Formula Sheet. December 8, 2011

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1 Formula Sheet December 8, 2011 Abtract I type thi for your coveice. There may be error. Ue at your ow rik. It i your repoible to check it i correct or ot before uig it. 1 Decriptive Statitic 1.1 Cetral Tedecy 1. (Arithmetic) Mea: P N Populatio: = x i P N Sample: x = x i 2. Weighted mea:x w = P w ix i P w i 3. Media: The 0:5 ( + 1)th data. If i eve, the we take 0:5 ( + 1)th data directly If i odd, the we take mea of 0:5th ad (0:5) + 1th data 4. Mode: The mot frequet data. (However, o mode if every data oly appear oce.) 1.2 Diperio 1. Rage: Rage = X Larget X Smallet 2. Iterquartile Rage: IQR = Q 3 Q 1 where Q 3 = 0:75( + 1)th data ad Q 1 = 0:25( + 1)th data. If + 1 i multiple of 4, the we take 0:75( + 1)th data ad the 0:25( + 1)th data a Q 3 ad Q 1 : If + 1 i ot multiple of 4, the we have to iterpolate. 3. Variace: P N Populatio: 2 = (X i ) 2 N 1

2 P Sample: 2 = (x i x) Stadard Deviatio rp N Populatio: = (x i ) 2 N P Sample: = (x i x) Coe ciet of variatio: CV = X 100% 1.3 Relatiohip betwee two variable 1. Covariace: P N Populatio: XY = (x i X ) (y i Y ) P N Sample: XY = (x i x) (y i y) 1 2. Correlatio coe ciet Populatio: XY = XY X Y Sample: r XY = XY X Y Note: 1 XY 1 ad 1 r XY 1 2 Probability Axiom of probability: 1. For ay evet E, 0 P (E) 1: 2. For ay evet E coitig outcome O 1 ; : : : ; O K, P (E) = P K P (O K) : 3. For ample pace S, P (S) = 1. Mutually excluive evet: A ad B are mutually excluive if ad oly if P (A \ B) = 0 Collectively exhautive evet: A 1, A 2,... ; A K are collectively exhautive if Complemet rule: Odd ratio i favor of A i Additio rule: P (A 1 [ A 2 [ : : : [ A K ) = 1 P (A) = 1 P A ; P A = 1 P (A) Odd (A) = P (A) P A = P (A) 1 P (A) P (A \ B) = P (A) + P (B) P (A [ B) P (A [ B) = P (A) + P (B) P (A \ B) 2

3 Coditioal probability: P (AjB) = P (A \ B) ; P (BjA) = P (B) P (A \ B) P (A) Multiplicatio rule: P (A \ B) = P (AjB) P (B) = P (BjA) P (A) Idepedece: Evet A ad B are idepedet if ad oly if P (A \ B) = P (A) P (B) Note: Thi implie that evet A ad B are idepedet if P (AjB) = P (A) P (BjA) = P (B) Law of total probability: If E 1, E 2,... ; E K are mutually excluive ad collectively exhautive, the P (A) = P (A \ E 1 ) + P (A \ E 2 ) + + P (A \ E K ) Baye theorem: = P (AjE 1 ) P (E 1 ) + P (AjE 2 ) P (E 2 ) + + P (AjE K ) P (E K ) P (BjA) = P (E i ja) = P (AjB) P (B) P (A) P (AjE i ) P (E i ) P (AjE 1 ) P (E 1 ) + P (AjE 2 ) P (E 2 ) + + P (AjE K ) P (E K ) 3 Dicrete Radom Variable For a radom variable X, probability ditributio fuctio i p (x) = P (X = x) ad the cumulative probability fuctio: F (x) = P (X x) = X p (x) ax Expectatio value Variace: E (X) = X xp (x) V ar (X) = X [x E (X)] 2 p (x) 3

