Download 7.8 Normal Approximation to the Binomial Distribution

Lesson 7.8 Normal Approximation and the Binomial
Distribution Notes
Statistics
Page 1 of 3
Normal Approximation to the Binomial Distributions
 Consider a binomial distribution where
o n = the number of trials
o r = number of successes
o p = probability of success on a single trial
o q = 1 – p = probability of failure on a single trial
 If np > 5 and nq > 5, then r has a binomial distribution that is approximated
by a normal distribution with µ = np and σ = npq .
o As n increases, the approximation becomes better.
Example 1: The owner of a new apartment building must install 25 water heaters.
From past experience in other apartment buildings, she knows that Quick Hot is a
good brand. A Quick Hot heater is guaranteed for only 5 years, but from the
owner’s past experience, she knows that the probability that it will last 10 years is
0.25.
a. What is the probability that 8 or more of the 25 water heaters will last at
least 10 years? Define success to mean a water heater lasts at least 10 years.
b. How does this result compare with the result we can obtain by using the
formula for binomial probability distribution with n = 25 and p = 0.25?
c. How do the results of parts (a) and (b) compare?
Lesson 7.8 Normal Approximation and the Binomial
Distribution Notes
Statistics
Page 2 of 3
Continuity correction
 Needs to be made when approximating a binomial distribution with a normal
distribution
o If the discrete variable is a left point of an interval, subtract 0.5 to
obtain the corresponding normal variable.
o If the discrete variable is a right point of an interval, add 0.5 to obtain
the corresponding normal variable.
Example 2: For many years of observation, a biologists knows that the probability
is only 0.65 that any given Artic tern will survive the migration from its summer
nesting area to its winter feeding grounds. A random sample of 500 Arctic terns
were banded at their summer nesting area. Use the normal approximation to the
binomial and the following to find the probability that between 310 and 340 of the
banded Arctic terns will survive the migration. Let r be the number of surviving
terns.
a. To approximate P(310 ≤ r ≤340), we use the normal curve with µ =
_________ and σ = _________.
b. P(310 ≤ r ≤340), is approximately equal to P( _____ ≤ x ≤ ____ ), where x
is a variable from the normal distribution described in part (a).
c. Convert the condition P( _____ ≤ x ≤ ____ ) to a condition in standard units.
(Find the z scores.)
d. Find P( _____ ≤ r ≤ ____ ).
e. Will the normal distribution make a good approximation to the binomial for
this problem? Explain your answer.
Lesson 7.8 Normal Approximation and the Binomial
Distribution Notes
Statistics
Page 3 of 3
Example 3: Throw a fair coin for 200 times and let r be the number of Tail
occurring in the 200 trials. Since the coin is fair, p = 0.5 and q = 0.5.
a. To approximate P( 90 ≤ r ≤ 120 ), we use the normal curve with µ =
_________ and σ = _________.
b. P( 90 ≤ r ≤ 120 ) is approximately equal to P( _____ ≤ x ≤ ____ ), where x
is a variable from the normal distribution described in part (a).
c. Convert the probability about x in part (b) into a probability about the
standard normal distribution z: P( _____ ≤ z ≤ ____ ).
d. Find the approximate value using the result of part (c).
e. Will the normal distribution make a good approximation to the binomial for
this problem? Explain your answer.
Assignment: p. 294 # 1, 5, 11; p. 298 # 5, 8, 10, 17, 19