Download Sampling distributions. Sampling distribution of sample proportion

CH18 Sampling Distribution Models
Sampling distributions.
Sampling distribution of sample
proportion and sample mean.
Central Limit Theorem.
Motivation
• Who is the next president of the U.S.?
• The opinion of the whole population is not known until
the election day.
• However, we can draw random samples to estimate the
population.
Motivation
Sample
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Sample Proportion for R
ORRROROOOORRROROORRR
RROORROOOOORROROOORO
RROROOROORRRRROOOOOR
OOROORROORORRRROORRR
ROROOOOOROORORRORORR
OROOROROORRROORROORO
ROROOOORRORORRRRRROO
OOOOROROOORORRROOROR
RORRORORORRRROOOROOO
RRRROOOROOORRROOORRO
RRRROORRRORORRORRORR
ROOORROOOROORROROROR
RROORORORRRRROORRROR
OORROROROOOROOROROOO
RROOROROORRRROROOORR
------11/20=0.55
------8/20=0.4
------10/20=0.5
------11/20=0.55
------9/20=0.45
------9/20=0.45
------11/20=0.55
------8/20=0.4
------10/20=0.5
------10/20=0.5
------14/20=0.7
------9/20=0.45
------13/20=0.65
------7/20=0.35
------11/20=0.55
Motivation
• The histogram of 15 sample proportions (sample size 20)
Motivation
• The histogram of 150 sample proportions (sample size 20)
Motivation
• The histogram of 1500 sample proportions (sample size
20)
Motivation
• The histogram of 1500 sample proportions (sample size
200)
Motivation
• The histogram of 1500 sample proportions (sample size
400)
Motivation
• The histogram of 1500 sample proportions (sample size
800)
• What distribution fits to the data well?
Sampling distribution
• For a random sample, the sample proportion is random.
• In other words, sample proportion is a random variable.
What is the distribution of this random variable?
• The distribution of the sample proportions is
approximately Normal.
Review
• In a population, there are usually parameters of interest
whose values are unknown.
e.g. The population proportion who supports Romney
• We use sample estimators to estimate the values of those
parameters.
e.g. The sample proportion who supports Romney
• The sample estimators are called sample statistics.
e.g. Sample proportion is a statistic
Sampling distribution
• The sampling distribution of a statistic is the distribution
of values taken by that statistic in all possible samples of
the same size from the same population.
e.g. histogram of the
sample proportion
favoring Romney
(1500 samples with
sample size 800)
Sampling distribution of the sample proportion
The sampling distribution of p̂ is never exactly normal. But as the sample size
increases, the sampling distribution of p̂ becomes approximately normal.
The normal approximation is most accurate for any fixed n when p is close to
0.5, and least accurate when p is near 0 or near 1.
Sampling distribution
• Caution: sampling distribution is not the same as the
distribution of a sample.
The sampling distribution of a sample proportion is
approximately Normal.
For the sample “ORRROROOOORRROROORR”, the
distribution of this sample is
Romney
11
0.55
Obama
9
0.45
Sampling distribution of sample proportion
• Population proportion
p=
number of individual s of interest in the population
total number of individual s in the population
• Sample proportion
p̂ =
number of individual s of interest in the sample
total number of individual s in the sample
Sampling distribution of sample proportion
• Provided that the sampled values are independent (e.g. a
simple random sample from a large population) and the
sample size is large enough, the sampling distribution of
the sample proportion ^
p of samples of size n is
approximately Normal with mean p and
standard deviation
• Or
pˆ ~ɺ N ( p,
p (1 − p )
.
n
p(1 − p)
)
n
Calculating Probability with TI calculator
P (a < pˆ < b) = normalcdf (a, b, p,
P ( pˆ < b) = normalcdf (−1E 99, b, p,
P ( pˆ > a ) = normalcdf (a,1E 99, p,
p (1 − p )
))
n
p (1 − p )
))
n
p (1 − p )
))
n
Example
pˆ ~ɺ N ( p,
p(1 − p )
0.13(1 − 0.13)
) = N (0.13,
= 0.035)
n
90
Example
• Suppose the population proportion of supporting
Romney is 50%. How likely will we get a sample of size
600 with sample proportion at least 48%?
