The_Normal_Distribution

The_Normal_Distribution - The Normal Distribution The The...

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Unformatted text preview: The Normal Distribution The The Normal Distribution The A theoretical or mathematically ideal distribution It can be used to estimate relative frequency It and/or probabilities for a wide variety of continuous variables continuous Actual variables measured on a population, Actual often approximate a normal distribution often Bernoulli & Gauss Not a perfect fit But, close enough to be very useful Mathematical approximations for probabilities in Mathematical games of chance or the distributions of expected errors in observations errors Population Parameters and the Normal Distribution Normal The larger the number of members of a The population, the more closely it will approximate a normal distribution approximate Parameters: Statistics that measure all Parameters: members of a population members Mean = μ (meu) Mean Standard Deviation = σ (sigma) Standard Sample data is seldom distributed Sample normally normally Properties of the Normal Distribution Distribution Symmetrical Scores cluster around the vertical axis Asymptotic (approaches the base line, but Asymptotic never reaches…) never Continuous .40 μ = 70 σ = 10 .30 .20 .10 40 100 -3σ 50 -2σ 60 -1σ 70 μ 80 +1σ 90 +2σ +3σ Standard Normal Distribution Standard A normal distribution on which all values normal on the X-axis are expressed as z scores on z scores = standard deviation unit μ =0, σ=1 Finding the z score: z = X- μ σ Standard Normal Distribution Standard Zobs = (X - μ)/ σ μ=0 σ = 1.0 -3 -2 -1 0 1 2 3 If population had μ = 75, σ = 10, X to Z If conversion z of μ = z75 = (75-75)/10 = 0 of z of 90 = z90 = (90-75)/10 = 1.5 z of 60 = z60 = (60-75)/10 = -1.5 z of 105 = z105 = (105-75)/10 = 3.0 Relative Frequency Distribution Relative The area underneath the curve is the The underneath relative frequency distribution! relative rf = # of occurrences total # of scores total Cumulative relative frequency has maximum Cumulative value of 1.0, minimum value of 0 value Group Frequency Distribution Group Class Real Limits Mid- tally freq Class Interval lower upper point Interval 30-32 30-32 27-29 24-26 21-23 18-20 15-17 12-14 9-11 6-8 3-5 0-2 29.5 32.5 26.5 29.5 23.5 26.5 20.5 23.5 17.5 20.5 14.5 17.5 11.5 14.5 8.5 11.5 5.5 8.5 2.5 5.5 -.5 2.5 31 II 2 28 0 25 II 2 22 I 1 19 III 3 16 IIII 4 13 III 3 10 IIII III 8 7 IIII IIII III 13 4 IIII IIII 9 1 IIII 5 rf .04 0 .04 .02 .06 .08 .06 .16 .26 .18 .10 %f 4 0 4 2 6 8 6 16 26 18 10 cf crf 50 1.00 48 .96 48 .96 46 .92 45 .90 42 .84 38 .76 35 .70 27 .54 14 .28 5 .10 .0013 .02 .34 .14 -2 -1 μ .68 .96 .34 .14 1 2 .02 .0013 3 -3 Total area under curve = 1.0 .50 to left of μ, .50 to right of μ .50 .50 .997 Summary Summary Normal curve a mathematical ideal Area under curve is a cumulative relative Area frequency distribution frequency Mathematical formula specifies how much Mathematical cumulative area falls at various points around the mean around Points around mean defined in standard Points deviation units or… deviation z scores Zobs = (X - μ)/ σ Known areas of the Normal Distribution Worked out and Tabled See Table A-1, p. 487-490 Value of +z -z .95 .96 .97 .98 .99 1.00 1.01 1.02 .3289 .3315 .3340 .3365 .3389 .3413 .3438 .3461 .1711 .1685 .1660 .1635 .1611 .1587 .1562 .1539 Determining relative frequency of scores in a population scores Known: IQ has μ =100, σ=10 Known: Question: What proportion of the scores Question: fall below 98? fall Answer: 1. Draw dist 2. Convert x to z 98 100 σ=10 Determining relative frequency of scores in a population scores Known: IQ has μ =100, σ=10 Known: Question: What proportion of the scores Question: fall below 98? fall Answer: σ=10 1. Draw dist 98 100 2. Convert x to z Z98 = (X98 – μ)/ σ = (98-100)/10 = -2/10 = -.20 )/ 3. Table A-1, Column C for +/-.20 Area = proportion = relative frequency = .4307 Question: What proportion of IQ scores Question: fall above 112? fall μ =100, σ=10 Answer: 1. Draw dist 100 112 2. Convert x to z Z112 = (X112 – μ)/ σ = (112-100)/10 = 12/10 = )/ 1.2 1.2 3. Table A-1, Column C 3. Area = proportion = relative frequency = .1151 Question: What proportion of IQ scores Question: fall between 90 and 100? fall μ =100, σ=10 Question: What proportion of IQ scores Question: fall between 90 and 100? fall μ =100, σ=10 Answer: 1. Draw dist 2. Convert x to z 90 100 Z90 = (X90 – μ)/ σ = (90-100)/10 = -10/10 = -1.0 )/ 3. Table A-1, Column b Area from z=0 to z=-1.0 = .3413 Question: What proportion of IQ scores Question: fall between 90 and 110? fall μ =100, σ=10 Answer: Answer: 1) Draw the distribution!!! 