Download Chapter 6: The Normal Probability Distribution

Chapter 6: The Normal Probability
Distribution
This chapter is to introduce you to the concepts
of normal distributions.
E.g. if a large number of students took a
statistics exam and their marks were recorded
in a frequency distribution table, we might
expect a normal distribution. Some students
will do very well, some will do very badly, but
the majority of the students will fall close to
either side of the mean.
• Normal probability distribution characteristics
The normal curve is bell-shaped.
The normal distribution is symmetric about its
mean(50% of all values lie either site of the
mean).
Mean, Median, and Mode are equal
• The Standardized Normal
Any normal distribution can be transformed
into the standardized normal distribution(Z).
The standardized normal distribution has 0
mean and 1 standard deviation.
Any normal distribution can be converted into
the standardized normal distribution by
subtracting the mean from each data point
and dividing by the standard deviation. The
result from this operation is called z value or z
score.
The standard normal value=Z value=(x-µ)/ơ
Where
 X is any data value
 µ is mean of distribution
 Ơ standard deviation
Example: The monthly incomes of managers
normally distributed with a mean of 1200$
and a standard deviation of 150$. What is the
z value for an income X of 1400$? X of 1000$?
Solution: the z value can be computed by using
the formula: Z value=(x-µ)/ơ
For x=1400:
z=(1400-1200)/150
=1.33
For x=1000:
z=(1000-1200)/150
=-1.33
The z value of 1.33 means that the monthly
income of 1400$ for a manager is 1.33 standard
deviation above the mean, and z value of -1.33
shows that the monthly income of 1000$ is -1.33
standard deviation below the mean.
• The Probability under the normal curve
 By computing the z value, we can determine the area
or the probability under the standard normal curve by
referring to the standard normal distribution table.
 The total area under the curve is 1.0
Example:
If z=2.84, what is the area between the z value and the
mean?
By looking at the z column and moving vertically down to
z value of 2.8 and then looking at the z row and moving
horizontally right to z value of 0.04. It is 0.4977. This is
the probability between the z value and the mean.
Example: Refer to the previous example. In
that example the mean income is 1000$ and
the standard deviation is 150$.
a. What is the probability that a particular
monthly income for a manager is between
800$ and 900$?
b. What is the probability that income is more
than 900$?
c. What is the probability that the income is
less than 800$?
• Solution
a. The probability of income between 800$ and
900$
For x=800:
z value=(1200-800)/150=-2.66
For x=900:
z value=(1200-900)/150=-2.0
The probability between the z value of -2.66 and
The mean, p(-2.66<=z<=0.0) is 0.4961.
The Probability between the z value of -2.0 and
The mean, p(-2.0<=z<=0.0) is 0.4772.
Therefore, the probability between the z value
of
-2.66(income=800$) and the z value of -2.0
(income=900$), p(-2.66<=z<=2.0) is the result of
subtracting p(-2.66<=z<=0) with p(-2.0<=z<=0).
p(-2.66<=z<=2.0)= p(-2.66<=z<=0)- p(-2.0<=z<=0)
=0.4961-0.4772
=0.0189
b. The probability that the income is more than 900$ (answer,
p=0.9772)
c. The probability that the income is less than 800$
(answer, p=0.0039)