FOR SAMPLE OF RAW DATA (E.G. 4, 1, 7, 5, 11, 6, 9, 7, 11, 5, 4, 7) BE ABLE TO COMPUTE MEAN G / STANDARD DEVIATION MEDIAN AND QUARTILES Σ ( Σ) / 1 GROUPED DATA E.G. AGE FREQ. 0-9 53 10-19 4...... 80-89 MEAN ( Σ f)/( Σf) STANDARD DEVIATION ( Σ f) ( Σ f) /( Σf) ( Σ f ) 1 MODE (HIGHEST-FREQUENCY CLASS MIDPOINT VALUE)
PROBABILITY WHEN SELECTING A RANDOM STUDENT FROM A TABLE LIKE FRESHMEN SOPHOMORE JUN. SENIOR MALE 14 197 171 156 FEMALE 55 03 180 169 BE ABLE TO ANSWER ANY PROBABILITY QUESTION (USING AND, OR, NOT, GIVEN), E.G. PROBABILITY OF: MALE AND JUNIOR, FEMALE GIVEN SOPHOMORE, JUNIOR OR SENIOR GIVEN FEMALE, NOT SENIOR GIVEN MALE, ETC. ALSO: ARE THE SENIOR AND FEMALE EVENTS INDEPENDENT? 35/1545 807/1545 = 0.1099, 169/1545 = 0.1094 (NO) RANDOM VARIABLE AND ITS DISTRIBUTION E.G. X = - -1 0 1 Pr: 0.13 0.19 0.3 0.7 0.18 BASED ON THIS, BE ABLE TO COMPUTE:
: = E p MEAN STANDARD DEVIATION F = ( Σ p) µ ANY PROBABILITY E.G. X > -1 ETC. SPECIAL DISTRIBUTIONS BINOMIAL: PROBABILITIES CAN BE COMPUTED FROM: Pr( X= i ) = Ci, pi q i THE CORRESPONDING MEAN AND STANDARD DEVIATION ARE p AND p q RESPECTIVELY. POISSON: FIRST COMPUTE 8 = t r INDIVIDUAL PROBABILITIES i λ Pr( X= i) = e λ i! THE CORRESPONDING MEAN AND STANDARD DEVIATION ARE: 8 AND λ
NORMAL: HAS MEAN : AND STANDARD DEVIATION F. CAN BE CONVERTED TO STANDARD NORMAL (OF OUT TABLES) BY Z = X σ µ E.G. X µ µ Pr( X< ) = Pr( σ < ) = Pr( Z < z) σ SIMILARLY, FIND SUCH THAT Pr(X<)=0.10 SOLVE IN TERMS OF z, THEN CONVERT: = z F + : X FINALLY, IS ALSO NORMAL, WITH MEAN : AND STANDARD DEVIATION (ERROR) OF σ, I.E. X µ Pr( X< ) = Pr( < µ σ ) = Pr( Z< z) σ
CONFIDENCE INTERVALS HYPOTHESES TESTING INTERVAL: TEST STATISTIC: NEEDED: ONE POPULATION MEAN ± tc µ 0 / z c E DISTRIBUTION: t -1 (z WHEN > 30 OR EXACT F GIVEN) DIFFERENCE OF TWO POPULATION MEANS LARGE SAMPLES (BOTH > 30) 1 1 ± z c + (NORMAL z ) 1 1 1 1 +
SMALL SAMPLES t 1 1 ( 1) + ( 1) 1 ± 1 1 c + + 1 1 1 1 1 ( 1) + ( 1) + 1 1 + 1 1 ( t WITH 1 + - DEGREES OF FREEDOM) PAIRED SAMPLES d S d ( t WITH - 1 DEGREES OF FREEDOM, OR z - LARGE SAMPLES) ONE POPULATION PROPORTION (r>5, -r>5) p$ ± z c p$ q$ $p p p q 0 0 0 p$ q$ z E c
OR IF NO PRELIMINARY ESTIMATE AVAILABLE 1 4 z c E DIFFERENCE OF TWO POP. PROPORTIONS pq $ $ pq $ $ p$ p$ p$ p$ ± z 1 1 1 1 c + 1 1 $ $ 1 + pq $p 1 IS THE POOLED SAMPLE PROPORTION ( r 1, r, 1 -r 1, -r ALL BIGGER THAN 5 ) REGRESSION AND CORRELATION SAMPLE SLOPE AND INTERCEPT b SP y = a= y b SS RESIDUAL STANDARD DEVIATION r = SS y b SP y
PREDICTION INTERVAL a+ b ± tc r 1+ 1 ( ) + 0 0 SS ( t WITH - DEGREES OF FREEDOM) SAMPLE CORRELATION COEFFICIENT r SS SP y SS y COEFFICIENT OF DETERMINATION: r TESTING H 0 : $ = 0... TEST STATISTIC: b r SS ( t WITH - DEGREES OF FREEDOM)
CHI-SQUARE TEST OF < INDEPENDENCE DATA GIVEN AS AN R (NUMBER OF ROWS) BY C (NUMBER OF COLUMNS) TABLE OF OBSERVED FREQUENCIES EXPECTED FREQUENCIES COMPUTED BY ( row um) ( colum um) table um TEST STATISTIC: Σ ( O E) E HAS, UNDER H 0 (INDEPENDENCE) CHI- SQUARE DISTRIBUTION WITH (R-1) (C-1) DEGREES OF FREEDOM (ALWAYS A RIGHT- TAIL TEST) < GOODNESS OF FIT DATA CONSISTS OF A SINGLE ROW OF k OBSERVED FREQUENCIES NULL HYPOTHESIS MUST SPECIFY THE
CORRESPONDING (THEORETICAL) PROBABILITIES p EXPECTED FREQUENCIES COMPUTED FROM p, WHERE IS THE TOTAL OBSERVED FREQUENCY TEST STATISTIC: Σ ( O E) E HAS, UNDER H 0, CHI-SQUARE DISTRIBUTION WITH k-1 DEGREES OF FREEDOM (ALWAYS A RIGHT- TAIL TEST) ANALYSIS OF VARIANCE IS TESTING WHETHER k POPULATION MEANS ARE ALL IDENTICAL (THE NULL HYPOTHESIS) OR NOT (ALTERNATE)
WE MUST FIRST COMPUTE AND SS w = SS TOT = Σ All ample TOT Σ ( Σ) Σ TOT ( ) N SS BET = SS TOT - SS W TEST STATISTIC: SS SS BET W N k k 1 HAS THE F DISTRIBUTION WITH k-1 (NUMERATOR) AND N-k (DENOMINATOR) DEGREES OF FREEDOM (ALWAYS A RIGHT- TAIL TEST)
NONPARAMETRIC TESTS SIGN TEST USED for PAIRED-SAMPLES COMPUTE SIGN OF THE DIFFERENCES DISCARD N.D. OBSERVATIONS REDUCE THE VALUE OF H 0 : p + = ½... TEST STATISTIC: z = 05. 05. RANK-SUM (MANN-WHITNEY) TEST USED WITH TWO INDEPENDENT SAMPLES H 0 : : 1 = :...
TEST STATISTIC: RANK POOLED SAMPLES SUM SAMPLE-ONE RANKS (R) R z = ( + + 1)/ 1 1 ( + + 1)/ 1 1 1 SPEARMAN (RANK) CORRELATION COEFFICIENT SUBTRACT y RANKS FROM RANKS (=d) r d 1 6 ( 1 ) USE TABLE 9 TO TEST H 0 : NO -y CORRELATION...