Transcript Summary Statistics When analysing practical sets of data

Sampling Theory
The procedure for drawing a random sample a distribution is that numbers 1, 2, … are
assigned to the elements of the distribution and tables of random numbers are then used to
decide which elements are included in the sample. If the same element can not be selected
more than once, we say that the sample is drawn without replacement; otherwise, the
sample is said to be drawn with replacement.
The usual convention in sampling is that lower case letters are used to designate the sample
characteristics, with capital letters being used for the parent population. Thus if the sample
size is n, its elements are designated, x1, x2, …, xn, its mean is x and its modified variance is
s2 =
 (xi - x )2 / (n - 1).
The corresponding parent population characteristics are N (or infinity), X and S2.
Suppose that we repeatedly draw random samples of size n (with replacement) from a
distribution with mean m and variance s2. Let x1, x2, … be the collection of sample
averages and let
xi’ =
xi - m
(i = 1, 2, … )
sn
The collection x1’, x2’, … is called the sampling distribution of means.
Central Limit Theorem.
In the limit, as n tends to infinity, the sampling distribution of means
has a standard normal distribution.
Attribute and Proportionate Sampling
If the sample elements are a measurement of some characteristic, we are said to have
attribute sampling. On the other hand if all the sample elements are 1 or 0 (success/failure,
agree/ no-not-agree), we have proportionate sampling. For proportionate sampling, the
sample average x and the sample proportion p are synonymous, just as are the mean m and
proportion P for the parent population. From our results on the binomial distribution, the
sample variance is p (1 - p) and the variance of the parent distribution is P (1 - P).
We can generalise the concept of the sampling distribution of means to get the sampling
distribution of any statistic. We say that a sample characteristic is an unbiased estimator of
the parent population characteristic, is the mean of the corresponding sampling distribution
is equal to the parent characteristic.
Lemma.
The sample average (proportion ) is an unbiased estimator of the parent
average (proportion):
E [ x] = m;
E [p] = P.
The quantity  ( N - n) / ( N - 1) is called the finite population correction (fpc). If the
parent population is infinite or we have sampling with replacement the fpc = 1.
Lemma.
E [s] = S * fpc.
Confidence Intervals
From the statistical tables for a standard normal
distribution, we note that
Area Under
Density Function
0.90
0.95
0.99
From
To
-1.64
-1.96
-2.58
1.64
1.96
2.58
n (0,1)
0.95
-1.96
0
+1.96
2
From the central limit theorem, if x and s are the mean and variance of a random sample of
size n (with n greater than 25) drawn from a large parent population, then we can make the
following statement about the unknown parent mean m
Prob { -1.64  x - m    
s/n
i.e.
Prob { x - 1.64 s /  n  m  x +  s /  n }  
The range x + 1.64 s /  n
is called a 90% confidence interval for the population mean m.
Example [ Attribute Sampling]
A random sample of size 25 has x = 15 and s = 2. Then a 95% confidence interval for m is
15 + 1.96 (2 / 5) (i.e.) 14.22 to 15.78
Example [ Proportionate Sampling]
A random sample of size n = 1000 has p = 0.40  1.96  p (1 - p) / (n) = 0.03.
A 95% confidence interval for P is 0.40 + 0.03 (i.e.) 0.37 to 0.43.
Small Sampling Theory
For reference purposes, it is useful to regard the expression
x ± 1.96 s /  n
as the “default formula” for a confidence interval and to modify it to suit particular
circumstances.
O If we are dealing with proportionate sampling, the sample proportion is the
sample mean and the standard error (s.e.) term s /  n simplifies as follows:
x -> p and
s /  n ->  p(1 - p) / (n).
O A 90% confidence interval will bring about the swap
1.96 -> 1.64.
O If the sample size n is less than 25, the normal distribution must be replaced by
Student’s t n - 1 distribution.
O For sampling without replacement from a finite population, a fpc term must be
used.
The width of the confidence interval band increases with the confidence level.
Example. A random sample of size n = 10, drawn from a large parent population, has a mean
x = 12 and a standard deviation s = 2. Then a 99% confidence interval for the parent mean is
x ± 3.25 s /  n (i.e.)
12 ± 3.25 (2)/3.2 (i.e.)
9.83 to 14.17
and a 95% confidence interval for the parent mean is
x ± 2.262 s /  n (i.e.)
12 ± 2.262 (2)/3.2 (i.e.) 10.492 to 13.508.
Note that for n = 1000, 1.96  p (1 - p) / n   for values of p between 0.3 and 0.7. This
gives rise to the statement that public opinion polls have an “inherent error of 3%”. This
simplifies calculations in the case of public opinion polls for large political parties.