Important Formulas. Expectation: E (X) = Σ [X P(X)] = n p q σ = n p q. P(X) = n! X1! X 2! X 3! X k! p X. Chapter 6 The Normal Distribution.

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1 Importat Formulas Chapter 3 Data Descriptio Mea for idividual data: X = _ ΣX Mea for grouped data: X= _ Σf X m Stadard deviatio for a sample: _ s = Σ(X _ X ) or s = 1 (Σ X ) (Σ X ) ( 1) Stadard deviatio for grouped data: _ s = _ (Σf Xm ) (Σf Xm ) ( 1) rage Rage rule of thumb: s _ 4 (Shortcut formula) Chapter 4 Probability ad Coutig Rules Additio rule 1 (mutually exclusive evets): P(A or B ) = P(A) + P(B) Additio rule (evets ot mutually exclusive): P(A or B) = P(A) + P(B ) P(A ad B) Multiplicatio rule 1 (idepedet evets): P(A ad B) = P(A) P(B) Multiplicatio rule (depedet evets): P(A ad B) = P(A) P(B A) Coditioal probability: P(B A) = P(A ad B) P(A) Complemetary evets: P( E ) = 1 P(E ) Fudametal coutig rule: Total umber of outcomes of a sequece whe each evet has a differet umber of possibilities: k 1 k k 3 k Permutatio rule: Number of permutatios of objects takig r at a time is P r = _! ( r)! Permutatio rule of objects with r 1 objects idetical, r objects idetical, etc.! r1! r! r p! Combiatio rule: Number of combiatios of r objects selected from objects is C r = _! ( r)!r! Chapter 5 Discrete Probability Distributios Mea for a probability distributio: μ = Σ [X P(X)] Variace ad stadard deviatio for a probability distributio: σ = Σ [X P(X)] μ _ σ= Σ [X P(X)] μ Expectatio: E (X) = Σ [X P(X)] Biomial probability: P(X ) = _! ( X )!X! px q X Mea for biomial distributio: μ = p Variace ad stadard deviatio for the biomial distributio: σ _ = p q σ = p q Multiomial probability: P(X) = _! X1! X! X 3! X k! p X 1 1 X p X p 3 3 X p k k Poisso probability: P(X; λ) = e λ λ X X! X = 0, 1,,... Hypergeometric probability: P(X ) = ac X b C X a+bc Geometric probability: P() = p(1 p) 1 is the umber of the trial i which the first success occurs ad p is the probability of a success Chapter 6 The Normal Distributio Stadard score: Populatio: z = _ X μ σ or Sample: z = X X s Mea of sample meas: μ _ X = μ Stadard error of the mea: σ _ X = σ X μ Cetral limit theorem formula: z = σ Chapter 7 Cofidece Itervals ad Sample Size z cofidece iterval for meas: X zα ( σ ) < μ < X + z α ( t cofidece iterval for meas: σ ) X t α ( s ) < μ < X+ t α ( s ) Sample size for meas: = ( z α σ E ) E is the margi of error Cofidece iterval for a proportio: ˆp ˆq ˆp (z α ) < p < ˆp + (z ˆp ˆq α )

2 z α Sample size for a proportio: = p ˆ q ˆ ( _ E ) p ˆ = X ad ˆ q = 1 p ˆ Cofidece iterval for variace: ( 1)s < σ ( 1)s < χ right χ left Cofidece iterval for stadard deviatio: ( 1)s < σ < ( 1)s χ right χ left Chapter 8 Hypothesis Testig X μ z test: z = for ay value. If < 30, σ populatio must be ormally distributed. t test: t = X μ (d.f. = 1) s p p z test for proportios: z = pq Chi-square test for a sigle variace: ( 1)s = (d.f. = 1) ˆ Chapter 9 Testig the Differece Betwee Two Meas, Two Proportios, ad Two Variaces z test for comparig two meas (idepedet samples): z = ( X 1 _ X ) (μ 1 μ ) _ σ 1 + _ σ 1 Formula for the cofidece iterval for differece of two meas (large samples): ( X 1 X ) z α _ σ _ σ < μ 1 μ σ < ( X 1 X ) + z α _ σ _ σ Formula for the cofidece iterval for differece of two meas (small idepedet samples, variaces uequal): _ ( X 1 X ) t α s 1 + s 1 < μ 1 μ _ < ( X 1 X ) + t α s 1 + s 1 (d.f. = smaller of 1 1 ad 1) t test for comparig two meas for depedet samples: D μ t = D s D D = _ ΣD s D = ΣD (ΣD) (d.f. = 1) ( 1) ad Formula for cofidece iterval for the mea of the differece for depedet samples: s D t α D < μ D < D + t α s D (d.f. = 1) z test for comparig two proportios: ( p ˆ z = 1 p ˆ ) (p 1 p ) _ p _ q ( ) p = _ X 1 + X 1 + p ˆ 1 = _ X 1 1 X _ q = 1 p p ˆ = _ Formula for the cofidece iterval for the differece of two proportios: _ p ˆ ( p ˆ 1 p ˆ ) z α 1 q ˆ 1 p ˆ + ˆ q 1 < p 1 p _ p ˆ < ( p ˆ 1 p ˆ ) + z α 1 q ˆ 1 p ˆ + ˆ q 1 F test for comparig two variaces: F = s 1 s 1 is the s larger variace ad d.f.n. = 1 1, d.f.d. = 1 t test for comparig two meas (idepedet samples, variaces ot equal): t = ( X 1 _ X ) (μ 1 μ ) _ s _ s (d.f. = the smaller of 1 1 or 1)

