1. 6.6 NORMAL APPROXIMATION TO
^ TO P
BINOMIAL DISTRIBUTION AND
DISTRIBUTION
Chapter 6:
Normal Curves and Sampling Distributions
2. Page 308 – 309
Normal Approximation to the Binomial
Distribution
Under the conditions stated below, the normal
distribution can be used to approximate the
binomial distribution.
Consider a binomial distribution where
n = number of trials
r = number of successes
p = probability of success on a single trial
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
and
Note: as n increases, the approximation becomes
better
3. Page
Example 14 – Binomial Distribution 309
Graphs
Notice that as n increases, the normal
approximation
to the binomial distribution improves
4. Page
311
Continuity Correction
The normal distribution is for a continuous
random variable. The binomial distribution is
for a discrete random variable. So, in order to
use the normal distribution to approximate the
binomial distribution, we need to make a
continuity correction.
5. How to Make the Continuity
Correction
Convert the discrete random variable r
(number of successes) to the continuous
normal random variable x by doing the
following:
1. If r is a left point of an interval, subtract 0.5 to
obtain the corresponding normal variable x.
x = r – 0.5
2. If r is a right point of an interval, ass 0.5 to
obtain the corresponding normal variable x.
x = r + 0.5
6. How to Make the Continuity
Correction
Example:
P(6 ≤ r ≤ 10) would be approximated by P(5.5 ≤ r ≤
10.5)
7. Not in
Textbook!
How to Find Probabilities
Given a binomial distribution where
n = number of trials
r = number of successes
p = probability of success on a single trial
q = 1 – p = probability of failure on a single trial
np > 5
nq > 5
1. Define what you are trying to find
2. Make the continuity correction
3. Convert to z scores Note in order to do this, you must find μ
and σ.
4. Use the standard normal distribution to find the
corresponding probabilities
8. Page
Example 15 – Normal 310
Approximation
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 5 years only, but from the
owner’s past experience, she knows that the
probability it will last 10 years is 0.25.
9. Example 15 – Normal
Approximation
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 that lasts at least
10 years.
Solution: We want: = P(r≥7.5)
n = 25 P(r≥8)
r = binomial random
variable
corresponding to the =
number Normalcdf(.58, E99)
= .280957
of successes
≈ .2810
p = 0.25
q = 0.75 The probability that 8 or more of the 25 water
np = 6.25 heaters will last at least 10 years is approximately
10. Page
Sampling Distributions for the ^
313
Proportion p
Given
n = number of binomial trials (fixed constant)
r = number of successes
p = probability of success on each trial
q = 1 – p = probability of failure on each trial
If np > 5 and nq > 5, then the random variable
can be approximated by a normal random
variable (x) with mean and standard deviation
11. Sampling Distributions for the ^
Proportion p
The standard error for the distribution is the
standard deviation
We do not use a continuity correction for the
distribution.
is an unbiased estimator for p, the population
proportion of success.
12. Page
Example 16 – Sampling Distribution ^
313
of p
The annual crime rate in the Capital Hill
neighborhood of Denver is 111 victims per 1000
residents. This means that 111 out of 1000
residents have been the victim of at least one
crime. These crimes range from relatively minor
crimes (stolen hubcaps or purse snatching) to
major crimes (murder). The Arms is an apartment
building in this neighborhood that has 50 year
round residents. Suppose we view each of the n =
50 residents as a binomial trial. The random
variable r (which takes on values 0, 1, 2, . . . , 50)
represents the number of victims of at least one
crime in the next year.
13. Example 16 – Sampling Distribution^
of p
a) What is the population probability p that a
resident in the Capital Hill neighborhood will be
the victim of a crime next year? What is the
probability q that a resident will not be a
victim?
Solution:
14. Example 16 – Sampling Distribution^
of p
b) Consider the random variable
Can we approximate the distribution with a
normal distribution? Explain.
Solution:
Since both np and nq are greater than 5, we can
approximate the distribution with a normal
distribution.
15. Example 16 – Sampling Distribution^
of p
c) What are the mean and standard deviation for
the distribution?
Solution:
