Chapter 5 -- The Normal Distribution
In the world around us, we observe a wide
variety of variables – height, weight, test scores,
life of a brand of batteries, etc. Surprisingly,
these variables share an important characteristic:
their distributions have roughly the shape of a
bell-shaped curve (normal curve) like the one
shown below.
A continuous random variable X is said to be
normally distributed or to have a normal
distribution if its distribution has the shape of a
normal curve.
A normal distribution is completely determined
by its mean and standard deviation. The
probability density function of a normal random
variable X is of the form
f(x) =
1

e
2
( x   )2
/ 2 2
Where  = mean = E(X) and  = S.D(X)
Notation: X ~ N(, )
How to find probabilities using the graph of f?
Standard Normal Random Variable A normal
random variable with a mean of 0 and a standard
deviation of 1. Such a variable is denoted by Z.
That is, Z ~ N(0, 1).
Using the TI-83 to find probabilities
Example 1: Let Z ~ N(0, 1). Find the following
probabilities:
1.
2.
3.
4.
5.
6.
P(Z  -1.15)
P(Z  1.15)
P(0  Z  0.83)
P(-2.45  Z  1.36)
P(Z  -0.34)
P(Z < -1.5 or Z > 1.5)
0.1251
0.1251
0.2967
0.9060
0.6331
0.1336
Example 2: Let Z ~ N(0, 1). Find z if
1. P(Z < z) = 0.0594
2. P(Z > z) = 0.5
3. P(Z > z) = 0.1635
-1.56
0.0
0.98
Example 3: Let X ~ N(240, 30). Find the
following probabilities:
1. P(X  200)
2. P(X  300)
3. P(200  X  300)
Example 4: The personnel manager of a large
company requires job applicants to take a certain
test and achieve a score of at least 500. If the test
scores are normally distributed with a mean of
485 and standard deviation of 30, what
percentage of the applicants pass the test?
(0.3085)
Example 5: Experience indicates that the
development time for a photographic printing
paper is normally distributed with a mean of 30
seconds and standard deviation of 1.1 seconds.
Find the probability that it will take between
28.5 and 31.2 seconds for a randomly selected
piece of photographic printing paper to develop.
(0.7760)
Example 6: The diameter of ball bearings
manufactured at a factory is normally distributed
with a mean of 3 mm and a standard deviation of
0.1 mm. A customer has specification that
require that ball bearings have diameter between
2.85 and 3.1 mm. (a) What fraction of ball
bearings manufactured meet specifications? (b)
What fraction of ball bearings manufactured do
not meet specifications? (0.7745, 0.2255)
Grading by the Curve (will discuss this in
class)
HW: (5.2-5.4) 9-31 (odd) pp. 219-220;
7-15(odd) pp. 225-226; 1, 5, 9, 13, 17, 21, 25,
27, 29, 31, and 33 pp. 234
5.5 The Central Limit Theorem
Population – the set of ALL elements of interest
for a particular study.
Sample – A subset of a population, usually
chosen so that it is representative of population.
Reasons for Sampling
If you want information about a population, two
alternatives are available:
1. Census – sample entire population
Exact, takes time, and costly
2. Sample
Inexact, quicker, and less expensive.
Sampling Error: error resulting from using a
sample to estimate a population characteristics.
Sampling Distribution of the Mean: For a
variable X, and a given sample size n, the
distribution of the variable X (that is, of all
possible sample means) is called the sampling
distribution of the mean.
Notations:
N=
Population size,
n =
Sample size,
X =
=
=
 =
 =
X
X
RV associated with the population,
Population Mean,
Population S. D,
Mean of X = E( X ),
S.D. of X .
THEOREM
1.  =
2.  =
X
X

/
n
Note: The standard deviation of X (/ n ) is also
called the standard error of the mean.
Shape of the Distribution of X .
1. Suppose that a variable X of a population is
normally distributed with a mean of  and a
standard deviation of . Then, for samples
of size n, the variable X is also normally
distributed with a mean of  and a standard
deviation of / n . That is,
X ~ N(, / n )
2. (The Central Limit Theorem) For a
relatively large sample size, the variable X is
approximately normally distributed,
regardless of the distribution of X. The
approximation becomes better and better
with increasing sample size.
Note: A sample size of 30 or more is generally
considered large.
Example 7: Weights of men are normally
distributed with a mean of 170 lbs and a
standard deviation of 20 lbs. A sample of 100
men is taken and their weights were observed (a)
What is the distribution of X , the sample
average? (b) Find the probability that a
randomly selected man will weigh less than 166
lbs? (c) What is the probability that the average
weight for the sample selected will be less than
166 lbs? (d) What is the standard error of the
mean?
Ans. (a) N(170, 2) (b) 0.4207 (c) 0.0228 (d) 2
Example 8: An electrical component is designed
to provide a mean service life of 3000 hours
with a standard deviation of 800 hours. A
customer purchases a batch of 50 components.
What is the probability that the mean life for this
sample will be a least 2750 hours?
Ans. (0.9864)
Example 9: One hundred small bolts are packed
in a box. These bolts are randomly selected from
a population with a mean weight of 1 ounce and
a standard deviation of 0.01 ounces. Find the
probability that the average weight of the bolts
in the box is less than 1.0015 ounces.
Ans. (0.9332)
Example 10: The scores on the ACT college
entrance examination in a recent year had a
mean of 18.6 and a standard deviation of 5.9.
The average score of 76 students at Northside
High who took the test was 20.4 What is the
probability that the mean score for 76 students
chosen randomly from all who took the test
nationally is 20.4 or higher? Ans. (0.004)
HW: 3, 4, 5-15 (odd) pp. 246-248