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```FOUNDATIONS OF STATISTICAL
INFERENCE
DEFINITIONS


Statistical inference is the process of reaching
conclusions about characteristics of an entire
population using data from a subset, or sample,
of that population.
Simple random sampling is a sampling method
which ensures that every combination of n
members of the population has an equal chance
of being selected.
Statistical Inference
The process of making guesses about the truth about a
population parameter from a sample statistic.
Sample statistics
n
̂  X n 
x
i 1
n
n
Truth (not
observable)
(x  X
i
ˆ 2  s 2 
n)
2
i 1
n 1
Sample
(observation)
*hat notation ^ is often used to
indicate “estimate”
Population
parameters
N


i 1
N
N
x
(x  )
2
i
2 
i 1
N
the whole
population
Sampling Distributions
A sampling distribution is the distribution of sample statistics
computed on the set of all possible random samples of size n
that could be drawn from a population.
Most experiments are one-shot deals. So, how do we know if an
observed effect from a single experiment is real or is just an
artifact of sampling variability (chance variation)?
Probability distributions important here.
Because they form the basis of describing the
distribution of a sample statistic.
Statistical Inference is based
on Sampling Variability
Sample Statistic – we summarize a sample into one number; e.g., could
be a mean, a difference in means or proportions, an odds ratio, or a
correlation or regression coefficient
– E.g.: Average support for gun control among women and men.
– E.g.: Proportion of women and men who supported the war in Iraq.
Sampling Variability – If we could repeat an experiment many, many
times on different samples with the same number of subjects, the
resultant sample statistic would not always be the same (because of
chance!).
Standard Error – a measure of the sampling variability. It is the
standard deviation of the sampling distribution.
•
•
•
For large enough sample sizes, the shape of
the sampling distribution will be
approximately normal.
The sampling distribution is centered on ,
the mean of the population.
The standard deviation of the sampling
distribution can be computed as the
population standard deviation divided by the
square root of the sample size.
Examples of Sample Statistics:
Single population mean μ (known population
standard deviation )
Single population mean μ (unknown
population standard deviation )
Single population proportion p
Difference in means μ1,μ2 (t-test)
Difference in proportions p1,p2 (Z-test)
Odds ratio/risk ratio
Correlation coefficient
Regression coefficient
…
The Central Limit Theorem:
If all possible random samples, each of size n, are
taken from any population with a mean  and a
standard deviation , the sampling distribution of
the sample means (averages) will:
1. Have mean:
x  
2. Have standard deviation (also called
standard error for sampling distribution):
x 

n
3. Be approximately normally distributed regardless of the shape
of the parent population (normality improves with larger n).
Symbol Check
x
x
The mean of the sample means.
The standard deviation of the sample means. Also called
“the standard error of the mean.”
INTUITIVE TREATMENT OF SAMPLING
DISTRIBUTION




Suppose we have a population of size 100. We then draw a sample of
100 people from the population of 100. We then compute the mean.
How confident could we be about the computed sample statistic? How
much sampling error would there be?
Suppose we have a population of size 100. We then draw every
sample of size 99 from this population. We compute means for all of
these samples. How many different samples could we draw? C99100
=100? How much sampling error would there be in the computed
means?
Suppose we have a population of size 100. We then draw a sample of
50 people from the population of 100. We then compute the means on
each sample. How many different samples could we draw? C50100
=1.089X1029 . How much sampling error would there be in the
computed means?
The principle is that the larger the sample size, relative to the
population we are drawing from, the lower the sampling error. The
smaller the sample size, relative to the population we are drawing
from, the larger the sampling error.
Alternative Region
Null Region
Alternative Region
HYPOTHESIS TESTING USING THE NORMAL
(Z) DISTRIBUTION
Calculate the estimated statistic from the
sample.
 Record the sample standard deviation  and N.
 Then calculate the standard error of the
sampling distribution from the preceding.



Then calculate Z
Compare the calculated value for Z to the table of
Z statistics.
EXAMPLE:



. Suppose we draw a sample with mean, variance, and N
as follows:
How confident could we be that the mean was not actually
10 (the a null hypthesis). We might then ask how many
standard deviations (Z units) away 12.5 is from 10. We can
then calculate a p value from the Z-statistic.
Using the preceding table, there is only a .0016 chance
that with a sample of size 50 and variance 36 we could
have drawn a sample with mean 12.5 when the actual
population mean was 10.
EXAMPLE:



With the NES92, we draw a sample of 1500 respondents. On the
variable, liking for Clinton we find a mean of 4.1 with a variance of 1.6 .
What is the probability that the real liking for Clinton in the population
is only 3, rather than the calculated 4.1?
Using the earlier table, the probability is less than 0.001 that the real
liking for Clinton is 3.0.
What factors determine this probability?
1) The magnitude of the hypothesized difference(the numerator)
2) The variance of the sample (1.6)
3) The N of the sample (1500)

Note that we can also think of these three quantities as distances in
standard deviation units on the sampling distribution. See slide 13
again.
THE CONFIDENCE INTERVAL APPROACH
Let UCL and LCL refer respectively to upper and
lower confidence limits. Let μ be the estimated
parameter. Let Z be the Z-statistic associated
with the desired p-value. Let σe be the standard
error. Then, calculate the confidence limits as
follows.
EXAMPLE:


Construct a 99 percent confidence interval around the
point estimate 12.5 from the preceding example with
the given information.
The interval does not contain zero. Therefore, we can
be at least 99 percent confident the estimated mean is
not zero. It also does not contain 10, so we can be at
least 99 percent confident that the true estimate is
not 10.
USING THE T-DISTRIBUTION




In actuality, we seldom know the population variance or standard
deviation. Under these circumstances we use the t distribution,
rather than the Z (normal distribution) for our tests of significance.
Unlike the Z distribution of which there is only one, there are many t
distributions. One for each possible degree of freedom for the test.
(Degrees of freedom refer to N minus the number of parameters
estimated.) Note, however that as N becomes large, say 100, the t
distribution equals the z distribution.
The t-distribution is used in precisely the same way as the Z in
conducting the preceding tests. Simply substitute in the numbers for
the t-distribution where you have the numbers for the Z distribution.
The t-distribution takes into account that we do not have full
information about the population variability. With small N, the tdistribution is somewhat more conservative than the Z. It gives the
same answer if N is larger than about 1,000. It is also quite close
when N is larger than about 100.
See the next table.
THE P-VALUE
The p-value is the probability that we would have
observed our sample statistic (or something more
unexpected) just by chance if the null hypothesis (null
value) is true.
For example, we might estimate as above 12.5, but
posit a null value of 10.
Small p-values mean the null value is unlikely given
our data.
```
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