Download 9. Normal Distribution

The Normal Distribution
BUSA 2100, Sections 3.3, 6.2
Introduction to the Normal
Distribution
The normal distribution is the most
widely used probability distribution.
 Reason: Most variables observed in
nature and many variables in business
are “normally distributed.”
 The normal distribution is a bell-shaped
curve.

Characteristics of the Normal
Distribution
The normal distribution has 3 major
characteristics.
 (1) Most important characteristic -Large frequencies near the mean, and
small frequencies at the extremes.
 (2) It is symmetric about the mean.
 (3) It is infinite in extent (in theory) -doesn’t touch the x-axis.

Examples of the Normal
Distribution
Women’s heights, men’s weights, IQs,
daily sales. (Explain)
 The size of the standard deviation
affects the shape of the normal curve.
 For all normal curves:
68+% of values are within +-1 std.dev.;
95+% of values are within +-2 std. dev.;
99+% of values are within +-3 std. dev.

z-Values
Definition: A z-value represents the
number of standard deviations that an
item is from the mean.
 A normal distribution has mean (mu) =
150 & std. dev. (sigma) = 20. Find the
z-values for X = 170, 150, 130, & 195.
 What is the z-value for X = 163? What
is the formula for z?

z-Values, Page 2
A negative z-value means that the item
is to the left of the mean.
 Items with z-values beyond +-3 std.
deviations are called outliers.
 z-values, together with a normal curve
table, can be used to find probabilities.

Procedure for Calculating
Normal Curve Probabilities
Step 1: Draw a sketch.
 Step 2: Calculate the z-value(s).
 Step 3: Look up the table value(s) in a
normal curve table.
 Step 4: Calculate the final answer.

Normal Curve Example



Example 1: Suppose that daily sales for a
product are normally distributed with mean
220 and standard deviation 36.
What is the approx. range for sales?
(a) What is the prob. that sales are less than
268?
Normal Curve Example, p. 2

(b) What is P(190 <= X <= 240)?
Normal Curve Example, p. 3

(d) What is the probability that sales are
larger than 265?

The normal curve table measures all
probabilities (areas) from the lower
end.
Types of Normal Curve
Problems

All the problems we have just done are
called regular normal curve problems:

Now we will do a backwards normal
curve problem:
Backwards Normal Curve Ex.

Ex. Mileage for a tire is normally distributed
with mean 36,500 and std. dev. 5,000.
 A customer refund will be given for the 10%

of tires that get the least mileage.
What mileage (X-value) qualifies?
Backwards Normal Curve
Example, Page 2

Solve z = (X - mu) / sigma for X.