Topics for Today

Recap and Summary of Probability Distributions

Introduction to the Normal Distribution

Characterizing the Normal Distribution

Standardization

Stat203 Page 1 of 25Fall 2011 – Week 4, Lecture 2

Probability Distributions

Last lecture, we defined a probability distribution as the __________ probability of an individual taking a particular value of a variable.

Like, an individual coin flip has a probability of being heads.

Individual = _______________Variable = _______Event = _______________

… we know the probability distribution of this variable (side facing up), so we can calculate the probability:

____________________

Stat203 Page 2 of 25Fall 2011 – Week 4, Lecture 2

Are the events:- two flipped coins show 1 head- two flipped coins show 2 heads

… __________________?

To decide, there is one simple question …

Can both of these events occur ________________?

__.

So, we can restructure the original question:

P(1 or more Heads) = ____________________

Now can which ________________ can we use to calculate this probability?

Stat203 Page 3 of 25Fall 2011 – Week 4, Lecture 2

Applying the Addition rule:

P(1 or more Heads) = P(1 Head or 2 Heads)= P(______) + P(_______)

So, now we can use the probability distribution of the _______________:

Stat203 Page 4 of 25Fall 2011 – Week 4, Lecture 2

Prob

ability

# Heads from flipping 2 coins

From this: P(1 Head) = ___P(2 Heads) = ____

So, we have

P(1 or more Heads) = P(1 Head or 2 Heads)= P(1 Head) + P(2 Heads)= ___ + ____= ____

So, what was the purpose of this example?

We showed that we can use the ________________________ of a variable to determine the _____________________ related to that variable.

Stat203 Page 5 of 25Fall 2011 – Week 4, Lecture 2

Try other events:

P(no heads) =

[ hint: what is the converse of the event: 1 or more heads ? ]

P(2 heads) =

Stat203 Page 6 of 25Fall 2011 – Week 4, Lecture 2

Probability Distributions for more Coins(probability in % on the y-axis)

Stat203 Page 7 of 25Fall 2011 – Week 4, Lecture 2

# Heads from flipping 3 coins

# Heads from flipping 6 coins

As we flip more and more coins, we begin to see a natural shape appear – this is the ___________________.

[ Images: http://records.viu.ca/~johnstoi/maybe/maybe5.htm ]

Stat203 Page 8 of 25Fall 2011 – Week 4, Lecture 2

# Heads from flipping 20 coins

This is most amusingly demonstrated by Plinko!

If you did lots and lots of Plinko discs, you’d get:http://youtu.be/9xUBhhM4vbM orhttp://youtu.be/5_HVBhwhwV8 or watch a simulationhttp://www.stattucino.com/berrie/dsl/Galton.html

Events for Continuous Variables?Stat203 Page 9 of 25Fall 2011 – Week 4, Lecture 2

What is the probability the next man you meet is _______ 5’6” and 6’1”?

What is the probability a randomly selected Vancouverite is ___________?

What proportion of trees on Burnaby mountain are ___________ 30 cm in diameter?

What part of the carpet is most likely to wear out first?

Stat203 Page 10 of 25Fall 2011 – Week 4, Lecture 2

What is the Normal Distribution andwhy is it so important?

It just so happens that (by far!) the most ______ probability distribution for __________ (interval or ratio) variables.

It approximates the distribution of many discrete variables as well (as in the coin example earlier)

Some things that are normally distributed:

- ___ of people- ______ (ie: marks on a midterm)- Most biological/physiological

measurements (body temperature, height, __, ______)

Stat203 Page 11 of 25Fall 2011 – Week 4, Lecture 2

Exploiting the Normal Distribution

When we determine that a variable follows a Normal distribution, we can use that knowledge to calculate _______________________ occurring, such as:

•Whether the height of the next person to walk in the door is > 5’8”

•Whether your commute will be ____________________

•The number of arrests that VPD might make _______________

•Whether Dave’s computer will last more than 1 year

…

The normal distribution can enable us to understand the chances of all of these things.

