The Practice of StatisticsThird Edition

Chapter 2:Describing Location in a Distribution

Copyright © 2008 by W. H. Freeman & Company

Empirical Rule

• All Normal distributions are the same if we

measure in units of size σ about the mean μ

as center (better known as converting to z-

scores).

• If the variable we standardize has a Normal

distribution, then so does the new variable z.

• This is a Linear Transformation (Ch 1).

An area under a density curve is a proportion of the observations

in a distribution.

Any question about what proportion of observations lie in some

range of values can be answered by finding an area under the

curve.

• Say Hello to Table A.

• It shows the area under the curve to the

LEFT of z.

Try This

• Find Proportion of observations from a

standard Normal distribution that are less

than 2.22.

• Go to Table A

– Find 2.2 in left column.

– Find .02 in top row.

– What did you find?

The area under the curve to the left of z = 2.22 is .9868.

What does this mean in terms of Standard Deviations and total

area?

Another Example

• Find the proportion of observations from a

standard Normal distribution that are

GREATER than -2.15.

• Go to Table A.

– Find -2.1 in left column

– Find .05 in top row

– What do you find?

This means at .0158 of the observations are to the left of z = -2.15.

But we want GREATER (to the right). What do we do?

1 - .0158 = .9842 is greater than -2.15.

Be Careful. Make sure you know what is being asked.

Table A gives the proportion that is to the LEFT. Make

sure you understand the difference between Less and

Greater!!!!!

We can now answer any question about proportions of

observations in a Normal distribution by standardizing and using

Table A.

Just follow these 4 steps:

Cholesterol in Young Boys

• 14 year old boys cholesterol levels

• Levels above 240 mg/dl require medical

attention

• μ = 170 mg/dl

• σ = 30 mg/dl

• What proportion of 14 year old boys have

cholesterol above 240?

Step 1: State the problem. What is our variable x?

N(170,30)

Finding x > 240

Step 2: Standardize and draw a picture. What is our z-score?

z > 2.33

Step 3: Use Table A. What do you find for z = 2.33? What does

this mean?

Step 4: What do you Conclude?

• z = 2.33

• Table A gives .9901

– 99.01% of 14 year old boys have cholesterol

levels less than 240.

• So area to right is 1 - .9901 or .0099 or

about 1%.

• Only about 1% of 14 year old boys have

cholesterol levels above 240 mg/dl.

An Interval

• What percentage of 14 year old boys have

cholesterol levels between 170 and 240?

• Follow the 4 steps.

• State the problem: 170 ≤ x ≤ 240

Standardize and draw a picture.

0 ≤ z ≤ 2.33

Draw a Normal curve with what we know.

Use Table to find area between 0 and 2.33.

.9901 - .5 = .4901

Conclusion

• State your conclusion in context of the

problem.

• 49.01% of 14 year old boys have

cholesterol levels between 170 and 240

mg/dl.

Find a Value

• SAT verbal N(505, 110) distribution

• How high do you have to score to be in the

top 10%

• State problem & draw picture

– x has area to right of .1.

– What does that imply for area to the left?

• Use the table.

– What is different about this?

Unstandardize to find x.

We have to do some Algebra.

x = 645.8

Conclusion

• We got x = 645.8

• What does this mean?

• You must score 646 to be in the top 10% on

the SAT Verbal.

Assignment

• Exercises 2.18 (b, c, d), 2.30, 2.32, 2.33

• Read pages 148 – 152

• Watch: www.youtube.com/watch?v=bo5STaOUOgI&index=11&l

ist=PLC8478000586FA6F9

• Watch: www.youtube.com/watch?v=_86q-hn_3DQ