5.1
Sampling Distributions for Counts and Proportions
link between Chapter 4 (probability theory) and rest of book
“Statistical inference draws conclusions about a population or process on the basis of data.
The
data are summarized by statistics such as means, proportions, and the slopes of leastsquares
regression lines.
When the data are produced by random sampling or randomized
experimentation, a statistic is a random variable that obeys the laws of probability theory.
The
link between probability and data is formed by the sampling distributions of statistics.
A
sampling distributions shows how a statistic would vary in repeated data production.
That is, a
sampling distribution
is a probability distribution that answers the question ‘What would happen
if we did this many times?’”
Count and sample proportion
sample 2500 adults, ask if agree or disagree with:
“I like shopping.”→
1650 agreed
random variable X is a count
of the occurrences of some outcome in a fixed # of observations
sample proportion
:
p
ˆ
= X/
n
p
ˆ
= (1650/2500) = 0.66
The binomial distributions for sample counts
The binomial setting
 four components
1.
There are a fixed number
n
of observations.
2.
The
n
observations are all independent
3.
Each observation falls into one of just two categories, which for convenience we
call “success” and “failure.”
4.
The probability of success, call it
p
, is the same for each observation.
Binomial distributions
The distribution of the count X of successes in the binomial setting is called the binomial
distribution with parameters
n
and
p
.
The parameter
n
is the number of observations, and
p
is the probability of a success on any one observation.
The possible values of X are the
whole numbers from 0 to
n
.
As an abbreviation, we say that X is B(
n
,
p
).
EX
1) Toss a coin 10 times and count the number X of heads.
The # of heads we observe has B(10, 0.5)
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 Spring '11
 Bill
 Statistics, Normal Distribution, Probability, Probability theory, Binomial distribution, Bernoulli Distribution

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