n p Variance and standard deviation for the binomial distribution: s 2 n p q s n p q Multinomial probability: n! PX X 1!X 2!X 3!... X k! px 1 1 p X 2 2 p X 3 3 p X k k Poisson probability: P(X; l) X 0, 1, 2,... Hypergeometric probability: Chapter 6 Standard score z X where The Normal Distribution Mean of sample means: m X m Standard error of the mean: s X n Central limit theorem formula: Chapter 7 Confidence Intervals and Sample Size z confidence interval for means: X z 2n X z2 t confidence interval for means: X s t 2n X t2 z X n s n Sample size for means: n z2 where E is the E 2 maximum error of estimate Confidence interval for a proportion: ˆp ˆq ˆp z2 n p ˆp ˆp ˆq z2 n e X X! PX a C X b C nx abc n or X X z s n

2 blu38582_if_1-8.qxd 9/27/10 9:19 PM Page 2 Sample size for a proportion: where ˆp X and ˆq 1 ˆp n Confidence interval for variance: n 1s 2 2 right Confidence interval for standard deviation: n 1s 2 1s2 right n left 2 Chapter 8 Hypothesis Testing z test: z X for any value n. If n 30, n population must be normally distributed. t test: t X (d.f. n 1) sn ˆp p z test for proportions: z pqn Chi-square test for a single variance: (d.f. n 1) Chapter 9 2 n 1s2 Testing the Difference Between Two Means, Two Proportions, and Two Variances z test for comparing two means (independent samples): z X 1 X Formula for the confidence interval for difference of two means (large samples): X 1 X 2 z2 2 1 n 1 2 n n X 1 X 2 z2 2 1 n 1 t test for comparing two means (independent samples, variances not equal): left n ˆp ˆq z2 E n n 1s n 2 Formula for the confidence interval for difference of two means (small independent samples, variance unequal): X 1 X 2 t2 s2 1 n 1 s2 2 n (d.f. smaller of n 1 1 and n 2 1) t test for comparing two means for dependent samples: t D D s Dn Formula for confidence interval for the mean of the difference for dependent samples: D S D t2 n D D S D t2 n (d.f. n 1) z test for comparing two proportions: where where s D nd 2 D 2 nn 1 z ˆp 1 ˆp 2 p 1 p 2 p_ q _ 1 n 1 1 n 2 X 1 X 2 t2 s2 1 n 1 s2 2 n 2 D D n d.f. n 1 p _ X 1 X 2 n 1 n 2 ˆp 1 X 1 n 1 q _ 1 p _ ˆp 2 X 2 n 2 Formula for the confidence interval for the difference of two proportions: ˆp 1 ˆp 2 z2 ˆp 1 ˆq 1 n 1 ˆp 2 ˆq 2 n 2 p 1 p 2 ˆp 1 ˆp 2 z2 ˆp 1 ˆq 1 n 1 ˆp 2 ˆq 2 n 2 F test for comparing two variances: F s2 1 where is the larger variance and d.f.n. n 1 1, d.f.d. n 2 1 s 2 2 and s 2 1 t X 1 X s2 1 s2 2 n 1 n 2 (d.f. the smaller of n 1 1 or n 2 1)

3 blu38582_if_1-8.qxd 9/27/10 9:19 PM Page 3 Chapter 10 Correlation coefficient: t test for correlation coefficient: (d.f. n 2) Correlation and Regression nxy xy r [nx 2 x 2 ][ny 2 y 2 ] The regression line equation: y a bx where Coefficient of determination: Standard error of estimate: s est y2 a y b xy n 2 Prediction interval for y: y t2s est 1 1 n nx X 2 n x 2 x 2 y y t2s est 1 1 n nx X 2 n x 2 x 2 (d.f. n 2) nxy xy b nx 2 x 2 Formula for the multiple correlation coefficient: R r 2 yx 1 r 2 yx 2 2r yx1 r yx2 r x 1 x 2 1 r 2 x 1 x 2 Formula for the F test for the multiple correlation coefficient: R F 2 k 1 R 2 n k 1 (d.f.n. n k and d.f.d. n k 1) Formula for the adjusted R 2 : a yx2 xxy nx 2 x 2 1 R R 2 adj 1 2 n 1 n k 1 n 2 t r 1 r 2 explained variation r 2 total variation Chapter 11 Other Chi-Square Tests Chi-square test for goodness-of-fit: O x 2 E2 a E (d.f. no. of categories 1) Chi-square test for independence and homogeneity of proportions: O x 2 E2 a E [d.f. (rows 1)(col. 1)] Chapter 12 ANOVA test: Analysis of Variance d.f.n. k 1 where N n 1 n 2 n k d.f.d. N k where k number of groups Scheffé test: F s 2 B s 2 W s 2 B n i X i X GM 2 k 1 s 2 W n i 1s 2 i n i 1 F S Xi X j 2 s 2 W 1n i 1n j F (k 1)(C.V.) Tukey test: q Xi X j swn 2 Formulas for two-way ANOVA: MS A SS A a 1 MS B SS B b 1 SS AB MS AB a 1b 1 MS W SS W abn 1 where X GM X N and F A MS A MS W F B MS B MS W F AB MS AB MS W

