Objective 1: I Can Interpret Graphs of Normal Probability Distributions

Statistics5.1: Introduction to Normal Distributions and Standard Normal Distributions

Objective 1: I can interpret graphs of normal probability distributions.

In chapter 4 we looked at discrete probability distributions. In chapter 5, we are going to focus on a very important ______probability distribution, called the ______distribution. Normal distributions can be used to model many types of data sets. For example, lifetime of tv sets, housing costs, and blood pressure measurements are all normally distributed variables.

Properties of a Normal Distribution

A normal distribution is a ______probability distribution for a random variable x. The graph of a normal distribution is called a ______curve. A normal distribution has the following properties:

1)

2)

3)

4)

The normal curve looks like:

A normal distribution can have any ______and any positive ______. These two parameters, _____ and _____, completely determine the shape of the normal curve. The mean gives the location of the ______and the SD describes how much the data are ______.

Which two curves above have the same mean? ______

Which two curves above have the same SD? ______

Example 1, pg 241

TIY 1:

Consider the normal curves shown below. Which normal curve has the greatest mean? ______

Which normal curve has the greatest SD? ______Explain.

Ex 2:

TIY 2:

Objective 2: I can find areas under the standard normal distribution.

There are many normal distributions—weight of newborn babies, height of adult males, IQ scores, etc. But there is only ONE ______. In chapter 2, we you learned that a z-score is a measure of position that indicates the number of standard deviations a value lies from the mean. We used the formula below to transform any x-value in that data set into a standard score.

Read paragraphs on page 243 below this diagram.

So basically, we can now use a table in the back of the book to find the area under the normal curve for ANY z-score, not just -3, -2, -1, 0, 1, 2, 3 like we did in Chapter 2 using the Empirical Rule.

Properties of the Standard Normal Distribution:

1)

2)

3)

4)

Ex 3: Using the Standard Normal Table (pages ______and ______in the orange section)

a) Find the cumulative area that corresponds to a z-score of 1.15

b) Find the cumulative area that corresponds to a z-score of -0.24.

TIY 3:

a) Find the area under the curve to the left of a z-score of -2.19.

b) Find the area under the curve to the left of a z-score of 2.17.

FINDING AREAS UNDER THE STANDARD NORMAL CURVE (pg 245)

1) The area in the table is the area to the ______of any z-score from ______to ______.

What is the area to the left of a z-score of -4 or smaller? ______

What is the area to the left of a z-score 4 or larger? ______

2) To find the area to the right of any z-score, find the area to the ______of the z-score.

Then ______.

What is the area to the right of a z-score of -4 or smaller? ______

What is the area to the right of a z-score 4 or larger? ______

3) To find the area between two z-scores, find the area to the ______of each z-score.

Then ______.

Ex 4: Finding area under the standard normal curve.

a) Find the area to the left of z = -0.99

b) Find the area to the right of z = 2.13.

Ex 5: Finding area under the standard normal curve.

a) Find the area to the right of z = 1.06

b) Find the area to the right of z = -2.16.

Ex 6: Find each area under the standard normal curve.

a) Between z = -1.5 and z = 1.25.

b) Between z = -2.16 and z = -1.35.

c) Between z = 0.19 and z = 2.87

There are three ways that I can give you a z-score and ask you to find the area

1) With ______

2) With ______

3) With ______