Download Lesson Three Normal Distribution File

The Normal
Distribution
DeMoivre-Laplace Theorem:
Let X be a binomial random variable defined on n independent trials
each having success probability p. For any number c and d:


X  np
1
lim P  c 
d


n 
npq
2


d
e
2
 x2
dx
c
If this integral is a continuous probability density function, then:
1
2

e

2
 x2
dx  1
Standard Normal Distribution:
A Random Variable Z is said to have a standard normal distribution
given by the following continuous pdf:
1  12 z 2
f ( z) 
e
2
A Standardized Normal Distribution is a normal distribution with a
mean of 0 and a standard deviation of 1.
Cumulative Distribution Function for a Standard
Normal Distribution
There is a special notation used for a cumulative distribution of a
standard normal distribution or P ( X  x ).
P( X  x) or P( X  x)
P( X  x)  ( z )
Let the random variable X have a normal distribution. Then:
E( X )  
Var ( X )   2
Find the probability of getting a z-value less than 0.43 in a standard
normal distribution.
Answer: 0.6664
Find the area between z = 0 and z = 1 in a standard normal curve.
Answer: 0.3413
Find the probability of getting a z-value in a standard normal distribution
that is greater than 1.82.
Answer: 0.0344
If the probability of getting less than a certain z-value is 0.1190, what is
the z-value?
Answer: -1.18
Find the probability of getting a z-value in a standard normal distribution
that is between 0.46 and 1.75.
Answer: 0.2827
Find the z-value for which the area under the standard normal curve to
the right of that value is 0.025.
Answer: 1.96
Properties of the Standard Normal Distribution:
1. The probability that a z-score falls within 1 standard deviation of the mean
on either side, that is, between z = -1 and z = 1, is approximately 68%.
2. The probability that a z-score falls within 2 standard deviations of the mean
on either side, that is, between z = -2 and z = 2, is approximately 95%.
3. The probability that a z-score falls within 3 standard deviations of the mean
on either side, that is, between z = -3 and z = 3, is approximately 99.7%.
A Random Variable X is said to have a normal distribution with standard
mean and variance is given by:
X
N ( ,  2 )
or
 12  x 
1
f ( x) 
e
 2
This function can be used for normal distributions with a mean different
from 0 and a standard deviation different from 1.
We can convert this into a standardized normal
variable by converting each of its scores into
standard scores, given by:
z
x

2
In a Normal Distribution, the mean and standard deviation are given
below. What is the probability of obtaining a value greater than 30?
  25
 5
Answer: 0.1587
In a Normal Distribution, the mean and standard deviation are given
below. What is the probability of obtaining a value less than 20?
  15
  2.3
Answer: 0.9850
In a Normal Distribution, the mean and standard deviation are given
below. What is the probability of obtaining a value greater than 120?
  100
  16
Answer: 0.1056
In a Normal Distribution, the mean and standard deviation are given
below. What is the probability of obtaining a value between -6 and 9?
 3
 4
Answer: 0.9210
In a Normal Distribution, the mean and standard deviation are given below.
Find the x-value for which the area to the left of that value is 0.04.
  100
  16
Answer: 72
In a Normal Distribution, the mean and standard deviation are given
below. Find the x-value for which the area to the right of that value is 0.6.
 3
 4
Answer: 1.99
In a 1905 study, R. Pearl determined that the brain weights of Swedish
men are approximately normally distributed with a mean and standard
deviation given below. Determine the percentage of Swedish men with
brain weights between 1.50 and 1.70 kg.
  1.4
  0.11
Answer: 0.178 or 17.8%
The annual wages, excluding board, of US farm laborers in 1926 are
normally distributed with a mean and standard deviation given below.
In 1926, what is the probability that a US farm laborer will have an
annual wage of at least $400?
  $586
  $97
Answer: 0.972
The masses of a certain species of bird have a normal distribution
with mean 50 grams. If 10 percent of the birds weigh more than 60
grams, find the variance of their masses.
Answer: 61 grams
The annual rainfall in a town has a normal distribution with standard
deviation of 5 inches. If the rainfall is over 20 inches for a third of the
years, find the mean rainfall.
Answer: 17.8 inches
The weights of a certain species of bird are normally distributed with
mean 0.8 kg and standard deviation 0.12 kg. Find the probability that
the weight of a randomly chosen bird of the species lies between 0.74
kg and 0.95 kg.
Answer: 0.586
M02/HL1/11
The random variable X is normally distributed and
P( X  10)  0.670
P( X  12)  0.937.
Answer: 9.19
Find E(X).
M03/HL1/14
Z is the standard normal variable with mean 0 and variance 1. Find
the value of a such that:
P( Z  a)  0.75
Answer: a = 1.15
M01/HL1/13
The speed of cars at a certain point on a straight road are normally
distributed with mean and standard deviation. 15% of the cars
travelled at speeds greater than 90 km/hr and 12% of them at
speeds less than 40 km/hr. Find the mean and standard deviation.
Answer:
Mean = 66.6
Standard Deviation = 22.6
SPEC06/HL1/8
A certain type of vegetable has a weight which follows a normal
distribution with mean 450 grams and standard deviation 50 grams.
a) In a load of 2000 of these vegetables, calculate the expected
number with a weight greater than 525 grams.
b) Find the upper quartile of the distribution.
Answers: 134, 484
N06/HL1/9
The weights in grams of bread loaves sold at a supermarket are
normally distributed with mean 200 grams. The weights of 88% of the
loaves are less than 220 grams. Find the standard deviation.
Answer:
Standard Deviation = 17.0
M06/HL1/8