Grant MacEwan University STAT 151 Formula Sheet Final Exam Dr. Karen Buro

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1 Grat MacEwa Uiverity STAT 151 Formula Sheet Fial Exam Dr. Kare Buro Decriptive Statitic Sample Variace: = i=1 (x i x) 1 = Σ i=1x i (Σ i=1 x i) 1 Sample Stadard Deviatio: = Sample Variace = Media: Order the data from mallet to larget. The media M i either the uique middle value or the mea of the two middle value. Lower Quartile: Order the data from mallet to larget. The lower quartile Q 1 i the media of the maller half of the value. Upper Quartile: Order the data from mallet to larget. The upper quartile Q 3 i the media of the upper half of the value. Iterquartile Rage (iqr) = Upper Quartile Lower Quartile =Q 3 Q 1 Outlier: lower fece =Q 1 1.5iqr ad upper fece=q iqr Probability Theory Additio Rule: P (A or B) = P (A) + P (B) P (A&B) Complemet Rule: P (A doe ot occur) = P (ot A) = 1 P (A) Multiplicatio Rule: P (A&B) = P (A B)P (B) Multiplicatio Rule for Idepedet Evet: If A ad B are idepedet, the P (A&B) = P (A)P (B) Coditioal Probability of A give B, if P (B) > 0 : P (A B) = P (A&B) P (B) Populatio Ditributio The mea (expected value) of a dicrete radom variable: µ = xp(x). The variace of a dicrete radom variable: = (x µ) p(x) The tadard deviatio of a dicrete radom variable: = Biomial Ditributio Probability to oberve k uccee i trial: p(k) = P (x = k) = ( ) k p k (1 p) k ( ) k =! k!( k)! Mea ad tadard deviatio of a biomial ditributio: µ = p ad =

2 Samplig Ditributio Samplig Ditributio of a Sample Mea, x: µ x = µ, x = Samplig Ditributio of the differece of two Sample Mea, x 1 x : µ x1 x = µ 1 µ, x1 x = Samplig Ditributio of a Sample Proportio, ˆp: µˆp = p, ˆp = Samplig Ditributio of the differece of two Sample Proportio, ˆp 1 ˆp : p1 (1 p 1 ) µˆp1 ˆp = p 1 p, ˆp1 ˆp = + p (1 p ) 1 Etimatio Parameter Etimator SE(Etimator) Approximate Cofidece Iterval µ x x ± t α/ p ˆp µ 1 µ x 1 x 1 + ( x 1 x ) ± t 1 α/ ˆp ± z α/ ˆp(1 ˆp) µ 1 µ x d d x d ± t α/ d p1 (1 p 1 ) p 1 p ˆp 1 ˆp + p (1 p ) ˆp1 (1 ˆp 1 ) (ˆp 1 ˆp ) ± z α/ + ˆp (1 ˆp ) 1 1 Chooig the Sample Size (formula) Etimate a mea µ with a (1 α) cofidece iterval withi a amout of m. ( z(1 α/) m Etimate a proportio p with a (1 α) cofidece iterval withi a amout of m. ( z(1 α/) m ) )

3 Tet Statitic Tet Statitic for large ample z-tet cocerig p z = ˆp p 0 p0 (1 p 0 ) Tet Statitic for Large-Sample z Tet for comparig p 1 ad p : z = ˆp 1 ˆp ˆpc, with ˆp c = 1ˆp 1 + ˆp (1 ˆp c ) 1 + ˆp c(1 ˆp c ) 1 + Tet Statitic for 1-ample t-tet cocerig µ if i ukow t = x µ 0 / df = 1 Tet Statitic for two ample t-tet for comparig two populatio mea: t = x 1 x, df = mi( 1 1, 1) Tet Statitic for paired t-tet for comparig two populatio mea: t = x d ( d / ) df = 1 Goode-of-Fit Tet ad Tet for Idepedece of two categorical variable Tet Statitic for Goode of Fit Tet: χ = all categorie (oberved cout expected cout) expected cout Expected cell cout = (hypotheized value of correpodig populatio proportio) df = k 1 where k i the umber of categorie. χ Tet for Idepedece: The tet tatitic i χ = all cell (row total)(colum total) Expected cell cout = grad total df =(umber of row - 1)(umber of colum - 1) (oberved cout expected cout) expected cout 3

4 Regreio Aalyi Sum of Square SS xy = x i y i ( x i )( y i ), = x i ( x i ), SS yy = yi ( y i ) Correlatio Coefficiet (r), Coefficiet of Determiatio (R ) r = SS xy SSxx SS yy, R = r Leat Square Regreio lie ŷ = b 0 + b 1 x, with Etimatio of e = b 1 = SS xy, ad b 0 = ȳ b 1 x SSE, with SSE = (ŷ i y i ) = SS yy b 1 SS xy Cofidece iterval for β 1 b 1 ± t e SSxx tet tatitic for a tet about β 1 t 0 = b 1 e /, df = Cofidece iterval for the mea of y, E(y), at x = x p Predictio Iterval for y at x = x p ŷ ± t e 1 + (x p x) ŷ ± t e (x p x) 4

5 The Aalyi of Variace (ANOVA) Total Sum of Square SST = ij (x ij x) = ij x ij CM (df = 1) with x = ample mea of all meauremet, G = ij x ij ad CM = G Sum of Square for group SST R = i i ( x i x) = i T i i CM (df = k 1) with x i = ample mea of obervatio i ample i, T i = Total of obervatio i ample i. Sum of Square for Error SSE = i ( i 1) i = SST SSG (df = k) with i i the tadard deviatio of obervatio from ample i. ANOVA Table Source df SS M S=SS/df F Treatmet/Group k 1 SST R MST R = SST R/(k 1) MST R/MSE Error k SSE MSE = SSE/( k) Total 1 SST 5