1 Normal Distribution

7/31/2019 1 Normal Distribution

1/12

HYPOTHESIS

TESTING- NORMALDISTRIBUTION


2/12

is symmetrical about the mean.

is asymptotic. That is the curve gets closer and

closer to theX-axis but never actually touches it.Has its mean, , to determine its location and itsstandard deviation, , to determine its dispersion.

The Normal probability distribution

is bell-shaped and has a single peak at the

center of the distribution.


3/12

-

5

0

.

4

0

.

3

0

.

2

0

.

1

.

0

x

f

(

x

r

a l

i

t

r b u i

o n :

m = 0 ,

s2 = 1

Mean, median, and

mode are equal

Theoretically, curve

extends to

infinity

a

Characteristics of a Normal Distribution

Normal curve issymmetrical

7-10


4/12

Standard Normal (Z) Table

There are infinite number of combination of and exist, an infinite number of normal

distributions exist and an infinite number of

tables would be required.

However, by standardizing the data, we need

only one table- Z table


5/12

The Standard NormalProbability Distribution

Xz

It is also called the

z distribution.

The standardnormal distributionis a normal distribution

with a mean of 0 and astandard deviation of 1.


6/12

Uses of normal distribution

Finding probabilities corresponding to knownvalue of X or Z

Given X or Z value: find the probability of its

occurrence Finding values of X or Z corresponding to

known probabilities

Given the probability of occurrence: find the X orZ that would occur


7/12

Finding probability corresponding toknown values of X or Z

Example: An engineering firms contracts receivedper year follows a normal distribution with = 50and = 5

Area under a normal bell shaped curve for this firmrelate to the entire history of the firm representing thepopulation

The probabilities or proportion of the area under the

curve must add up to 1

Q) Firm wants to know what is the probability that itwill receive between 50 - 55 contracts next year?


8/12

Finding probability correspondingto known values of X or Z

Converting to standard normal value

From the Z table for Z=1.00 the associatedprobability = 0.3413

Az-value of 1 indicates that the value of 55 isone standard deviation above the mean of 50.

00.1

5

5055

Z


9/12

Finding Probabilities Corresponding to KnownValues

-3 -2 -1 +1 +2 +3

35

-3

40

-2

45

-1

50

0

55

+1

60

+2

65

+3

Area is 0.3413

Z Scale

Z Scale

(=50, =5)

Area between and + 1= 0.3431


10/12


If the firm wants to find out what is theprobability that it will get between 45 to 50contracts next year?

Z = -1.00

Area between and - 1 = -0.3431


11/12


What is the probability for the firm to getcontracts between 45 and 55 next year?

Area between 1 = (2 x 0.3431) = 0.6826

Similarly area between 2 = 0.9544

Area between 3 = 0.9973


12/12

Areas Under the Normal

Curve

Practically all is within three standard

deviations of the mean.

+ 3

About 68 percent of

the area under the

normal curve is withinone standard deviation

of the mean.

+ 1

About 95 percent is within two standard

deviations of the mean.

+ 2

Documents

1 Normal Distribution