GROUPED DATA E.G. FOR SAMPLE OF RAW DATA (E.G. 4, 12, 7, 5, MEAN G x / n STANDARD DEVIATION MEDIAN AND QUARTILES STANDARD DEVIATION

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1 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 MEAN ( Σ f)/( Σf) STANDARD DEVIATION ( Σ f) ( Σ f) /( Σf) ( Σ f ) 1 MODE (HIGHEST-FREQUENCY CLASS MIDPOINT VALUE)

2 PROBABILITY WHEN SELECTING A RANDOM STUDENT FROM A TABLE LIKE FRESHMEN SOPHOMORE JUN. SENIOR MALE FEMALE 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 = , 169/1545 = (NO) RANDOM VARIABLE AND ITS DISTRIBUTION E.G. X = Pr: BASED ON THIS, BE ABLE TO COMPUTE:

3 : = 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

4 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

5 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 )

6 SMALL SAMPLES t 1 1 ( 1) + ( 1) 1 ± 1 1 c ( 1) + ( 1) ( t WITH 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$ $p p p q p$ q$ z E c

7 OR IF NO PRELIMINARY ESTIMATE AVAILABLE 1 4 z c E DIFFERENCE OF TWO POP. PROPORTIONS pq $ $ pq $ $ p$ p$ p$ p$ ± z c $ $ 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

8 PREDICTION INTERVAL a+ b ± tc r 1+ 1 ( ) 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)

9 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

10 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)

11 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)

12 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 = RANK-SUM (MANN-WHITNEY) TEST USED WITH TWO INDEPENDENT SAMPLES H 0 : : 1 = :...

13 TEST STATISTIC: RANK POOLED SAMPLES SUM SAMPLE-ONE RANKS (R) R z = ( + + 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...