Formulas and Tables. for Elementary Statistics, Tenth Edition, by Mario F. Triola Copyright 2006 Pearson Education, Inc. ˆp E p ˆp E Proportion

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1 Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola Copyright 2006 Pearson Education, Inc. Ch. 3: Descriptive Statistics x Sf. x x Sf Mean S(x 2 x) 2 s Å n 2 1 n(sx 2 ) 2 (Sx) 2 s Å n(n 2 1) Mean (frequency table) Standard deviation n3s(f. x 2 )4 2 3S(f. x)4 2 s Å n(n 2 1) variance s 2 Ch. 4: Probability P(A or B) 5 P(A) 1 P(B) if A, B are mutually exclusive P(A or B) 5 P(A) 1 P(B) 2 P(A and B) if A, B are not mutually exclusive P(A and B) 5 P(A). P(B) if A, B are independent P(A and B) 5 P(A). P(B 0A) if A, B are dependent P(A) P(A) Rule of complements n! np r 5 Permutations (no elements alike) (n 2 r)! n! Permutations (n 1 alike,...) n 1! n 2!... n k! n! nc r 5 Combinations (n 2 r)! r! Ch. 5: Probability Distributions x. P(x) [ x 2. P(x)] 2 Mean (prob. dist.) Standard deviation (prob. dist.) n! P(x) Binomial probability (n x)! x!. p x. q n x n. p 2 n. p. q n. p. q x. e P(x) x! Ch. 6: Normal Distribution x Sx n n Central limit theorem (Standard error) Mean (binomial) Variance (binomial) z x x or x Standard score s x Central limit theorem Standard deviation (shortcut) Standard deviation (frequency table) Standard deviation (binomial) Poisson Distribution where e Ch. 7: Confidence Intervals (one population) ˆp E p ˆp E Proportion pˆ qˆ where E 5 z a>2 Å n x 2 E,m,x 1 E Mean s where E 5 z a>2 (s known )!n s or E 5 t a>2 (s unknown)!n (n 2 1)s 2 (n 2,s 2 1)s2, x 2 R Ch. 7: Sample Size Determination n 5 3za> E 2 n 5 3za>242 pˆ qˆ E 2 n 5 B za>2s E R 2 Proportion Variance Proportion (ˆp and ˆq are known) Mean Ch. 9: Confidence Intervals (two populations) (pˆ 1 2 pˆ 2) 2 E, (p 1 2 p 2 ), (pˆ 1 2 pˆ 2) 1 E pˆ 1qˆ 1 where E 5 z a>2 1 pˆ 2qˆ 2 Å n 1 n 2 (x 1 2 x 2 ) 2 E, (m 1 2m 2 ), (x 1 2 x 2 ) 1 E where E 5 t a>2 Å n 1 s2 2 1 n 2 (s 1 and s 2 unknown and not assumed equal) s 2 p E 5 t a>2 1 s2 p (df 5 n Å n 1 n 1 1 n 2 2 2) 2 sp 2 5 (n 1 2 1)s (n 2 2 1)s 2 2 (n 1 2 1) 1 (n 2 2 1) (s 1 and s 2 unknown but assumed equal) s 2 1 s 2 1 x 2 L (df smaller of n 1 1, n 2 1) E 5 z a>2 1 s 2 2 Å n 1 n 2 (s 1, s 2 known) d 2 E,m d, d 1 E (Matched Pairs) s d where E 5 t a>2 (df n 1)!n (Indep.)

