introduction to behavioral statistics probability, the binomial distribution and the normal curve

Introduction to Behavioral Introduction to Behavioral StatisticsStatistics

Probability, The Binomial Probability, The Binomial Distribution and the Normal CurveDistribution and the Normal Curve

Probability, The Binomial Distribution and Probability, The Binomial Distribution and the Normal Curvethe Normal Curve

Introduction to ProbabilityIntroduction to Probability– We all think in terms of probability– We all compute and use probability– If we have a coin, there is a 1/2 probability that it

will land on heads/tails when we flip it. (.5 heads + .5 tails gives coin a total probability of 1)

– What is the probability of drawing a particular card from a deck. (1/52 or .019 or 1.9 per 100)

– What is the probability of drawing any two (2) cards from a deck. (1/52 + 1/52=2/52 or .038)


Additive Theorem -– The probability that any one of a set of The probability that any one of a set of

mutually exclusive events will occur is the mutually exclusive events will occur is the sum of the probability of the separate sum of the probability of the separate events.events.

– What is the probability of drawing either of What is the probability of drawing either of two cards from a deck.two cards from a deck.

– (1/52 + 1/52) = 2/52 or .038)


Multiplication Theorem -– The joint probability of obtaining both of two events is the product of The joint probability of obtaining both of two events is the product of

their separate probabilities.their separate probabilities. What is the probability of drawing two aces from a deck of cards.What is the probability of drawing two aces from a deck of cards.

– (4/52 * 3/51) = 12/2652 or .0045)(4/52 * 3/51) = 12/2652 or .0045)

What is the probability of first a 5 and then a 6.What is the probability of first a 5 and then a 6.– (1/6 * 1/6) = 1/36(1/6 * 1/6) = 1/36

With two dice, what is the probability of rolling a 7 or 11?With two dice, what is the probability of rolling a 7 or 11?– How many ways to get 7 (4 +3) (5+2) (6+1) on each dieHow many ways to get 7 (4 +3) (5+2) (6+1) on each die

» using the additive theorem we see that there are using the additive theorem we see that there are 6/36 ways to get 7 and 2/36 to get 116/36 ways to get 7 and 2/36 to get 11

» Thus there are 1/9 ways to to roll a 7 or 11.Thus there are 1/9 ways to to roll a 7 or 11.


Multiplication Theorem - What is the probability of drawing two aces from a deck of cards.What is the probability of drawing two aces from a deck of cards.

1 2 3 4 5 61 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12


Permutations of r objects taken r at a Permutations of r objects taken r at a timetime

N

NN ! 9!= 362880

Permutations of N objects taken r at a Permutations of N objects taken r at a timetime

N

r

N

N r

N

!

!

!

!9 272


Combination of N objects taken r at a Combination of N objects taken r at a time.time.

CN

r

N

r N r

!

! !

!

! !

10

2 10 2

3628800

2 4032045


Binomial DistributionBinomial Distribution– This distribution is very important to psychology.This distribution is very important to psychology.

The chi square distribution is based on it…The chi square distribution is based on it… The normal curve is based on it….The normal curve is based on it…. F and t distributions are The based on it...F and t distributions are The based on it...


Binomial DistributionBinomial Distribution Bernoulli TrialBernoulli Trial

Experiments often have only two possible Experiments often have only two possible outcomes.outcomes.

– true falsetrue false– effective not effectiveeffective not effective

Flipping a coin one time and noting whether it Flipping a coin one time and noting whether it lands heads or tails, or randomly drawing one lands heads or tails, or randomly drawing one sample from a distribution is called a sample from a distribution is called a Bernoulli Bernoulli trial or Bernoulli experiment.trial or Bernoulli experiment.


Bernoulli TrialBernoulli Trial– Characteristics of a Bernoulli trialCharacteristics of a Bernoulli trial

A trial can result in one of two outcomesA trial can result in one of two outcomes The probability of success remains constant from trial to trial.The probability of success remains constant from trial to trial. The outcomes of successive trials are independent.The outcomes of successive trials are independent.

– In reality very few In reality very few real situationsreal situations meet these requirements since probability doesn’t remain constant when we remove an item meet these requirements since probability doesn’t remain constant when we remove an item from the distribution.from the distribution.


Binomial DistributionBinomial Distribution– In this distribution, the In this distribution, the random variablerandom variable is a is a

sum (the number of successes observed sum (the number of successes observed on n greater than or equal to two Bernoulli on n greater than or equal to two Bernoulli trials.trials.

This distribution is a relatively simple example This distribution is a relatively simple example of an important class of theoretical distributions of an important class of theoretical distributions or models that are referred to as or models that are referred to as sampling sampling distributions.distributions.


Binomial DistributionBinomial Distribution– Sampling distribution is the special name Sampling distribution is the special name

that is given to a probability distribution that is given to a probability distribution where the random variable is a statistic where the random variable is a statistic based on the result of n greater than or based on the result of n greater than or equal to 2 trials.equal to 2 trials.

– The Binomial Distribution is one of the The Binomial Distribution is one of the distributions used by psychologists.distributions used by psychologists.


Binomial DistributionBinomial Distribution– The number of successes observed on n greater than or The number of successes observed on n greater than or

equal to 2 identical Bernoulli trials is called a equal to 2 identical Bernoulli trials is called a binomial binomial random variablerandom variable, and its probability distribution is called a , and its probability distribution is called a binomial distributionbinomial distribution..