4 Propertie of expectatio ad variace fuctio: E (g (X)) = X g (x) p (x) E (a + bx) = a + bex V ar (a + bx) = b 2 V arx Joit probability fuctio for radom variable X ad Y Covariace Correlatio p (x; y) = P (X = x; Y = y) Cov (X; Y ) = E [(X E (X)) (Y E (Y ))] = X X [x E (X)] [y E (Y )] p (x; y) x y = Corr (X; Y ) = Propertie of expectatio ad variace fuctio: Cov (X; Y ) p V arx p V ary E (ax + by ) = ae (X) + be (Y ) V ar (ax + by ) = a 2 V arx + b 2 V ary + 2abcov (X; Y ) 4 Cotiuou Radom Variable For a radom variable X, the cumulative probability fuctio: F (x) = P (X x) = X ax p (x) Hece, P (a < X < b) = F (b) F (a) Propertie of expectatio ad variace fuctio for radom variable X ad Y E (a + bx) = a + bex V ar (a + bx) = b 2 V arx E (ax + by ) = ae (X) + be (Y ) V ar (ax + by ) = a 2 V arx + b 2 V ary + 2abcov (X; Y ) If X i ormally ditributed with mea ad variace 2, we deote X N ; 2 Normal ditributio i ymmetric about mea, bell-haped, mea=media=mode. Theorem: If X N (; 2 ), the Z = X N (0; 1) 4

5 5 Samplig Let X 1 ; X 2 ; :::; X be radom ample from a populatio with mea ad variace 2. Sample mea X = 1 X X i ad tadard error of the mea Fact: If populatio i ormal, the X = X N ; 2 Fact: If poulatio i NOT ormal, the by cetral limit theorem, whe ample ize i large, the X N ; 2 6 Etimatio Coider a populatio parameter ad it poit etimator ^. A etimator ^ i ubiaed if E ^ = : Fact: ample mea X i a ubiaed etimator of populatio mea. Fact: ample variace 2 i a ubiaed etimator of populatio variace 2. For two ubiaed etimator ^ 1 ad ^ 2, etimator ^ 1 i more e ciet if V ar V ar ^2 Iterval etimator: 100(1 ) % co dece iterval etimator with kow 2 i ^1 X Z =2 p where ME = Z =2 p, UCL = X + Z =2 p ad LCL = X Z =2 p 100(1 ) % co dece iterval etimator with ukow 2 uder ample ize ad i X t 1;=2 where ME = t 1;=2, UCL = X + t 1;=2 ad LCL = X t 1;=2 5

6 7 Hypothei Tetig Give populatio variace 2, hypothei tetig with igi cace level with radom ample of ize ad ample mea x. Kow 2 Two-tail Tet Upper Tail Lower Tail Null Hypothei H 0 = 0 = 0 or 0 = 0 or 0 Alterative Hypothei H 1 6= 0 > 0 < 0 X critical value 0 z =2 p 0 + z p 0 z p Deciio: Reject if x > 0 + z =2 p or x < 0 z =2 p x > 0 + z p x < 0 z p tet tatitic z = x 0 = z = x 0 = z = x 0 = Z critical value z =2 +z z Deciio: Reject if z > z =2 or z < z =2 z > z z < z p-value 2P (Z > jzj) P (Z > z) P (Z < z) Deciio: Reject if > p Without kowig populatio variace 2, hypothei tetig with igi cace level with radom ample of ize, ample mea x ad ample tadard deviatio 2 : Ukow 2 Two-tail Tet Upper Tail Lower Tail Null Hypothei H 0 = 0 = 0 or 0 = 0 or 0 Alterative Hypothei H 1 6= 0 > 0 < 0 X critical value 0 t 1;=2 0 + t 1; 0 t 1; Deciio: Reject if x > 0 + t 1;=2 x > 0 + t 1; p x < or x < 0 t 1;=2 0 t 1; t-tet tatitic t = x 0 = t = x 0 = t = x 0 = T critical value t 1;=2 +t 1; t 1; Deciio: Reject if t > t 1;=2 or t < t 1;=2 t > t 1; t < t 1; p-value 2P (T > jtj) P (T > t) P (T < t) Deciio: Reject if > p 6

Tables and Formulas for Sullivan, Fundamentals of Statistics, 2e Pearson Education, Inc.

Tables and Formulas for Sullivan, Fundamentals of Statistics, 2e Pearson Education, Inc. Table ad Formula for Sulliva, Fudametal of Statitic, e. 008 Pearo Educatio, Ic. CHAPTER Orgaizig ad Summarizig Data Relative frequecy frequecy um of all frequecie Cla midpoit: The um of coecutive lower