4 blu38582_if_1-8.qxd 9/27/10 9:19 PM Page 4 Chapter 13 Nonparametric Statistics X 0.5 n2 z test value in the sign test: z n 2 where n sample size (greater than or equal to 26) X smaller number of or signs Wilcoxon rank sum test: where R n 1 n 1 n z R m R s R R 1n 2 n 1 n 2 1 n 12 R sum of the ranks for the smaller sample size (n 1 ) n 1 smaller of the sample sizes n 2 larger of the sample sizes n 1 10 and n 2 10 Wilcoxon signed-rank test: z nn 1 w s 4 nn 12n 1 A 24 where n number of pairs where the difference is not 0 w s smaller sum in absolute value of the signed ranks Kruskal-Wallis test: H 12 NN R2 1 R2 2 R2 k 1 n 1 n 2 n k 3N 1 where R 1 sum of the ranks of sample 1 n 1 size of sample 1 R 2 sum of the ranks of sample 2 n 2 size of sample 2 R k sum of the ranks of sample k n k size of sample k N n 1 n 2 n k k number of samples Spearman rank correlation coefficient: r S 1 6 d 2 nn 2 1 where d difference in the ranks n number of data pairs 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. Step 3 Compute the test value. 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. ISBN-13: ISBN-10:

5 Appendix C Tables Table A Factorials Table B The Binomial Distribution Table C The Poisson Distribution n n! Table D Random Numbers 0 1 Table E The Standard Normal Distribution 1 1 Table F The t Distribution 2 2 Table G The Chi-Square Distribution 3 6 Table H The F Distribution Table I Critical Values for the PPMC 4 24 Table J Critical Values for the Sign Test Table K Critical Values for the Wilcoxon Signed-Rank Test Table L Critical Values for the Rank Correlation Coefficient Table M Critical Values for the Number of Runs Table N Critical Values for the Tukey Test Table A Factorials , , , ,628, ,916, ,001, ,227,020, ,178,291, ,307,674,368, ,922,789,888, ,687,428,096, ,402,373,705,728, ,645,100,408,832, ,432,902,008,176,640,000 A 17

6 770 Appendix C Tables Table B The Binomial Distribution n x p A 18

7 Appendix C Tables 771 Table B (continued) n x p A 19

8 772 Appendix C Tables Table B (continued) n x p A 20

9 Appendix C Tables 773 Table B (continued) n x p A 21

10 774 Appendix C Tables Table B (continued) n x p A 22

11 Appendix C Tables 775 Table B (concluded) n x Note: All values of or less are omitted. Source: J. Freund and G. Simon, Modern Elementary Statistics, Table The Binomial Distribution, 1992 Prentice-Hall, Inc. Reproduced by permission of Pearson Education, Inc. p A 23

12 768 Appendix B 3 Alternate Approach to the Standard Normal Distribution Table B 1 The Standard Normal Distribution z For z values greater than 3.49, use Area given in table 0 z A 16

13 776 Appendix C Tables Table C The Poisson Distribution L x L x L x L x A 24

14 Appendix C Tables 777 Table C (continued) L x L x L x A 25

15 778 Appendix C Tables Table C (continued) L x L x A 26

16 Appendix C Tables 779 Table C (continued) x x L L A 27

17 780 Appendix C Tables Table C (continued) L x L x A 28

18 Appendix C Tables 781 Table C (continued) L x L x A 29

19 782 Appendix C Tables Table C (concluded) L x Reprinted with permission from W. H. Beyer, Handbook of Tables for Probability and Statistics, 2nd ed. Copyright CRC Press, Boca Raton, Fla., A 30

20 Appendix C Tables 783 Table D Random Numbers Reprinted with permission from W. H. Beyer, Handbook of Tables for Probability and Statistics, 2nd ed. Copyright CRC Press, Boca Raton, Fla., A 31

21 784 Appendix C Tables Table E The Standard Normal Distribution Cumulative Standard Normal Distribution z For z values less than 3.49, use Area z 0 A 32

22 Appendix C Tables 785 Table E (continued) Cumulative Standard Normal Distribution z For z values greater than 3.49, use Area 0 z A 33

23 786 Appendix C Tables Table F The t Distribution Confidence intervals 80% 90% 95% 98% 99% One tail, A d.f. Two tails, A (z) a b c d a This value has been rounded to 1.28 in the textbook. b This value has been rounded to 1.65 in the textbook. c This value has been rounded to 2.33 in the textbook. d This value has been rounded to 2.58 in the textbook. Source: Adapted from W. H. Beyer, Handbook of Tables for Probability and Statistics, 2nd ed., CRC Press, Boca Raton, Fla., Reprinted with permission. One tail t Area Two tails Area Area 2 2 t t A 34

24 Appendix C Tables 787 Table G The Chi-Square Distribution Degrees of A freedom Source: Owen, Handbook of Statistical Tables, Table A 4 Chi-Square Distribution Table, 1962 by Addison-Wesley Publishing Company, Inc. Copyright renewal Reproduced by permission of Pearson Education, Inc. Area 2 A 35

25 A 36 Table H The F Distribution A d.f.d.: degrees of d.f.n.: degrees of freedom, numerator freedom, denominator Appendix C Tables 1 16,211 20,000 21,615 22,500 23,056 23,437 23,715 23,925 24,091 24,224 24,426 24,630 24,836 24,940 25,044 25,148 25,253 25,359 25,