Download Multinomial, Hypergeometric, Poisson Distributions

Normal Distribution
Dhon G. Dungca, PIE
Normal Distribution
• Also known as the Gaussian
Distribution
• A distribution where the mean, median
& mode are equal.
Normal Curve
Mean, Median
& Mode
Characteristics of the Normal
Distribution
• The curve has a single peak.
• It is bell-shaped.
• The mean lies at the center of the distribution.
The distribution is symmetrical about the
mean.
• Two two tails extend indefinitely in both
directions coming closer and closer to the
horizontal axis but never quite touching it.
Additional characteristics of
the Normal Distribution
• The curve has a single maximum at x=.
• The curve is symmetrical about the vertical
line x= . Thus, the height of the curve at
some point say, x= + is the same as the
height of the curve at x= -.
• The curve is concave downward between x=
- and x= +, and concave upward for
value of x outside that interval.
Additional characteristics of
the Normal Distribution
• The total area under the curve is 1.00,
and since the curve is symmetrical
about x= , it follows that the area on
either side of the vertical line x= is 0.50
• As x moves away on either side of the
mean , the height of the curve
decreases but remains non-negative for
all real values of x.
Normal Distribution Formula
Z=x-

for the population
Z=x-x
s
for the sample
Normal Distribution Formula
• Where:
• Z = the number of standard deviations
from x to the mean of this distribution
• x = the value of the random variable
•  or x = the mean of the distribution
•  or s = the standard deviation
Areas under the Normal Curve
P(x)
P = 0.50
Z=0
Example 1
A random variable has a normal distribution with
the mean = 80 and the standard deviation
of 4.8. What are the probabilities that this
random variable will take on a value:
a) Less than 87.2
b) Greater than 76.4
c) Between 81.2 and 86
0.9332, 0.7734, 0.2957
Example 2
Records show that in a certain hospital the
distribution of the length of stay of its
patients is normal with a mean of 10.5 days
and a standard deviation of 2 days.
a) What percentage of the patients stayed 8
days?
b) What is the probability that a patient stays in
the hospital between 9 and 11 days?
10.56%, 37.21%
Example 3
Given a normal distribution with  = 40
and  = 6, find the value of x that has
a) 45% of the area to the left.
b) 14% of the area to the right.
39.22, 46.48
Example 4
A certain type of storage battery lasts, on
average, 3.0 years with a standard
deviation of 0.5 year. Assuming that the
battery lives are normally distributed, find
the probability that a given battery will last
less than 2.3 years.
0.0808
Example 5
In an industrial process the diameter of a ball
bearing is an important component part.
The buyer sets specifications on the
diameter to be 3.0 ± 0.01 cm. The
implication is that no part falling outside
these specifications will be accepted. It is
known that in the process the diameter of a
ball bearing has a normal distribution with
mean 3.0 and a standard deviation 0.005.
On the average, how many manufactured
ball bearings will be scrapped?
4.56%
Example 6
Gauges are used to reject all components
where a certain dimension is not within the
specification 1.50 ± d. It is known that this
measurement is normally distributed with
mean 1.50 and standard deviation 0.2.
Determine the value d such that the
specifications “cover” 95% of the
measurements.
0.392
Example 7
The average grade for an exam is 74, and the
standard deviation is 7. If 12% of the class
are given A’s, and the grades are curved to
follow a normal distribution, what is the
lowest possible A and the highest possible
B?
lowest A = 83, highest B = 82