• What if the population proportion is in fact 40%?
• What if the population proportion is actually 60%?
Assumptions and conditions
• Independence Assumption: the sampled values must be
independent of each other
Randomization Condition: a random sample
10% Condition: the sample size n must be no larger
than 10% of the population
• Sample Size Assumption: the sample size n must be large
enough
Success/Failure Condition: the sample size has to be
big enough so that we expect at least 10 successes and
at least 10 failures.
Sampling distribution of sample mean
• A fair die is tossed 10,000 times. Below is the histogram
of the outcomes.
Sampling distribution of sample mean
• Two fair dice are tossed 10,000 times. Below is the
histogram of average of the outcomes.
Sampling distribution of sample mean
• Four fair dice are tossed 10,000 times. Below is the
histogram of average of the outcomes.
Sampling distribution of sample mean
• Eight fair dice are tossed 10,000 times. Below is the
histogram of average of the outcomes.
Sampling distribution of sample mean
• Sixteen fair dice are tossed 10,000 times. Below is the
histogram of average of the outcomes.
Central limit theorem
• The mean of a random sample has a sampling distribution
whose shape can be approximated by a Normal model.
The larger the sample, the better the approximation will
be.
• If the sample is from a population with Normal
distribution, then the approximation is exact.
The central limit theorem
Central Limit Theorem: When randomly sampling from any population
with mean µ and standard deviation σ, when n is large enough, the
sampling distribution of
Population with
strongly skewed
distribution
Sampling
distribution of
x for n = 10
observations
x is approximately normal: ~ N(µ, σ/√n).
Sampling
distribution of
x for n = 2
observations
Sampling
distribution of
x for n = 25
observations
Income distribution
Let’s consider the very large database of individual incomes from the Bureau of
Labor Statistics as our population. It is strongly right skewed.
We take 1000 SRSs of 100 incomes, calculate the sample mean for
each, and make a histogram of these 1000 means.
We also take 1000 SRSs of 25 incomes, calculate the sample mean for
each, and make a histogram of these 1000 means.
Which histogram
corresponds to
samples of size
100? 25?
Sampling distribution of sample mean
• When a random sample is drawn from any population
with mean µ and standard deviation σ, its sample mean x,
has a sampling distribution with the same mean µ but
σ
.
standard deviation
n
• No matter what population the random sample comes
from, the shape of the sampling distribution is
approximately Normal as long as the sample size is large
enough.
• The larger the sample used, the more closely the Normal
approximates the sampling distribution for the mean.
Calculating Probability for sample mean with TI
calculator
P(a < x < b) = normalcdf (a, b, µ ,
σ
n
σ
P( x < b) = normalcdf (−1E 99, b, µ ,
P( x > a ) = normalcdf (a,1E 99, µ ,
)
n
σ
n
)
)
The amount of soda in cans of a particular
brand has a mean of 12 oz and a standard
deviation of .2 oz. If you select random
samples of 50 cans, what percentage of the
sample means would be less than 11.95
oz?
SODA
• Assume that the systolic blood pressure of 30-year-old
males is normally distributed, with an average of 122
mmHg and a standard deviation of 10mmHg. A random
sample of 16 men from this age group is selected.
• Calculate the probability that the average blood pressure
of the sample will be greater than 125mmHg?
• Calculate the probability that the average blood pressure
of this sample will be between 118 and 124 mmHg?
• Calculate the probability that the blood pressure of an
individual male from this population will be between 118
and 124mmHg?
Suggested exercises from the textbook:
Ch18 5, 7, 11, 15, 17, 19, 25, 27, 33, 37, 39,
41, 43