Make sure you take notes on all the information on the lesson pages.  The hippocampus lesson is of high importance.  The YouTube videos and other links are mandatory and you are accountable for the information in/on them.

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Lesson 03-11: Finding Scores for Probabilities

Standard:

 

Introduction:

In this lesson you will learn

• Given a probability, find the z score.
• How to read a table

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Instruction:

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URL: http://www.montereyinstitute.org/courses/Statistics for Social Sciences/course files/multimedia/lesson07/lessonp.html?showTopic=6

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Additional Instruction:

 

We will find the z scores from a probability 2 ways.  1 is using the applet

and the other way is to use a table.  You will be expected to know how to use both methods.

 

Let's start with the table. 

Example:

Find the z score that corresponds to a cumulative area of 0.3632

Note that the word cumulative means it starts at negative infinity. So you are looking at a LEFT tail distribution.

It is a good idea to sketch your normal curve before you start.  Notice your probability is less than 0.5 so you know your z score is to the left of the mean and thus that it is negative.

Use the table http://www.math.unb.ca/~knight/utility/NormTble.htm 

 

Notice on the table that you only have the positive z scores... but we are expecting a negative z score.

This is okay because the normal graph is symmetrical (same on the right and left side).

So we are looking for a LEFT tail with a probability of 0.3632 but all of these values are over 50%!!!!

Since the table is symmetrical lets figure out the probability of the RIGHT tail...

SUBTRACT FROM 1

1 - 0.3632 = 0.6368

Look at the table for the value of 0.6368

Notice to the left this is a z score of 0.3... but wait, we can get one more decimal point out of this Look UP to get the hundredths place.

z = 0.35

But remember we found the complementary z score.  We expect our z score to be to the LEFT of the mean... we expect it to be negative.

So the answer is z = NEGATIVE 0.35 = -0.35

 

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Find the z score that corresponds to a cumulative area of 0.8925

 

Cumulative area means it is a LEFT tail

Make a sketch of what you expect this to look like. 

Your probability is more than 50% so you will shade in your left tail past the middle.

 

Now look for 0.8925 on the chart

 

Look to the left and you see your z score is 1.2 and look up to get your hundredths place and you get a z score of 1.24

Answer: z = 1.24

 

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Going backwards, what if you know the z score, what is the UNSCALED RAW value?

If you start with the z score formula

multiply both sides by the denominator of sigma

add mu to both sides

and you can find the value of the raw unscaled number from the z score.

 

Example

In N(10, 6) what is the raw score for a z score of 2?

x = (2)(6) + (10)

x = 12 + 10

x = 22

 

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You will need to save this z score table

RIGHT CLICK Over the table and SAVE IMAGE AS

Save this to your computer.

Now you can open this image in paint if you have a PC (if you have a mac http://www.apple.com/downloads/macosx/imaging_3d/paintbrush.html ) 

MAC users: download and install the free paintbrush application. Right click on the picture where you saved it and OPEN WITH paintbrush.  Make sure after you draw on it that you go file-save as so you don't mess up the original picture.

PC users: go to programs->Applications->Paint

OPEN the file in Paint.  Draw on it

Make sure you do FILE->Save As so you don't mess up the original file.

 

To show your work you can draw a circle around the important parts of the table and then save the new image and upload it to your google sites.

 

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Additional Instruction:

 

Note that the homework directions say to use the table.

If you want to use the http://davidmlane.com/hyperstat/z_table.html tool 

you will need to set it up as a standard normal distribution N(0,1)

The INTERPOLATE until you find the right z score.

 

Example if you want to find the z score for a cumulative area of 0.3228

start with N(0,1)

Switch to BELOW and GUESS at what you think the z score is. 

Since my cumulative area is 32%... I will guess a negative z score.

Notice the shaded area is only 0.15 and not 0.3228

I need MORE area so I will need to move my z score to the right a little bit.

Still not enough.  I need to move my z score further to the right

And even when I took my z score over to -0.7 it is STILL not enough area.  I need to move over farther to the right.

With a z score of -0.5 I am getting closer... but not there yet...

keep INTERPOLATING (guessing until you narrow in on the exact value)

Oops, went too far. So I know my answer is between a z score of -0.5 and -0.3

So I will now try something in between.

Oh, I am sooo close.  I need 0.3228 and this is 0.3263.  I have a little too much. 

And BINGO, I interpolated my way to getting the z score of -0.46 for a cumulative area of 0.3228

 

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Going Backwards.  What if I give you the z scores and want to know the probabilities?

 

 

Example: Find the area under the standard normal curve between z = -1 and z = 3

First notice the table does not have z scores of -1, so you will need to find the z score of positive 1 and use the complement.

P(Z < -1) = 1 - P(x < 1) = 1 -0.8413 = 0.1587

So now we know the left tail of Z < -1

 

Now the table will give us the left tail of Z < 3

From the table we can see the probability for a z score of 3.0 is 0.9987

The problem is we do not want a left tail we want IN BETWEEN

So you need to CUT OFF (subtract) the left tail value of 0.1587

P(-1<Z<3) = 0.84

 

So you can use the table or you can use the between function on the normal applet using N(0,1)