2 Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola Copyright 2006 Pearson Education, Inc. Ch. 8: Test Statistics (one population) z 5 pˆ 2 p pq Å n z 5 x 2m s>!n t 5 x 2m s>!n (n 2 x 2 1)s2 5 Proportion one population Ch. 9: Test Statistics (two populations) z 5 (pˆ 1 2 pˆ 2) 2 (p 1 2 p 2 ) pq Å n 1 pq 1 n 2 Two proportions df smaller of t 5 (x 1 2 x 2 ) 2 (m 1 2m 2 ) n 1 1, n 2 1 s s2 2 Å n 1 n 2 Two means independent; s 1 and s 2 unknown, and not assumed equal. t 5 (x 1 2 x 2 ) 2 (m 1 2m 2 ) F 5 s2 1 s 2 2 s 2 Two means independent; s 1 and s 2 unknown, but assumed equal. z 5 (x 1 2 x 2 ) 2 (m 1 2m 2 ) t 5 d 2m d s d >!n s 2 p 1 s2 p Å n 1 s 2 1 Å n 1 s n 2 Ch. 11: Multinomial and Contingency Tables (O 2 x 2 E)2 5 g Mean one population ( known) Mean one population ( unknown) Standard deviation or variance one population n 2 Two means matched pairs (df n 1) Standard deviation or variance two populations (where s 2 1 s 2 2 ) Multinomial (df k 1) (df n 1 n 2 2) s 2 p 5 (n 1 2 1)s (n 2 2 1)s 2 2 n 1 1 n Two means independent; 1, 2 known. E (O 2 Contingency table x 2 E)2 5 g E [df (r 1)(c 1)] (row total) (column total) where E 5 (grand total) McNemar s test x 2 5 ( 0 b 2 c 0 2 1)2 b 1 c for matched pairs (df 1) Ch. 10: Linear Correlation/Regression nsxy 2 (Sx)(Sy) Correlation r 5 "n(sx 2 ) 2 (Sx) 2 "n(sy 2 ) 2 (Sy) 2 nsxy 2 (Sx)(Sy) b 1 5 n(sx 2 ) 2 (Sx) 2 b 0 5 y 2 b 1 x or b 0 5 (Sy)(Sx2 ) 2 (Sx)(Sxy) n(sx 2 ) 2 (Sx) 2 ŷ 5 b 0 1 b 1 x Estimated eq. of regression line explained variation r 2 5 total variation s e 5 Å S(y 2 ŷ) 2 n 2 2 ŷ E y ŷ E where or Å Sy 2 2 b 0 Sy 2 b 1 Sxy n 2 2 Prediction interval E t 2 s e 1 1 n n(x 0 x) 2 n( x 2 ) ( x) 2 Ch. 12: One-Way Analysis of a Variance Procedure for testing H 0 : m 1 5m 2 5m 3 5 c 1. Use software or calculator to obtain results. 2. Identify the P-value. 3. Form conclusion: If P-value a, reject the null hypothesis of equal means. If P a, fail to reject the null hypothesis of equal means. Ch. 12: Two-Way Analysis of Variance Procedure: 1. Use software or a calculator to obtain results. 2. Test H 0 : There is no interaction between the row factor and column factor. 3. Stop if H 0 from Step 1 is rejected. If H 0 from Step 1 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.

3 Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola Copyright 2006 Pearson Education, Inc. Ch. 13: Nonparametric Tests (x 1 0.5) 2 (n>2) z 5!n>2 T 2 n(n 1 1)>4 z 5 n(n 1 1)(2n 1 1) Å 24 z 5 R2m R2 n 1(n 1 1n 2 11) R 2 5 s R n 1 n 2 (n 1 1n 2 11) Å 12 H 5 Sign test for n N(N 1 1) ar2 1 n 1 1 R2 2 n R2 k n k b 2 3(N 1 1) Kruskal-Wallis (chi-square df k 1) r s Sd2 n(n 2 2 1) acritical value for n. 30: z 5 G 2m G s G 5 Å 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: x 1 A 2 R Centerline: x LCL: x 2 A 2 R Rank correlation 6 z!n 2 1 b G 2 a 2n 1n 2 1 1b n 1 1 n 2 (2n 1 n 2 )(2n 1 n 2 2 n 1 2 n 2 ) (n 1 1 n 2 ) 2 (n 1 1 n 2 2 1) p chart: Plot sample proportions pq UCL: p 1 3 Å n Centerline: p pq LCL: p 2 3 Å n Wilcoxon signed ranks (matched pairs and n 30) Wilcoxon rank-sum (two independent samples) Runs test for n 20 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 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 2 D 3 D

4 FINDING P-VALUES Start Left-tailed Left What type of test? Two-tailed Is the test statistic to the right or left of center? Right Right-tailed P-value area to the left of the test statistic P-value twice the area to the left of the test statistic P-value twice the area to the right of the test statistic P-value area to the right of the test statistic P- value P- value is twice this area. P-value is twice this area. P-value Start Test statistic Test statistic Test statistic Test statistic Does the original claim contain the condition of equality? HYPOTHESIS TEST: WORDING OF FINAL CONCLUSION No (Original claim does not contain equality and becomes H 1 ) Yes (Original claim contains equality) Do you reject H 0? Yes (Reject H 0 ) No (Fail to reject H 0 ) Do you reject H 0? Yes (Reject H 0 ) No (Fail to reject H 0 ) Wording of final conclusion There is sufficient evidence to warrant rejection of the claim that... (original claim). There is not sufficient evidence to warrant rejection of the claim that... (original claim). The sample data support the claim that... (original claim). There is not sufficient sample evidence to support the claim that... (original claim). (This is the only case in which the original claim is rejected.) (This is the only case in which the original claim is supported.) 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 distrubted and n 30

5 NEGATIVE z Scores z 0 TABLE A-2 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 POSITIVE z Scores 0 z TABLE A-2 (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 TABLE A-3 t Distribution: Critical t Values Area in One Tail Degrees of Area in Two Tails Freedom Large

8 Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola Copyright 2006 Pearson Education, Inc. TABLE A-4 Chi-Square (x 2 ) Distribution Area to the Right of the Critical Value Degrees of Freedom From Donald B. Owen, Handbook of Statistical Tables, 1962 Pearson Addison-Wesley, an imprint of Pearson Education, Inc.

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