Download Continuous Probability Distribution This chapter continues our study

Lecture 13 Standard Normal Distribution
Continuous Probability Distribution
This chapter continues our study of probability distributions by examining the continuous probability
distribution. Recall that a continuous probability distribution can assume an infinite number of values
within a given range. As an example, the weights for a sample of small engine blocks are: 54.3, 52.7,
53.1 and 53.9 pounds.
The normal probability distribution is described by its mean and standard deviation. Suppose the life of
an automobile battery follows the normal distribution with a mean of 36 months and a standard
deviation of 3 months. We can determine the probability that a battery will last between 36 and forty
months. Life of a battery is measured on a continuous scale.
Normal Probability Distribution
A continuous probability distribution uniquely determined by µ and σ. The Greek letter µ (mu),
represents the mean of a normal distribution and the Greek letter σ (sigma) represents the standard
deviation.
The major characteristics of the normal distribution are:
1. The normal distribution is "bell-shaped" and the mean, median, and mode are all equal and are
located in the center of the distribution. Exactly one-half of the area under the normal curve is
above the center and one-half of the area is below the center.
2. The distribution is symmetrical about the mean. A vertical line drawn at the mean divides the
distribution into two equal halves and these halves possess exactly the same shape.
3. It is asymptotic. That is, the tails of the curve approach the X-axis but never actually touch it.
4. A normal distribution is completely described by its mean and standard deviation. This
indicates that if the mean and standard deviation are known, a normal distribution can be
constructed and its curve drawn.
5. There is a "family" of normal probability distributions. This means there is a different normal
distribution for each combination of µ and σ.
Instructor: Ms. Azmat Nafees
15-Oct-09
These characteristics are summarized in the graph.
The Standard Normal Probability Distribution
Since there are an infinite number of probability distributions, it would be awkward to construct tables
of probabilities for so many different normal distributions. An efficient method for overcoming this
difficulty is to standardize each normal distribution. Therefore, a normal distribution with a mean of 0
and a standard deviation of 1 is known as standard normal distribution.
An actual distribution is converted to a standard normal distribution using a z value.
The formula for a specific standardized z value is text formula:
Where:
X is the value of any particular observation or measurement.
µ is the mean of the distribution.
σ is the standard deviation of the distribution.
z is the standardized normal value, usually called the z value.
Applications of the Standard Normal Distribution
To obtain the probability of a value falling in the interval between the variable of interest (X) and the
mean (µ), we first compute the distance between the value (X) and the mean (µ). Then we express that
difference in units of the standard deviation by dividing (X - µ) by the standard deviation. This process
is called standardizing.
Example 1: Suppose the mean useful life of a car battery is 36
months, with a standard deviation of 3 months. What is the
probability that such a battery will last between 36 and 40
months?
z
0.00 0.01
0.02
0.03
0.04 0.05
!
!
!
!
!
!
!
!
1.0
1.1
The first step is to convert the 40 months to an equivalent
standard normal value, using formula [7–5]. The computation 1.2
0.3665 0.3686 0.3708 0.3729
1.3
0.4049 0.4066 0.4082 0.4099
1.4
0.4207 0.4222 0.4236 0.4251
is:
Instructor: Ms. Azmat Nafees
0.3869 0.3888 0.3907 0.3925
15-Oct-09
Next refer to a table for the areas under the normal curve. A part of the table is shown at the right.
To use the table, the z value of 1.33 is split into two parts, 1.3 and 0.03. To obtain the probability go
down the left-hand column to 1.3, then move over to the column headed 0.03 and read the probability.
It is 0.4082.
The probability that a battery will last between 36 and 40 months is 0.4082.
Example 2: The mean weekly income of a shift foreman in the glass industry is normally distributed
with a mean of $1,000 and a standard deviation of $100.
(a)
What is the likelihood of selecting a foreman whose weekly income is between $1,000 and
$1,100?
(b) What is the probability of selecting a shift foreman in the glass industry whose income is
between $790 and $1,000?
(c)
What is the probability of selecting a shift foreman in the glass industry whose income is
between $840 and $1,200?
Empirical Rule
Before examining various applications of the standard normal probability distribution, three areas
under the normal curve will be considered.
1. About 68 percent of the area under the normal curve is within plus one and minus one standard
deviation of the mean. This can be written as µ ± 1σ.
2. About 95 percent of the area under the normal curve is within plus and minus two standard
deviations of the mean, written µ ± 2σ.
3. Practically all of the area under the normal curve is within three standard deviations of the
mean, written µ ± 3σ.
The estimates given above are the same as those shown on the diagram.
Example 3: The daily water usage per person in Gulberg, Lahore is normally distributed with a mean
of 80 litres and a standard deviation of 10 litres. About 68 percent of those living in Gulberg will use
how many litres of water?
About 68% of the daily water usage will lie between 70 and 90 litres (using µ ± 1σ).
Instructor: Ms. Azmat Nafees
15-Oct-09