Transcript 8.2 Sampling Distributions

8.2 SAMPLING DISTRIBUTIONS
Distribution of the Sample Proportion
Obj: Use sample data distribution to approximate
probability
SAMPLE DISTRIBUTIONS

Distribution of the Sample Mean

Whether the spread is normal or not, as long as the
population is greater than 30,
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μx = μ
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σx =

n
DISTRIBUTION OF THE SAMPLE PROPORTION
The sample proportion estimates the population
proportion.
If 16 of a population of 100 have a certain
characteristic, the population proportion is 16/100
= 0.16.
x
The sample proportion p =
where x is the number
n
of individuals and n is the random sample size.
PROPERTIES OF THE SAMPLING
DISTRIBUTION OF P


If n < 0.05N, then…
The shape of the distribution is approximately
normal as long as np(1 - p) > 10
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The mean of p is μp = p

The standard deviation σp =
p(1  p)
n
EXAMPLE
Describe the sampling distribution of p. Assume the
size of the population is 25000.
n = 500 and p = 0.4
Is n < .05N?
Is np(1 – p) > 10?
500 < .05(25000)?
500(.4)(.6) > 10?
Yes, so the sampling distribution is approximately
normal
μp = 0.4
σp = .4(.6)  0.022
500
FINDING PROBABILITY
A nationwide study in 2003 indicated that about 60% of
college students with cell phones send and receive
text messages with their phones. Suppose a simple
random sample of n = 1136 college students with cell
phones is obtained.
a) Describe the sampling distribution of p.
Normal? Mean? Standard Deviation?
b)
What is the probability that 665 or fewer college
students in the sample send and receive text
messages with their cell phones?
c)
What is the probability that 725 or more send or
receive messages?
PRACTICE
Peanut and tree allergies are considered to be the
most serious food allergies. According to the
National Institute of Allergy and Infectious
Diseases, roughly 1% of Americans are allergic to
peanuts or tree nuts. Suppose a random sample
of 1500 Americans is obtained.
(There are approximately 295 million Americans.)
Describe the sampling distribution of p.
What is the probability that more than 1.5% are
allergic to peanuts or tree nuts?
PRACTICE
According to the National Center for Health Statistics
(2004), 22.4% of adults are smokers. Suppose a
random sample of 300 adults is obtained.
Describe the sampling distribution of p.
In a random sample of 300, what is the probability
that at least 50 are smokers?
Would it be unusual if a random sample of 300 results
in 18% or less being smokers?
ASSIGNMENT
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