Introduction to Behavioral Statistics Probability, The Binomial Distribution and the Normal Curve

n Introduction n Introduction 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)

Probability, The Binomial Distribution and the Normal Curve n n Additive Theorem - –The –The probability that any one of a set of mutually exclusive events will occur is the sum of the probability of the separate events. –What –What is the probability of drawing either of two cards from a deck. – –(1/52 + 1/52) = 2/52 or.038)

Probability, The Binomial Distribution and the Normal Curve n n Multiplication Theorem - –The –The joint probability of obtaining both of two events is the product of their separate probabilities. n What n What is the probability of drawing two aces from a deck of cards. –(4/52 –(4/52 * 3/51) = 12/2652 or.0045) n What n What is the probability of first a 5 and then a 6. –(1/6 –(1/6 * 1/6) = 1/36 n With n With two dice, what is the probability of rolling a 7 or 11? –How –How many ways to get 7 (4 +3) (5+2) (6+1) on each die »using »using the additive theorem we see that there are 6/36 ways to get 7 and 2/36 to get 11 »Thus »Thus there are 1/9 ways to to roll a 7 or 11.

Probability, The Binomial Distribution and the Normal Curve n n Multiplication Theorem - n What n What is the probability of drawing two aces from a deck of cards.

Probability, The Binomial Distribution and the Normal Curve n Permutations n Permutations of r objects taken r at a time of N objects taken r at a time

Probability, The Binomial Distribution and the Normal Curve n Combination n Combination of N objects taken r at a time.

Probability, The Binomial Distribution and the Normal Curve n Binomial n Binomial Distribution –This –This distribution is very important to psychology. _ The _ The chi square distribution is based on it… normal curve is based on it…. _ F _ F and t distributions are The based on it...

Probability, The Binomial Distribution and the Normal Curve n Binomial n Binomial Distribution dBernoulli dBernoulli Trial n Experiments n Experiments often have only two possible outcomes. –true –true false –effective –effective not effective n Flipping n Flipping a coin one time and noting whether it lands heads or tails, or randomly drawing one sample from a distribution is called a Bernoulli trial or Bernoulli experiment.

Probability, The Binomial Distribution and the Normal Curve n Bernoulli n Bernoulli Trial –Characteristics –Characteristics of a Bernoulli trial ¶ A ¶ A trial can result in one of two outcomes · The · The probability of success remains constant from trial to trial. ¸ The ¸ The outcomes of successive trials are independent. –In –In reality very few real situations situations meet these requirements since probability doesn’t remain constant when we remove an item from the distribution.

Probability, The Binomial Distribution and the Normal Curve n Binomial n Binomial Distribution –In –In this distribution, the random variable variable is a sum (the number of successes observed on n greater than or equal to two Bernoulli trials. n This n This distribution is a relatively simple example of an important class of theoretical distributions or models that are referred to as sampling distributions.

Probability, The Binomial Distribution and the Normal Curve n Binomial n Binomial Distribution –Sampling –Sampling distribution is the special name that is given to a probability distribution where the random variable is a statistic based on the result of n greater than or equal to 2 trials. –The –The Binomial Distribution is one of the distributions used by psychologists.

Probability, The Binomial Distribution and the Normal Curve n Binomial n Binomial Distribution –The –The number of successes observed on n greater than or equal to 2 identical Bernoulli trials is called a binomial random variable, variable, and its probability distribution is called a binomial distribution. –If –If we toss a fair coin 5 times, the probability of observing exactly r heads in n tosses is given by p(X=r)= n C r p r q n-r –This –This gives the probability that the random variable X equals r heads. n C r n C r is the combination of nobjects taken rat a time. Pis the probability of a success (a head), and q = (1-p). –Next –Next - lets look at a particular example:

Probability, The Binomial Distribution and the Normal Curve Binomial Distribution for N=5 and p=1/2

Probability, The Binomial Distribution and the Normal Curve Binomial Distribution for N=5 and p=1/2 Histogram for binomial distribution shown above…. Notice how much this resembles the form of a normal curve.

Probability, The Binomial Distribution and the Normal Curve  The  The normal curve is a limiting form of the binomial distribution normal curve occurs when we have an infinite number of events occurring according to the laws of chance.

Probability, The Binomial Distribution and the Normal Curve n We can write a formula to plot a set of points –In this case we have used the formula y=A+bX to plot a straight line. n In this same way we can generate a plot of the normal curve

Probability, The Binomial Distribution and the Normal Curve Where: f(X)=height of the distribution at X Pi = approximately e=base of natural logarithms) approximately Where: f(X)=height of the distribution at X Pi = approximately e=base of natural logarithms) approximately 2.718

Probability, The Binomial Distribution and the Normal Curve Fortunately, we don’t need to calculate the normal curve. We just use the table in the back of our book…..

Using the Normal Curve to normalize a distribution of scores

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= Using IQ data 1.29(20.2) = This is the normalized centile score.

Determining the % of cases which fall between any two scores K For our IQ data - suppose we want to know what % of scores fall between and K First we convert these scores to z scores ( )/20.2 = 1.00 ( )/20.2=.50 KThen we get the area from column 2 and subtract =.1498 or 15% K To see exactly where the above values came from, use the table in your book and work through this problem.

Using the table to determine the expected frequency of any given score n Lets suppose a shirt maker wants to determine how many shirts of a given neck size should be made. n We will assume: –Average neck size is 15 and the SD is 2. –formula: F e =(iN/ F e =(iN/F)y –Where »I=size of interval »N=number of shirts n fe=(1000/2).3532 =176 size 16 shirts.

The Normal Curve and Z scores: Some Final Considerations The term normal curve implies that this type of curve is normal. The term normal curve implies that this type of curve is normal. –Mathematicians did once believe that this curve was ‘normal’ and that is how it got its name. –We now know that this is a ‘chance’ distribution, not a normal distribution. n The normal curve extends from n The normal curve extends from ± infinity as the line never actually touches the base line n n ±1 sd locates the deflection point for the normal curve. (line moves out more than down)

Introduction to Behavioral Statistics Probability, The Binomial Distribution and the Normal Curve Well-that's it. Next we will look at correlation. Press Above to Return to Class Page!