Stats Chapter 6 Normal Probability Distributions

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NORMAL PROBABILITY DISTIBUTION

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1. MOST IMPORTANT DISTRIBUTION

2. USED WITH CONTNUOUS RANDOM VARIABLES

(THAT ARE EITHER NORMAL OR APPROIMATELY NORMAL DISTURBASTION)

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A random variable is said to have a normal distribution WHEN

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i it has a probability distribution that is symmetric and bell-shaped see figure 4.1.

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WHAT TWO PARAMETERS THAT ARE N IN THE SHAPE AND POSITION OF A NORMAL DISTRIBUTION CURVE

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THE MEAN
AND STANDARD DEVIATION

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WHAT IS THE AREA UNDER THE CURVE

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EQUAL TO 1 OR 100%

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THE MEAN

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IS ALSWAYS SMYMMETRIC

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THE SHAPE OF A NORMAL STANDARD DEVIATION IS WHAST

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BELL SHAPED

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THE NORMAL CURVE IS

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BELL SHAPED AND ALWAYS SYMMETRIC

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MEAN IS LOCATED

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IN THE MIDDLE

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MEAN IS THE SAME AS

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MEDIAN

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HOW DO YOU MEASURE THE DISTANCE UNDER THE CURVE

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IN Z SCORES

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TO THE RIGHT OF THE MEAN IS

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POSITIVE

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TO THE LEFT OF THE MEAN IS

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NEGATIVE

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REMEMBER IF AREA IS USED IT MEANS

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PROBAILITY

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When a graph of all such (X, y) points is drawn,

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the normal (bell-shaped) curve will appear.

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a random VARIABLE IS

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A CONTINUOUS DISTRIBUTION WHICH IS USED IN NORMAL DISTRIBUTION

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AREA UNDER THE CURVE IS EQUAL TO

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ONE AND IT IS SYMMETRIC

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When given any normal distribution, we can transform it
into the standard normal distribution by

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standardizing all
the data values.
• In other words, convert all x-values into z-scores (or
standard scores).

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ANOTHER NAME FOR Z- SCORE

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STANDARD SCORE

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NAME THREE CONCEPTS THAT MEAN THE SAME

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PERCENT
PROPORTION OR FRACTIONS

PROBABILITY OR AREA

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The z-score is the

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number of standard
deviations that a particular x-value is away
from the mean

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The formula for converting an x-value into a
z-score is:

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Z = VALUE - MEAN
__________________

STANDARD DEVIATION

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PROPERTIES OF THE STANDARD NORMAL DISTRIBUTION

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1. The total area under the normal curve is equal to 1.

2. The distribution is mounded and symmetrical; it
extends indefinitely in both directions, approaching
but never touching the horizontal axis.

3. The distribution has a mean of 0 and a standard
deviation of 1.

4. The mean divides the area in half (0.50 on each
side).

5. Nearly all the area is between z = -3.00 and z =
3.00. (Remember Empirical Rule)
(pg 316

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Given z-Scores

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We will learn how to find the area under the
standard normal distribution curve for any zscore.
Note: Area under curve = Probability
• Use Table E “The Standard Normal
Distribution Tables”, which you can print from
BlackBoard

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WHAT DOES AREA UNDER THE CURVE MEAN

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PROBABILITY

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THE STANDARD NORMAL DISRIBUTION OF THE STANDARD VARIABLE Z EQUALA

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THE STANDARD SCORE OR THE Z SCORE

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Standard Normal Distribution Tables
•Notice for Table E:

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1.The area given under the curve is the shaded region to the left of the z-score.

2.
The curve is symmetric, so the area below –z also equals the area above +z.


3.Total area under the curve equals 1 (which is also total probability).

4.Total area above the mean is 0.5 and total area below the mean is 0.5.

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Find Areas Given z-Scores (cont’d)

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Step 1: Draw a picture.
Step 2: Shade the area desired.
Step 3: Find the corresponding area by using the Standard Normal Distribution Table E.
(you may have to manipulate the result)

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total area under the curve

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is 1

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These Z-scores can then be used to find

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the area (and thus the probability) under the normal curve.

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A Z-score

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is the number of standard deviations that a given x value is above or below the mean. If z represents the Z-score for a given x value then

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Find z-Scores When Given Area (Probability) WHAT MUST YOU DO`

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•Remember to use the table in reverse!!!
•On the inside (or body) of Table E are the areas and on the outside are the z-scores.

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Find the z-score for the standard normal distribution


WHEN GIVEN AREA IS TO THE RIGHT TAIL



.

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1.NEED TO SUBTRACT FROM ONE AND TAKE THE ANSWER TO LOOK UP Z- SCORE

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DESCRIBE THE DISTRIBUTION OF THE STANDARD NORMAL SCORE OF Z?

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A BELL SHAPED CURVE DISTRIBUTION WITH A MEAN 0 AND A STANDARD DEVIATION OF 1

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WHY IS THIS DISTRIBUTION CALLED A STANDARD NORMAL

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THE VARIABLE IS Z; THE STANDARD SCHORE AND THIS DISTRIBUTION IS THE STANDARD OR REFERENCE USED TO DETERMINE THE PROBABIOITIES FOR OTHER NORMAL DITRIBUTIONS