Unit 6
Section 5.5
Section 5.5
5.5: The Normal Approximation
to Binomial Distributions
 Remember:
Binomial distributions have the
following characteristics…
 There
is a fixed number of trials
 The outcome of each trial is independent
 Each experiment has two outcomes or
outcomes that can be reduced to two
outcomes
 The probability of success remains the same
for each trial.
n = the number of trials, p = prob of success
Section 5.5
When to use a normal approximation?
 In
order to be able to use the normal
approximation to compute probabilities
for a binomial variable, the following
conditions must be true:
n times p is greater than or equal to 5
and
n times q is greater than or equal to 5
Section 5.5
Correction for continuity – is employed
when a continuous distribution is used to
approximate a discrete distribution.
(Remember, normal approximations are
continuous, binomial distributions are
discrete)
This correction will require an addition or
subtraction of 0.5 to specific values prior to
calculating probabilities.
Section 5.5
Summary of the Normal Approximation to the
Binomial Distribution
Binomial
Normal
When finding:
Use:
P(X = a)
P(a – 0.5 < X < a + 0.5)
P(X ≥ a)
P(X > a – 0.5)
P(X > a)
P(X > a + 0.5)
P(X ≤ a)
P(X < a + 0.5)
P(X < a)
P(X < a – 0.5)
Section 5.5
Also recall the following formulas associated
with a binomial distribution
Mean
μ= n * p
Standard Deviation
σ= √(n * p * q)
Section 5.5
Steps for the Normal Approximation to the
Binomial Distribution






Check to see whether the normal
approximation can be used.
Find the mean and the standard deviation
Write the problem in probability notation
(using X)
Rewrite the problem by using the continuity
correction factor, and show the
corresponding area under the normal
distribution
Find the corresponding z value
Find the solution
Section 5.5
Example 1:
A magazine reported that 6% of American
drivers read the newspaper while driving. If
300 drivers are selected at random, find the
probability that exactly 25 say they read the
newspaper while driving.
Section 5.5
Example 2:
Of the members of a bowling league, 10%
are widowed. If 200 bowling league
members are selected at random, find the
probability that 10 or more will be widowed.
Section 5.5
Example 3:
If a baseball player’s batting average is
0.320 (32%), find the probability that the
player will get at most 26 hits in 100 times at
bat.
Section 5.5
Example 4:
When n = 10 and p = 0.5, use the binomial
distribution table (Table B in Appendix C) to
find the probability that X = 6.
Now use the normal approximation to find
the probability that X = 6.
How do your answers compare?
Section 5.5
Homework:
 Pg
281-283
 #’s
1 – 29 ODD