Download NORMAL APPROXIMATION OF THE BINOMIAL DISTRIBUTION

NORMAL APPROXIMATION OF THE BINOMIAL DISTRIBUTION
A binomial variable, one kind of discrete variable, can take only two values, often termed successes and
failures. Examples include coin flips that come up either heads or tails, manufactured parts that either
continue working past a certain point or do not, and basketball tosses that either fall through the hoop or
do not.
The outcomes of binomial trials have a frequency distribution, just as continuous variables do. The more
binomial outcomes there are (for example, the more coins you flip simultaneously), the more closely the
sampling distribution resembles a normal curve.
Figure 1As the number of trials increases, the binomial distribution approaches the normal distribution.
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If we look at the binomial distribution as an approximation to the normal distribution, we can use the
z-score table to help determine probabilities.
Recall:
Standard Deviation (  ) is a measure of dispersion (spread).
Z-Score is the number of standard deviations above or below the mean.
xx
z

2
N(13,2.5 ) Normal distribution with a mean of 13 and a standard deviation of 2.5
For a normal approximation to the binomial distribution:
x  np
  np(1  p)
or
  npq
**Remember a normal distribution is CONTINUOUS and our binomial distribution is for a
DISCRETE random variable**
Therefore, when we use the normal distribution curve we need to consider a range of values instead of a
specific discrete value. For example, 4.5 – 5.5 would represent a discrete value of 5. Looking at a range
of 20 to 22 you would use 19.5 to 22.5 on the normal distribution curve.
Not all binomial distributions can be approximated using the normal distribution. If a distribution is right
skewed or left skewed it does not fit the normal distribution. To check if your binomial distribution is
“symmetrical enough” look at the following:
Is
np >5
AND
n(1-p) >5 … if yes, it can be approximated by a normal
distribution.
Read pages 305-311 (be sure to review Example 5 on page 310)
Examples
1. A bank found that 24% of its loans to new small businesses become delinquent. If 200 small
businesses are selected randomly from the bank’s files, what is the probability that at least 60 of them
are delinquent?
2. QuenCola, a soft drink company, knows that it has a 42% market share in one region of the province.
QuenCola’s marketing department conducts a blind taste test of 70 people at the local mall.
a. What is the probability that fewer than 25 people will choose QuenCola?
b. What is the probability that exactly 25 people will choose QuenCola?
3. A manufacturer of pencils has 60 dozen pencils randomly chosen from each day’s production and
checked for defects. A defect rate of 105 is considered acceptable. Assuming that 10% of all the
manufacturers’ pencils are actually defective, what is the probability of finding 80 or more defective
pencils in this sample?
4. A store manager believes that 42% of her customers are repeat business (have visited her store within
the last two weeks). Assuming she is correct, what is the probability that out of the next 500
customers, between 200 and 250 customers are repeat business?
More practice... page 312 #5,6,8,10,11,17