Stat203 Page 12 of 25Fall 2011 – Week 4, Lecture 2

But Which Normal Distribution?

Many variables follow a Normal distribution, but they’re not all the same

A Normal distribution is _________ by two things:

•The ____ of the distribution, _

•The __________________ of the distribution _

Let’s compare these.

Stat203 Page 13 of 25Fall 2011 – Week 4, Lecture 2

Two Normal Distributions with Different Meanshttp://www.stattucino.com/berrie/dsl/index.html

Stat203 Page 14 of 25Fall 2011 – Week 4, Lecture 2

Two Normal Distributions with Different shttp://www.stattucino.com/berrie/dsl/index.html

Stat203 Page 15 of 25Fall 2011 – Week 4, Lecture 2

Everything that is normally distributed has similar, but unique shapes

Stat203 Page 16 of 25Fall 2011 – Week 4, Lecture 2

Using the Normal Distribution

We know an event we’re interested in (eg: our nephew will read > 145 words / min).

We know/presume that kids reading speeds are ____________________ …

And we know the ____ (µ) and standard deviation () of reading speed …

Now what.

We know and can calculate the probability of certain ______ of a normal distribution.Stat203 Page 17 of 25Fall 2011 – Week 4, Lecture 2

Probability of Chunks

Here µ = ___ and = __ … and 68% of kids are within 1 standard deviation of the mean.

This is just a ____ of the Normal Distribution.

Stat203 Page 18 of 25Fall 2011 – Week 4, Lecture 2

More Chunks

• ___ of values are within 1 of the mean• ___ of values are within 2 of the mean• _____ of the observations fall within 3 of

http://www.pdnotebook.com/2010/06/statistical-tolerance-analysis-root-sum-square/

Stat203 Page 19 of 25Fall 2011 – Week 4, Lecture 2

Example of Using These Chunks

Scores on the Wechsler Adult Intelligence Scale (a standard IQ test) for the 20 to 34 age group are approximately normally distributed with _______ and ______. Use the 68-95-99.7 rule to answer these questions.

(a) About what percentage of people in this age group have scores above 110?

(b) About what percent have scores above 160?

(c) In what range do the middle 95% of all scores lie?

Stat203 Page 20 of 25Fall 2011 – Week 4, Lecture 2

What about other chunks?

What if you’re interested in a more specific event, how do you do other probabilities.

Conveniently, every normal distribution can be converted exactly to a _______________ distribution; a Normal distribution with ____ and _____.

… and probabilities for many many many different chunks of the standard normal are in tables or software.

Stat203 Page 21 of 25Fall 2011 – Week 4, Lecture 2

What does this mean?

It means that if we need a probability related to a variable which is Normally distributed, we can convert our variable to a Standard Normal, and look the probability in a table!

Stat203 Page 22 of 25Fall 2011 – Week 4, Lecture 2

We all Standardize

How are you doing compared to your classmates?

On one test, you got 90%. The class average was 88%.

On another test, you got 78%. The class average was 55%.

You’d certainly be much more relieved in the second one.

Think about why.

You just standardized the scores on the two tests.

More on Standardization next lecture.

Stat203 Page 23 of 25Fall 2011 – Week 4, Lecture 2

Today’s Topics

Probability distributions- Discrete and Continuous- Underlie the probabilities of events we observe

The Normal Distribution- Most common, naturally occurring probability

distribution- Most biological/physiological measurements

are normally distributed- Characterized by mean µ and standard

deviation

Standardization- Converting any Normal to a Standard Normal- For comparison of two different Normally

Distributed variables- Calculating probability of certain events

Stat203 Page 24 of 25Fall 2011 – Week 4, Lecture 2

Reading for next lecture

Chapter 5 – The Normal Curve

Stat203 Page 25 of 25Fall 2011 – Week 4, Lecture 2