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5.1 Sampling
Distributions for Counts
and Proportions
(continued)
Sample Proportions
We ended last class with the following
example:
A person has a 5% chance of winning a
free ticket in a state lottery. If she plays
the game 12 times, what is the probability
she will win more than 1 free ticket?
What if we had asked for the probability
that the sample proportion of wins was at
most 25%?
Mean and Standard Deviation
We can also find the mean and standard
deviation of a proportion.
 pˆ  p
 pˆ 
p(1  p)
n
Example
The US Forestry Service reports that 20%
of the oak trees in PA are infested with
leaf rot. What is the probability that in a
random sample of 20 trees, 10% or fewer
have leaf rot? Find the mean and
standard deviation of the sample
proportion.
What do the mean and standard deviation
of a proportion mean?
The mean of a proportion being equal to
the probability of success means that the
sample proportion is an unbiased
estimator of the population proportion p.
As n gets larger, the standard deviation
gets smaller.
Practice Problem with review
A Las Vegas casino has determined that 10% of
the dice it purchases are unsuitable for game
play. A random sample of 15 dice is taken.
1. Find the mean and standard deviation.
2. What is the probability this sample contains no
unsuitable dice?
3. What is the probability this sample contains at
most 20% unsuitable dice?
4. Find the mean and standard deviation of the
sample proportion.
Normal Approximation
As we hinted at before, we may be able to
approximate our discrete data with a normal
curve.
Take a SRS of size n from a population. Let X
be the count of successes and X/n the sample
proportion of successes. When n is large, these
statistics are approximately normal.
X is approximately
N (np, np(1  p) )
p
(
1

p
)
X/n is approximately N ( p,
)
n
This is exactly the mean and standard
deviation for a count and a proportion.
We use this if np≥10 and n(1-p)≥10.
Examples
Assume that we have a binomial
experiment with n=40 and p=.3.
Estimate the probability of
1. More than 10 successes.
2. 4 or less successes
3. Between 13 and 17 successes.
Practice Problem
A civil service multiple choice exam
consist of 100 questions with four choices
per question. Suppose an applicant
randomly guesses the answer to every
question. Estimate the probability that the
applicant will answer 35 or more questions
correctly. Estimate the probability that he
will guess less than 28% correct.