90 100 Question: What proportion of IQ scores fall Question: between 90 and 110? between Think! Need to add 2 areas together [100 to 110] + [90 to 100] Convert Xs to Zs Z90 = (90-100)/10 = -10/10 = -1.0 Z110 = (110-100)/10 = 10/10 = 1.0 Consult table A-1, column b Area 0 to -1.0 = .3413 Area 0 to +1.0 = .3413 Area from -1.0 to 1.0 = .6826 What proportion of IQ scores fall between What the values of 105 and 115? the μ =100, σ=10 Draw the distribution! What proportion of IQ scores fall between What the values of 105 and 115? the μ =100, σ=10 Draw the distribution! 100 105 115 Think! Need to subtract area between 100 & 105 Need from the area between 100 & 115. from Use column (b) of Table A-1 Convert Xs to Zs Z105 = (105-100)/10 = .50 Z115 = (115-100)/10=1.50 Table A-1 Area 0 to 1.5 = .4332 Area 0 to .5 = .1915 Area .5 to 1.5 = .2417 Probability Probability Subjective Empirical P(event A) = # of occurrences of A Total # of Possible events Total Define Event A Define 1) drawing an Ace of Clubs from a full 1) deck deck P(Ace of Clubs) = 1/52 = .0192 2) Winning CA State lottery if only one 2) winner for 10,000,000 tickets sold. winner P(win) = 1/10,000,000 = .0000001 Possible Range of Probabilities Possible 0 to 1.0 0: it never occurs No winners out of 100 P(win) = 0/100 = .000 1.0: it always occurs Object will hit ground if dropped P(hit) = 100/100 = 1.0 Drop 100 objects – all 100 will hit the ground Identity of Empirical Probability and Relative Frequency Relative rf = # of occurrences of score Total # of scores Total Prob(A) = # of occurrences of A Total # of events Total The Standard Normal Distribution is a probability density function density **** as well as **** **** a cumulative relative frequency distribution cumulative Summary Can use known areas of Standard Normal Can Distribution to estimate probabilities of any events that are approximately normally distributed distributed Usually large populations measured on variables Must know population parameters, μ & σ Must Must convert X values of interest to Z scores Use Table of area densities Cannot give probabilities for discrete values – only Cannot ranges of values ranges What is the probability that a male What selected at random from the US population will be 78″ tall or taller? population Answer: Answer: We need to have a good reason to assume We height is approximately normally distributed height Need μ & σ Need μ = 69″ σ = 4″ Draw distribution 78 69 73 77 +1σ +2σ Area = Prob Z78 = (X- μ)/ σ = (78-69)/4 = 9/4 = +2.25 )/ Consult Table A-1, Col C Area beyond +2.25 = .0122 Probability of male ≥ 78 ″ = .0122 Standard Scores Standard Provides logical and objective way to Provides combine scores combine Provides common base for comparison of Provides scores from different distributions scores Why shouldn’t we compare raw scores across different distributions? scores Max and min values may be different μ may be different σ may be different All of these may be different! Seagram’s Contest Seagram’s All Star Drinking Person 10 contestants Coors beer Charles Krag wines J&B Scotch X(coors) X(coors) Total 1 20 2 10 3 40 4 60 5 30 6 40 7 30 8 20 9 30 10 30 5 12 7 4 8 6 6 8 7 7 X(C.K.) X(J&B) 23 4 3 1 3 2 3 3 4 2 48 (3) 27 50 (2) 65 (1) 41 48 (3) 39 31 41 39 ∑X 310 X 31.0 70 7.0 48 4.8 Coors Coors 5 4 3 2 1 Std. Dev = 13.70 Mean = 31.0 0 10.0 20.0 30.0 40.0 50.0 60.0 N = 10.00 VAR00001 CK CK 10 8 6 4 2 Std. Dev = 2.16 Mean = 7.0 0 5.0 15.0 25.0 35.0 45.0 55.0 N = 10.00 VAR00002 J&B J&B 10 8 6 4 2 Std. Dev = 6.46 Mean = 4.8 0 2.7 13.1 23.5 34.0 44.4 54.8 N = 10.00 VAR00003 Solution? Solution? Convert all scores to z-scores Zobs = (X - μ)/ σ )/ Z(coors) Z(coors) Total 1 -.85 -.85 2 -1.62 -1.62 3 .69 .69 4 2.23 2.23 5 .08 .08 6 .69 .69 7 .08 .08 8 -.85 -.85 9 .08 .08 10 .08 .08 -.98 2.44 0 -1.46 .48 -.48 -.48 .97 0 0 Z(C.K.) Z(J&B) 2.96 -.13 -.29 -.62 -.29 -.45 -.29 -.29 -.13 -.45 1.13 (1) .69 (2) .40 (3) .15 (5) .27 (4) -.24 -.69 -.17 -.05 -.37 ...
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This note was uploaded on 01/16/2011 for the course PSYC 274 at USC.

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