Distributions

Chapter 7

GOALS

1.

2.

3.

4.

5.

6.

Understand the difference between discrete and continuous distributions. List the characteristics of the normal probability distribution.

Define and calculate z values.

Determine the probability an observation is between two points on a normal probability distribution.

Determine the probability an observation is above (or below) a point on a normal probability distribution.

Use the normal probability distribution to approximate the binomial distribution.

Probability Distributions: Review

As

we saw last week, a probability distribution is the result of a series of events such as several coin tosses.

One way to predict the outcome of a ‘best of

3’ coin tosses is to lay out the possible outcomes: What is a Probability Distribution?

PROBABILITY DISTRIBUTION A listing of all the outcomes of an experiment and the probability associated with each outcome.

Experiment:

Toss a coin three times.

Observe the number of heads. The possible results are: Zero heads,

One head,

Two heads, and

Three heads.

What is the probability distribution for the number of heads?

What are the outcomes?

There

are 8 scenarios

Based on the outcomes in each scenario, you could determine the likelihood of each.

Probability Distribution of Number of

Heads Observed in 3 Tosses of a Coin

Odds and distribution

In

this case, if you were to bet on Heads being the outcome at least twice, you would win 50% of the time, since 2 Heads and 1

Tail comes out in 3 of 8 scenarios, and 3

Heads comes out in 1 of 8 scenarios.

This distribution pattern is starting to resemble the ‘normal’ bell-shaped continuous distribution pattern, even though it is an example of a discrete distribution

Types of Random Variables

DISCRETE RANDOM VARIABLE A random variable that can assume only certain clearly separated values. It is usually the result of counting something. CONTINUOUS RANDOM VARIABLE can assume an infinite number of values within a given range. It is usually the result of some type of measurement Normal Distributions

At

the end of last week, we noted that distribution patterns tend to resemble the

‘normal’ – bell-shaped pattern of associated with continuous variables – as the number of observations increase.

We can even use the ‘normal’ curve to approximate the results of a calculated discrete distributions using a correction factor discussed on p. 242 of the text.

The Normal Curve

Not

only can we use the normal curve to determine the results of discrete probability distributions. We can take a short-cut with continuous probability distributions as well, drawing conclusions from our sample observations based on generally accepted universal laws applying to all ‘normal’ continuous distributions. Using the Normal Distribution

Instead of waiting until voting day to officially count how all people will vote you could draw conclusions from sampling at random.

Asking everyone multiple times is too costly.

Sampling can be cost effective and accurate, as long as it is done randomly.

A true random sample will approach the actual results from the whole population.

Random Sampling

Observations have shown that a random sample will approach a true 'normal' population distribution if the sample size is large enough.

Depending on the accuracy required, calculations can be made as to how many samples are required... this will be discussed in more detail next week.

Standard Deviation

As long as the sample size is large enough, a random sample will begin to fall into a 'normal' bell-shaped distribution with a relationship to its' standard deviation measurement.

The 'empirical rule' holds true in all occasions, but conclusions drawn from it tend to be more conservative. Normal Distribution

The normal distribution gives better conclusions, but sampling rules are more stringent.

The accurate measurement of probabilities under the normal curve are…