Probability Distribution- PART

3

Objectives

Introduce the Normal Distribution

Properties of the Standard Normal

Distribution

Introduce the Central Limit Theorem

Use Normal Distribution in an inferential

fashion

Theoretical Distribution

Empirical distributions

based on data

Theoretical distribution

based on mathematics

derived from model or estimated from data

Normal Distribution

Why are normal distributions so important?

Many dependent variables are commonly assumed to be normally distributed in the population

If a variable is approximately normally distributed we can make inferences about values of that variable

Example: Sampling distribution of the mean

So what?

Remember the Binomial distribution With a few trials we were able to calculate possible

outcomes and the probabilities of those outcomes

Now try it for a continuous distribution with an infinite number of possible outcomes. Yikes!

The normal distribution and its properties are well known, and if our variable of interest is normally distributed, we can apply what we know about the normal distribution to our situation, and find the probabilities associated with particular outcomes

Normal Distribution

Symmetrical, bell-shaped curve

Also known as Gaussian distribution

Point of inflection = 1 standard deviation

from mean

Mathematical formula

f (X ) 1

2(e)

(X )2

2 2

Since we know the shape of the curve, we can

calculate the area under the curve

The percentage of that area can be used to determine

the probability that a given value could be pulled

from a given distribution

The area under the curve tells us about the probability- in

other words we can obtain a p-value for our result (data)

by treating it as a normally distributed data set.

Key Areas under the Curve

For normal

distributions

+ 1 SD ~ 68%

+ 2 SD ~ 95%

+ 3 SD ~ 99.9%

Example IQ mean = 100 s = 15

Problem:

Each normal distribution with its own values of

and would need its own calculation of the area

under various points on the curve

Normal Probability Distributions Standard Normal Distribution – N(0,1)

We agree to use the

standard normal

distribution

Bell shaped

=0

=1

Note: not all bell

shaped distributions are

normal distributions

Normal Probability Distribution

Can take on an infinite

number of possible

values.

The probability of any

one of those values

occurring is essentially

zero.

Curve has area or

probability = 1

Normal Distribution

The standard normal distribution will allow us

to make claims about the probabilities of

values related to our own data

How do we apply the standard normal

distribution to our data?

Z-score

If we know the population mean and

population standard deviation, for any value

of X we can compute a z-score by subtracting

the population mean and dividing the result by

the population standard deviation

z X

Important z-score info

Z-score tells us how far above or below the mean a value is in terms of standard deviations

It is a linear transformation of the original scores

Multiplication (or division) of and/or addition to (or subtraction from) X by a constant

Relationship of the observations to each other remains the same

Z = (X-)/

then

X = Z +

[equation of the general form Y = mX+c]

Probabilities and z scores: z tables

Total area = 1

Only have a probability from width

For an infinite number of z scores each point has

a probability of 0 (for the single point)

Typically negative values are not reported

Symmetrical, therefore area below negative value

= Area above its positive value

Always helps to draw a sketch!

Probabilities are depicted by areas under the curve

Total area under the curve

is 1

The area in red is equal to

p(z > 1)

The area in blue is equal to

p(-1< z <0)

Since the properties of the

normal distribution are

known, areas can be looked

up on tables or calculated

on computer.

Strategies for finding probabilities for the

standard normal random variable.

Draw a picture of standard normal

distribution depicting the area of interest.

Re-express the area in terms of shapes like

the one on top of the Standard Normal

Table

Look up the areas using the table.

Do the necessary addition and subtraction.

Suppose Z has standard normal distribution

Find p(0<Z<1.23)

Find p(-1.57<Z<0)

Find p(Z>.78)

Z is standard normal

Calculate p(-1.2<Z<.78)

The Central Limit

Theorem

For simple random samples from any

population with finite mean and variance, as

n becomes increasingly large, the sampling

distribution of the sample means is

approximately normally distributed.

The Central Limit Theorem

The Central Limit Theorem states that under rather

general conditions, sums and means of samples of

random measurements drawn from a population tend

to possess, approximately, a bell-shaped (normal)

distribution in repeated sampling.

The Central Limit Theorem

If random samples of n observations are taken from

any population with mean X and standard deviation

of X , and if n is large enough ( n > 30 ), the

distribution of possible X values will be

approximately normal distribution.