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Chapter 7
Section 5
The Normal Approximation to
the Binomial Probability Distribution
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 1 of 21
Chapter 7 – Section 5
● Learning objectives
1

Approximate binomial probabilities using the normal
distribution
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 2 of 21
Chapter 7 – Section 5
● Recall from section 6.2 that a binomial
experiment is one where
 An experiment is performed n independent times
 Each time, or trial, has two possible outcomes –
success or failure
 The outcome of success – p – is the same for each
trial
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 3 of 21
Chapter 7 – Section 5
● The binomial probabilities are the probabilities of
observing x successes in n trials
● If X has a binomial distribution, then
P(X = x) = nCx px (1 – p)n-x
● We would like to calculate these probabilities, for
example when both n and x are large numbers
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 4 of 21
Chapter 7 – Section 5
● If both n and x are large numbers, then
calculating the number of combinations nCx in
P(X = x) = nCx px (1 – p)n-x
is time-consuming since we need to compute
terms such as n! and x!
● Calculations for probabilities such as
P(X ≥ x)
are even more time-consuming
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 5 of 21
Chapter 7 – Section 5
● For example, if n = 1000, then
P(X ≥ 700) = P(X = 700) + P(X = 701) + …
+ P(X = 1000)
which is the sum of 301 terms
● Each P(X = x) by itself is an involved calculation,
and we need to do this 301 times, so this
becomes an extremely long and tedious sum to
compute!
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 6 of 21
Chapter 7 – Section 5
● However, we can simplify this calculation
● As the number of trials, n, increases, then the
distribution of binomial random variables
becomes more and more bell shaped
● As a rule of thumb, if np(1-p) ≥ 10, then the
distribution will be reasonably bell shaped
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 7 of 21
Chapter 7 – Section 5
● To show that the approximation is pretty good,
the following chart shows the binomial with
n = 40, p = 0.5, and its corresponding normal
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 8 of 21
Chapter 7 – Section 5
● Even if the original binomial trial isn’t symmetric,
such as for p = 0.3, the normal is still a good
approximation (n = 60 in the chart below)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 9 of 21
Chapter 7 – Section 5
● Because the binomial distribution is
approximately normal when np(1-p) ≥ 10, the
normal distribution can be used to approximate
a binomial distribution
● Remember that a binomial distribution has
 Mean = n p
 Standard deviation = √ n p (1 – p)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 10 of 21
Chapter 7 – Section 5
● It would make sense to use an approximation
that has the same mean and standard deviation
● Thus to approximate a binomial random variable
X (with parameters n and p), we use a normal
random variable Y with parameters
 μ=np
 σ = √ n p (1 – p)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 11 of 21
Chapter 7 – Section 5
● This matches the mean and the standard
deviation between X and Y
● We could then approximate the probabilities of X
with probabilities of Y
P(X ≤ a) = P(Y ≤ a)
● However, there is a slightly better method
● This better method is called the correction for
continuity
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 12 of 21
Chapter 7 – Section 5
● First we will look at an example
● Consider a binomial variable X with
 n = 60 trials and
 p = 0.3 probability of success
● This variable has
 Mean = n p = 18
 Standard deviation = √ n p (1 – p) = 3.55
● Thus we use a normal random variable Y with
mean μ = 18 and standard deviation σ = 3.55
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 13 of 21
Chapter 7 – Section 5
● The graph shows the situation for P(X = 17)
 The pink curve shows the binomial probabilities for X
 The blue curve shows the normal approximation for Y
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 14 of 21
Chapter 7 – Section 5
● A better approximation for P(X = 17)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 15 of 21
Chapter 7 – Section 5
● A better approximation for P(X = 17) would be
P(16 ½ ≤ Y ≤ 17 ½)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 16 of 21
Chapter 7 – Section 5
● Thus a better approximation for
P(X = a)
is
P(a – ½ ≤ Y ≤ a + ½)
and a better approximation for
P(X ≤ a)
is
P(Y ≤ a + ½)
● This is the correction for continuity
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 17 of 21
Chapter 7 – Section 5
● Now we look back at the example
● We want to approximate
P(X ≤ 17)
for a binomial variable X with n = 60 trials and
p = 0.3 probability of success
● X has mean 18 and standard deviation 3.55
● We use a normal random variable Y with mean
18 and standard deviation 3.55
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 18 of 21
Chapter 7 – Section 5
P(X ≤ 17) ≈ P(Y ≤ 17.5)
● Since Y has mean μ = 18 and standard deviation
σ = 3.55, the Z-score of 17.5 is
17.5  18
 0.14
3.55
and P(Z ≤ – 0.14) = 0.4440
● Thus the normal approximation to the binomial is
P(X ≤ 17) ≈ 0.4440
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 19 of 21
Chapter 7 – Section 5
● From the normal, an approximation is
P(X ≤ 17) ≈ 0.4440
● The actual value (computed from Excel) is
P(X ≤ 17) = 0.4514
● These are quite close
● However, technology can compute the Binomial
probabilities directly, so the normal
approximation is not needed as much these
days
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 20 of 21
Summary: Chapter 7 – Section 5
● The binomial distribution is approximately bell
shaped for large enough values of np(1 – p)
● The normal distribution, with the same mean
and standard deviation, can be used to
approximate this binomial distribution
● With technology, however, this approximation is
not as needed as it used to be
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 21 of 21
Example: Chapter 7 – Section 5
● in the United States never use the Internet. (Source: gallup.com)
Suppose 200 people are randomly selected. Use the normal
approximation to the binomial to
 a. approximate the probability that exactly 54 of them will never use the
Internet. (0.0635)
 b. approximate the probability that more than 45 of them will never use
the Internet. (0.9121)
 c. approximate the probability that more than 60 of them will never use
the Internet. (0.1503)
 d. approximate the probability that the number of them who will never
use the Internet is between 46 and 60, inclusive. (0.7618)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 22 of 21
Example: Chapter 7 – Section 5
● Darren did not prepare for a multiple-choice test in his biochemistry class.
He is forced to randomly guess on each question. Suppose the test has 60
questions with 4 possible answers for each question. Use the normal
approximation to the binomial to
 a. approximate the probability that he will get exactly 12 questions right. (0.0797)
 b. approximate the probability that he will get more than 15 questions right.
(0.4407)
 c. approximate the probability that he will get more than 20 questions right.
(0.0505)
 d. approximate the probability that he will get more than 25 questions right.
(0.0009)
 e. approximate the probability that the number of questions he will get is between
12 and 17, inclusive. (0.6236)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 23 of 21
Example: Chapter 7 – Section 5
● Most nurseries guarantee their plants for one year. If the plant dies within
the first year of the purchase date, it can be returned for credit. Experience
has shown that about 5% of all plants purchased from Evergreen Nursery
will die in their first year. Evergreen Nursery sold 250 walnut trees this
season. Use the normal approximation to the binomial to
 a. approximate the probability that exactly 12 walnut trees will be returned for
credit. (0.1142)
 b. approximate the probability that more than 15 walnut trees will be returned for
credit. (0.1920)
 c. approximate the probability that more than 20 walnut trees will be returned for
credit. (0.0101)
 d. approximate the probability that more than 25 walnut trees will be returned for
credit. (0.0001)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 5 – Slide 24 of 21