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The Binomial Distribution The Binomial Distribution

The Binomial Distribution - PowerPoint Presentation

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The Binomial Distribution - PPT Presentation

Karl L Wuensch Department of Psychology East Carolina University A Binomial Experiment consists of n identical trials each trial results in one of two outcomes a success or a failure ID: 280033

pressure binomial fearful treatment binomial pressure treatment fearful pups trials blood basenji null strangers data cocker patients approximation good test significantly mothers

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Slide1

The Binomial Distribution

Karl L. Wuensch

Department of Psychology

East Carolina UniversitySlide2

A Binomial Experiment

consists of

n

identical trials.

each trial results in one of two outcomes, a “success” or a “failure.”

the probabilities of success (

p

) and of failure (

q

= 1 ‑

p

) are constant across trials.

trials are independent, not affected by the outcome of other trials.

Y is the number of successes in

n

trials.Slide3

P(Y = y) may also be determined by reference to

a binomial table

.

The binomial distribution has:Slide4

Binomial Hypotheses, Directional

H

0

: Mothers cannot identify their babies by scent alone, binomial

p

 .5H1: Yes they can, binomial p > .5The data: 18 of 25 mothers correctly identified their baby.P(Y 

18 | n = 25, p = .5) = 1 - P(Y

< 17 | n = 25, p = .5) = .022Slide5

Mothers were allowed to smell two articles of infant’s clothing and asked to pick the one which was their infant’s. They were successful in doing so 72% of the time, significantly more often than would be expected by chance, exact binomial

p

(one-tailed) = .022

.Slide6

The Basenji is fearful of strangers.Slide7

The cocker spaniel is not.Slide8

What About A Cockenji

?Slide9

Inheritance of Fearfulness

John Paul Scott and John Fuller

Basenji x Basenji

 fearful pups

Cocker x Cocker  fearless pups

Basenji x Cocker  fearful pupsDominant F gene codes for FearfulnessRecessive f gene codes for fearlessnessF1 dogs are heterozygous,

FfSlide10

Breed F1

Dogs With Each Other

Mother

Father

F

f

F

FF

Ff

f

fF

ffSlide11

Binomial Hypotheses: Nondirectional

H

0

: 75% of the babies will fear strangers, binomial

p

= .75.H1: binomial p  .75The data: 18 of 25 puppies were fearful of strangers.

Under the null, we expect 75% of pups to be fearful. 18/25 = 72% were.psig = 2

P(Y  18 | n = 25, p = .75)Slide12

“p =

2

*PROBBNML(

.75

,

25, 18);”p = .8778The high value of p indicates very good fit between the null hypothesis and the data.Slide13

Eighteen of 25 pups (72%) born to F

1

parents were fearful of strangers. The obtained proportion was not significantly different from the expected .75,

p

= .88 Slide14

Normal Approximation

If

  

falls within 0 to

n, then the binomial approximation should be good.We want

P(Y ≥ 18 | n = 25, p = .5).which is contained within 0  25, so approximation should be good.Slide15

Correction for Continuity

When computing the z, move the observed value of Y one-half point towards the mean under the null.

p

sig

= .0228 Slide16

The Binomial Sign Test

Design = Matched Pairs

Pre and post data for patients given a blood pressure treatment

Of 10 patients, 9 had lower pressure at post-test.

Under the null of no effect of treatment, we expect .5(10) = 5 lower and 5 higher.Slide17

H

0

: The treatment has no effect on blood pressure, binomial

p

= .5H

1: The treatment does affect blood pressure, binomial p  .52P(Y  9

 n = 10, p = .5) =2

(1-(P(Y ≤ 8  n = 10, p = .5)) = .0215 =2*P(Y ≤ 1

 n = 10, p = .5) Slide18

An exact binomial sign test indicated that the treatment significantly lowered blood pressure, 9 of 10 patients having post-treatment pressure lower than their pre-treatment pressure,

p

= .021.