# lecture6 - Psych 100A Winter 2009 Lecture 6 Normal...

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Psych 100A Winter 2009 Lecture 6: Normal Distribution and Statistical Inference Normal distribution 1. General properties . G n ra prop rt s 2. Standard normal distribution tandardization of units 3. Standardization of units 4. Finding areas under normal curves Statistical Inference 1. Statistical inference: techniques 2. Point estimates 1

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Example • Fair coin tossed four times. What is the chance of etting exactly 2 heads? getting exactly 2 heads? • Is this a binomial experiment? Yes 1. n = 4 iid trials (assume coin toss process OK) 2. Dichotomous outcome (“heads, H” or “tails, T”) 3. Pr(H) = Pr(T) = ½ , which is constant. 4. Variable = total number of successes (heads) 2
1 st 2 nd Toss: 3 rd 4 th Solution 1. • Draw a path diagram. 1 2 3 H T H H T HHTT HTHT 6 4 5 T T H H T H H HTTH 7 8 H T T 3 6 ) ' 2 Pr( s H THHT THTH 9 10 11 H T H T H H T 8 16 TTHH 14 12 13 T T H H T 15 16 H T T 3

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Formula: Pr(success) in a binomial experiment • Probability of x successes in n trials of a binomial experiment is x n x p p x n x n successes x ) 1 ( )! ( ! ! ) Pr( •where •n = number of trials • p = probability of success • 1-p = probability of failure • x = number of successes • x! = x(x-1)(x-2)…(2)(1) 4
4. Normal distribution Special continuous probability distribution 5

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Suppose r(x) .250 .375 ensity x 0 Pr(x) .125 75 0 .125 De 01 23 1 2 3 .375 .375 .125 N=3 Number of heads r(x) 75 =10 x 0 1 Pr(x) .0010 .0097 439 25 .250 .375 N=10 2 3 4 .0439 .1172 .2051 461 0 .125 D 01 2345678910 umber of heads 5 .2461 Number of heads 6
Suppose 25 x Pr(x) 000 50 .075 .100 .125 ensity 0 1 12 .0000 .0000 .0001 =50 0 .025 .050 D 0 1 02 03 04 05 0 15 20 .0020 .0418 N=50 Number of heads 25 .1123 nsity N=immense De n An ideal or normal curve should approximate well the binomial distribution Number of heads 7

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Normal curve deMoivre’s normal curve 2 2 1 ) ( x e f .50 y 2 .25 Densit 0 -4 -3 -2 -1 0 1 2 3 4 Standard units 8
1. General properties of normal dist. a. Family of continuous probability distributions. A particular member is defined by its center (mean, ) and spread (SD, ).

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lecture6 - Psych 100A Winter 2009 Lecture 6 Normal...

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