CK-12 Probability and Statistics - Advanced

7.1. Sampling Distribution http://www.ck12.org

First, the sample statistics resulting from the samples are distributed around the population parameter. Although
there is a wide range of estimates, more of them lie close to the 50% area of the graph. Therefore, the true value
is likely to be in the vicinity of 50%. In addition, probability theory gives a formula for estimating how closely the
sample statistics are clustered around the true value. In other words, it is possible to estimate the sampling error – the

degree of error expected for a given sample design. The formulas=

√

P·Q

n

contains three factors: the parameters

(PandQ), the sample size(n), and the standard error(s)

The symbolsPandQin the formula equal the population parameters for the binomial: If 60 percent of the student
body approves of the dress code and 40% disapprove,PandQare 60% and 40% respectively, or 0.6 and 0.4. Note
thatQ= 1 −PandP= 1 −Q. The square root of the product ofP and Qis actually the population standard deviation.
The symbolnequals the number of cases in each sample, andsis the standard error.

If the assumption is made that the true population parameter is 50 percent approving the dress code and 50 percent
disapproving the dress code while selecting samples of 100, the standard error obtained from the formula equals
5 percent or.05.

Q= 1 −P P= 1 −Q

Q= 1 − 0. 50 P= 1 − 0. 50

Q= 0. 50 P= 0. 50

s=

√

P·Q

n

s=

√

( 0. 50 ).( 0. 50 )

100

= 0 .05 or 5%

σ=

√

P·Q

σ=

√

( 0. 50 ).( 0. 50 )

σ= 0 .050 or 50%−→ This is the assumption that was made

as being the true population parameter.

This indicates how tightly the sample estimates are distributed around the population parameter. In this case, the
standard error is the standard deviation of the sampling distribution.

Probability theory indicates that certain proportions of the sample estimates will fall within defined increments- each
equal to one standard error-from the population parameter. Approximately 34 percent of the sample estimates will
fall within one standard error increment above the population parameter and another 34 percent will fall within one
standard error increment below the population parameter. In the above example, you have calculated the standard
error increment to be 5 percent, so you know that 34% of the samples will yield estimates of student approval
between 50% (the population parameter) and 55% (one standard error increment above). Likewise, another 34%
of the samples will give estimates between 50% and 45% (one standard error increment below the parameter).
Therefore, you know that 68% of the samples will give estimates within±5 percent of the parameter. In addition,
probability theory says that 95% of the samples will fall within±two standard errors of the true value and 99.9%
will fall within±three standard errors. With reference to this example, you can say that only one sample out of one
thousand would give an estimate below 35 percent or above 65 percent approval.

The size of the standard error is a function of the population parameter and the sample size. By looking at this

formula,s=

√

P·Q

n

it is obvious that the standard error will increase as a function of an increase in the quantityP

times Q. Referring back to our example, the maximum quantity forP times Qoccurred when there was an even split