3 Chapter 10 Correlatio ad Regressio Correlatio coefficiet: (Σ xy) (Σ x)(σy) r = [(Σ x ) (Σ x) ][(Σ y ) (Σ y) ] t test for correlatio coefficiet: t = r 1 r (d.f. = ) The regressio lie equatio: yʹ = a + bx (Σy)(Σx a = _ ) (Σx)(Σxy) (Σx ) (Σx) (Σxy) (Σx)(Σy) b = _ (Σx ) (Σx) Coefficiet of determiatio: r explaied variatio = total variatio Stadard error of estimate: _ s est = Σy a Σy b Σxy Predictio iterval for y: yʹ t α s est (x _ X ) Σx (Σx) < y < yʹ + t α s est (x _ X ) Σx (Σx) (d.f. = ) Formula for the multiple correlatio coefficiet: R = _ ryx 1 + ryx r yx1 r yx r x1 x 1 rx 1 x Formula for the F test for the multiple correlatio coefficiet: F = _ R k (1 R ) ( k 1) (d.f.n. = k ad d.f.d. = k 1) Formula for the adjusted R : R adj = 1 [ _ (1 R )( 1) k 1 ] Chapter 11 Other Chi-Square Tests Chi-square test for goodess-of-fit: = (O E ) E (d.f. = o. of categories 1) Chi-square test for idepedece ad homogeeity of proportios: = (O E ) E [d.f. = (rows 1)(colums 1)] Chapter 1 Aalysis of Variace ANOVA test: F = _ s B X GM = _ ΣX s W N d.f.n. = k 1 N = k d.f.d. = N k k = umber of groups s Σ B = i ( X i X GM ) k 1 s W = _ Σ( i 1)s i Σ( i 1) ( X i X j ) Scheffé test: F S = _ ad sw (1 i + 1 j ) Fʹ = (k 1)(C.V.) X i X j Tukey test: q = s W Formulas for two-way ANOVA: SS MS A = _ A a 1 F A = _ MS A MS W SS MS B = _ B b 1 F B = _ MS B MS W MS A B = _ SS A B F A B = MS A B (a 1)(b 1) MS W SS MS W = _ W ab( 1)

4 Chapter 13 Noparametric Statistics (X + 0.5) 0.5 z test value i the sig test: z = = sample size (greater tha or equal to 6) X = smaller umber of + or sigs R μ Wilcoxo rak sum test: z = R σ R μ R = _ 1 ( ) _ σ R = _ 1 ( ) 1 R = sum of the raks for the smaller sample size ( 1 ) 1 = smaller of the sample sizes = larger of the sample sizes 1 10 ad 10 w s _ ( + 1) Wilcoxo siged-rak test: z = 4 ( + 1)( + 1) 4 = umber of pairs the differece is ot 0 ad 30 w s = smaller sum i absolute value of the siged raks Kruskal-Wallis test: H = _ 1 N(N + 1) ( _ R 1 + _ R _ R k k ) 3(N + 1) R 1 = sum of the raks of sample 1 1 = size of sample 1 R = sum of the raks of sample = size of sample R k = sum of the raks of sample k k = size of sample k N = k k = umber of samples Spearma rak correlatio coefficiet: r s = 1 6Σd ( 1) d = differece i the raks = umber of data pairs Procedure Table Solvig Hypothesis-Testig Problems (Traditioal Method) Step 1 State the hypotheses ad idetify the claim. Step Fid the critical value(s) from the appropriate table i Appedix A. Step 3 Compute the test value. Step 4 Make the decisio to reject or ot reject the ull hypothesis. Step 5 Summarize the results. Procedure Table Solvig Hypothesis-Testig Problems (P-Value Method) Step 1 State the hypotheses ad idetify the claim. Step Compute the test value. Step 3 Fid the P-value. Step 4 Make the decisio. Step 5 Summarize the results.

5 TABLE E The Stadard Normal Distributio Cumulative Stadard Normal Distributio z For z values less tha 3.49, use z 0

6 TABLE E (cotiued ) Cumulative Stadard Normal Distributio z For z values greater tha 3.49, use z

7 TABLE F The t Distributio Cofidece itervals 80% 90% 95% 98% 99% Oe tail, α d.f. Two tails, α (z) 1.8 a b c.576 d a This value has bee rouded to 1.8 i the textbook. b This value has bee rouded to 1.65 i the textbook. c This value has bee rouded to.33 i the textbook. d This value has bee rouded to.58 i the textbook. Oe tail α Two tails α α t t +t

8 TABLE G The Chi-Square Distributio α Degrees of freedom Source: values calculated with Excel. α χ