– If we toss a fair coin 5 times, the probability of observing If we toss a fair coin 5 times, the probability of observing exactly r heads in n tosses is given by exactly r heads in n tosses is given by p(X=r)= p(X=r)= nnCCrrpprrqqn-rn-r

– This gives the probability that the random variable X equals r This gives the probability that the random variable X equals r heads. heads. nnCCr r is the combination of is the combination of nn objects taken objects taken rr at a time. at a time.

PP is the probability of a success (a head), and is the probability of a success (a head), and q = (1-p).q = (1-p).– Next - lets look at a particular example:Next - lets look at a particular example:


p X C( )

!

! !

4

5

4 5 4

5

32

5 4

4 5 4

4 1

12

12

12

12

No. of Heads0 1 2 3 4 5

p(X=r) 1/325/3210/3210/325/321/32

Binomial Distribution for N=5 and p=1/2


No. of Heads0 1 2 3 4 5

p(X=r) 1/325/3210/3210/325/321/32

Binomial Distribution for N=5 and p=1/2

Histogram for binomial distribution shown above….

0123456789

10

0123456789

10

Notice how much this resembles the form of a normal curve.


The normal curve is a The normal curve is a limiting form of the limiting form of the binomial distributionbinomial distribution

The normal curve The normal curve occurs when we have occurs when we have an infinite number of an infinite number of events occurring events occurring according to the laws of according to the laws of chance.chance.

0123456789

10

5 4 3 2 1 0


A Linear Graphy=A+bX

0

10

20

30

40

50

60

We can write a We can write a formula to plot a set formula to plot a set of pointsof points– In this case we have In this case we have

used the formula used the formula y=A+bX to plot a y=A+bX to plot a straight line.straight line.

In this same way we In this same way we can generate a plot can generate a plot of the normal curveof the normal curve


f X Xe 1

2

22

2

/

Where:

f(X)= height of the distribution at XPi = approximately 3.142e= base of natural logarithms)

approximately 2.718

Where:

f(X)= height of the distribution at XPi = approximately 3.142e= base of natural logarithms)

approximately 2.718


Fortunately, we don’t need to calculate the normal curve.

We just use the table in the back of our book…..(1)

StandardScore

(2)Area fromMean toStandard

Score

(3)Area in Larger

Portion

(4)Area inSmallerPortion

(5)y ordinate at X

0.00 .0000 .5000 .5000 .3989

1.00 .3413 .8413 .1587 .2420

1.96 .4750 .9750 .0250 .0584

Using the Normal Curve to normalize a Using the Normal Curve to normalize a distribution of scoresdistribution of scores

(1)Standard

Score


Score

(3)Area in Larger

Portion


(5)y ordinate at X

0.00 .0000 .5000 .5000 .3989

1.00 .3413 .8413 .1587 .2420

1.96 .4750 .9750 .0250 .0584

zx

z x z X X

so

z X X

so or

Using the Normal Curve to normalize a Using the Normal Curve to normalize a distribution of scoresdistribution of scores

(1)Standard

Score


Score

(3)Area in Larger

Portion


(5)y ordinate at X

0.00 .0000 .5000 .5000 .3989

1.00 .3413 .8413 .1587 .2420

1.96 .4750 .9750 .0250 .0584

z X X 1. We find 90th centile from column 3 of table.2. We then use the corresponding z score.3. In case of 90th centile z=1.294. Using IQ data 1.29(20.2)+103.29= 129.348

This is the normalized centile score.

Determining the % of cases which fall Determining the % of cases which fall between any two scoresbetween any two scores

For our IQ data - suppose we want to know what % For our IQ data - suppose we want to know what % of scores fall between 123.49 and 113.32.of scores fall between 123.49 and 113.32.

First we convert these scores to z scoresFirst we convert these scores to z scores

(123.49-103.29)/20.2 = 1.00(123.49-103.29)/20.2 = 1.00(113.32-103.29)/20.2= .50(113.32-103.29)/20.2= .50

Then we get the area from column 2 and subtractThen we get the area from column 2 and subtract

.3413-.1915=.1498 or 15%.3413-.1915=.1498 or 15%

To see exactly where the above values came from, To see exactly where the above values came from, use the table in your book and work through this use the table in your book and work through this problem.problem.

Using the table to determine the expected Using the table to determine the expected frequency of any given scorefrequency of any given score

Lets suppose a shirt maker wants to determine how Lets suppose a shirt maker wants to determine how many shirts of a given neck size should be made.many shirts of a given neck size should be made.

We will assume:We will assume:

– Average neck size is 15 and the SD is 2.Average neck size is 15 and the SD is 2.– formula:formula:

FFee=(iN/=(iN/)y– WhereWhere

» I=size of intervalI=size of interval» N=number of shirtsN=number of shirts

fe=(1000/2).3532 =176 size 16 shirts.fe=(1000/2).3532 =176 size 16 shirts.

The Normal Curve and Z scores: Some The Normal Curve and Z scores: Some Final ConsiderationsFinal Considerations

The term normal curve implies that this The term normal curve implies that this type of curve is normal.type of curve is normal.– Mathematicians did once believe that this curve Mathematicians did once believe that this curve

was ‘normal’ and that is how it got its name.was ‘normal’ and that is how it got its name.– We now know that this is a ‘chance’ distribution, We now know that this is a ‘chance’ distribution,

not a normal distribution.not a normal distribution. The normal curve extends from The normal curve extends from ± infinity as the line

never actually touches the base line ±1 sd locates the deflection point for the normal

curve. (line moves out more than down)

Introduction to Behavioral Introduction to Behavioral StatisticsStatistics

Probability, The Binomial Probability, The Binomial Distribution and the Normal CurveDistribution and the Normal Curve

Well-that's it. Next we will look at correlation.

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introduction to behavioral statistics probability, the binomial distribution and the normal curve

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