Formulas and Tables. for Essentials of Statistics, by Mario F. Triola 2002 by Addison-Wesley. ˆp E p ˆp E Proportion.

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1 Formulas and Tables for Essentials of Statistics, by Mario F. Triola 2002 by Addison-Wesley. Ch. 2: 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. 3: 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. 4: Probability Distributions x. P(x) Mean (prob. dist.) Standard deviation (prob. dist.) n! P(x) Binomial probability (n x)! x!. p x. q n x Mean (binomial) Variance (binomial) [ x 2. P(x)] 2 n. p 2 n. p. q n. p. q Ch. 5: Normal Distribution z x x or x Standard score s x Central limit theorem x Sx n n Central limit theorem (Standard error) Standard deviation (shortcut) Standard deviation (frequency table) Standard deviation (binomial) Ch. 6: Confidence Intervals (one population) x 2 E,m,x 1 E Mean s where E 5 z a>2 (s known or n 30)!n s or E 5 t a>2 (s unknown and n 30)!n ˆp E p ˆp E Proportion pˆ qˆ where E 5 z a>2 Å n (n 2 1)s 2 (n 2,s 2 1)s2, x 2 R Ch. 6: Sample Size Determination n 5 B za>2s E R 2 Mean Variance n 5 3za> Proportion E 2 n 5 3za>242 pˆ qˆ Proportion (ˆp and ˆq are known) E 2 Ch. 7: Test Statistics (one population) z 5 x 2m s>!n t 5 x 2m s>!n z 5 pˆ 2 p pq Å n (n 2 x 2 1)s2 5 s 2 Proportion one population Ch. 8: Test Statistics (two populations) z 5 (x 1 2 x 2 ) 2 (m 1 2m 2 ) t 5 d 2m d s d >!n Mean one population ( known or n 30) Mean one population ( unknown and n 30) s 2 1 Å n 1 s n 2 x 2 L Two means matched pairs (df n 1) z 5 (pˆ 1 2 pˆ 2) 2 (p 1 2 p 2 ) pq Å n 1 pq 1 n 2 Standard deviation or variance one population Two means independent ( 1, 2 known or n 1 30 and n 2 30) Two proportions

2 Formulas and Tables for Essentials of Statistics, by Mario F. Triola 2002 by Addison-Wesley. Ch. 8: Confidence Intervals (two populations) (x 1 2 x 2 ) 2 E, (m 1 2m 2 ), (x 1 2 x 2 ) 1 E s 2 1 where E 5 z a>2 1 s 2 2 Å n 1 n 2 (s 1, s 2 known or n and n 2. 30) d 2 E,m d, d 1 E (Matched Pairs) s d where E 5 t a>2!n (df n 1) (pˆ 1 2 pˆ 2) 2 E, (p 1 2 p 2 ), (pˆ 1 2 pˆ 2) 1 E (Indep.) Ch. 10: Multinomial and Contingency Tables (O 2 x 2 E)2 5 g E Ch. 10: One-Way Analysis of a Variance F 5 ns22 x s 2 p Multinomial (df k 1) (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) k samples each of size n (num. df k 1; den. df k(n 1)) pˆ 1qˆ 1 where E 5 z a>2 1 pˆ 2qˆ 2 Å n 1 n 2 Ch. 9: 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 F 5 MS(treatment) MS(error) MS(treatment) 5 SS(treatment) k 2 1 MS(error) 5 SS(error) N 2 k df k 1 df N k MS(total) 5 SS(total) N 2 1 SS(treatment) 5 n 1 (x 1 2 x) n k (x k 2 x) 2 SS(error) 5 (n 1 2 1)s (n k 2 1)s 2 k SS(total) 5S(x 2 x) 2 SS(total) 5 SS(treatment) 1 SS(error) explained variation r 2 5 total variation s e 5 Å S(y 2 ŷ) 2 n 2 2 ŷ E y ŷ E or Å Sy 2 2 b 0 Sy 2 b 1 Sxy n 2 2 where E t 2 s e 1 1 n n(x 0 x)2 n( x 2 ) ( x) 2 r s Sd2 Rank correlation n(n 2 2 1) 6 z acritical value for n. 30:!n 2 1 b

3 HYPOTHESIS TESTING 1. Identify the specific claim or hypothesis to be tested and put it in symbolic form. 2. Give the symbolic form that must be true when the original claim is false. 3. Of the two symbolic expressions obtained so far, let the null hypothesis H 0 be the one that contains the condition of equality; H 1 is the other statement. 4. Select the significance level a based on the seriousness of a type I error. Make a small if the consequences of rejecting a true H 0 are severe. The values of 0.05 and 0.01 are very common. 5. Identify the statistic that is relevant to this test, and identify its sampling distribution. 6. Determine the test statistic and either the P-value or the critical values, and the critical region. Draw a graph. 7. Reject H 0 : Test statistic is in the critical region or P-value # a. Fail to reject H 0 : Test statistic is not in the critical region or P-value. a. 8. Restate this previous conclusion in simple, nontechnical terms. FINDING P-VALUES Start Left-tailed What type of test? Two-tailed Right-tailed Left Is the test statistic to the right or left of center? Right 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 m m m m Test statistic Test statistic Test statistic Test statistic

4 0 z TABLE A-2 Standard Normal (z) Distribution z and.4999 higher NOTE: For values of z above 3.09, use for the area. *Use these common values that result from interpolation: z score Area From Frederick C. Mosteller and Robert E. K. Rourke, Sturdy Statistics, 1973, Addison-Wesley Publishing Co., Reading, MA. Reprinted with permission of Frederick Mosteller.

5 Left tail Student t distribution Right tail Two tails a a a/2 a/2 Critical t score (negative) Critical t scorecritical t scorecritical t score (positive) (negative) (positive) TABLE A-3 t Distribution Degrees (one tail) (one tail) (one tail) (one tail) (one tail) (one tail) of Freedom (two tails) (two tails) (two tails) (two tails) (two tails) (two tails) a Large (z)

6 TABLE A-4 Chi-Square (x 2 ) Distribution Formulas and Tables for Essentials of Statistics, by Mario F. Triola 2002 by Addison-Wesley. Area to the Right of the Critical Value Degrees of Freedom From Donald B. Owen, Handbook of Statistical Tables, 1962 Addison-Wesley Publishing Co., Reading, MA. Reprinted with permission of the publisher.

7 Formulas and Tables for Essentials of Statistics, by Mario F. Triola 2002 by Addison-Wesley. Linear Correlation (Section 9-2) Rank Correlation (Section 9-5) TABLE A-5 Critical Values of the Pearson Correlation Coefficient r n a.05 a TABLE A-6 Critical Values of Spearman s Rank Correlation Coefficient r s n a 0.05 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.

8 HYPOTHESIS TEST: WORDING OF FINAL CONCLUSION Start Does the original claim contain the condition of equality? Yes No (Original claim does not contain equality and becomes H 1 ) (Original claim contains equality and becomes H 0 ) Do you reject H 0? No (Fail to reject H 0 ) Do you reject H 0? No (Fail to reject H 0 ) Yes (Reject H 0 ) Yes (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.)