# Math Statistics

It is often not necessary to survey the entire population.

Instead, you can select a random sample of people/or firms from the population and survey just them.

You can then draw conclusions about how the entire population would respond based on the responses from this randomly selected group of people. This is exactly what political pollsters do - they ask a group of people a list of questions and based on their results, they draw conclusions about the population as a whole with those often heard disclaimers of "plus or minus 5%"

If your population consists of just a few hundred people, you might find that you need to survey almost all of them in order to achieve the level of accuracy that you desire. As the population size increases, the percentage of people needed to achieve a high level of accuracy decreases rapidly.

In other words, to achieve the same level of accuracy:

Larger population = Smaller percentage of people surveyed

Smaller population = Larger percentage of people surveyed

Probability Sampling

A probability sampling method is any method of sampling that utilizes some form of random selection.

In order to have a random selection method, you must set up some process or procedure that assures that the different units in your population have equal probabilities of being chosen.

Some basic terms

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N - the number of cases in the sampling frame

n - the number of cases in the sample

NCn - the number of combinations (subsets) of n from N

f = n/N - the sampling fraction

1. Simple Random Sampling

The simplest form of random sampling is called simple random sampling. •

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Objective: To select n units out of N such that each NCn has an equal chance of being selected.

Procedure: Use a table of random numbers, a computer random number generator or a mechanical device to select the sample.

Example

The problem

A small service agency wishes to assess client's views of quality of service over the past year.

A solution

- To set the sampling frame

To accomplish this, we'll go through agency records to identify every client over the past 12 months. If we're lucky, the agency has good accurate computerized records and can quickly produce such a list. Then, we have to actually draw the sample.

- To decide on the number of clients you would like to have in the final sample. For the sake of the example, let's say you want to select 100 clients to survey

and that there were 1000 clients over the past 12 months. Then, the sampling fraction is f = n/N = 100/1000 = .10 or 10%.

- To draw the sample using for example Excel

The function =RAND() which is Excel's way of putting a random number between 0 and 1 in the cells.

In Polish version of Excel =LOS()

Then, sort both columns - the list of names and the random number - by the random numbers. This rearranges the list in random order from the lowest to the highest random number. Then, you have to take the first hundred names in this sorted list. Advantages and disadvantages

1. Simple random sampling is simple to accomplish and is easy to explain to others. 2. Because simple random sampling is a fair way to select a sample, it is reasonable to generalize the results from the sample back to the population. 3. Simple random sampling is not the most statistically efficient method of sampling and you may, just because of the luck of the draw, not get good representation of subgroups in a population.

2. Stratified Random Sampling

Stratified Random Sampling, also sometimes called proportional or quota random sampling, involves dividing your population into homogeneous subgroups and then taking a simple random sample in each subgroup. In more formal terms: Objective: Divide the population into non-overlapping groups (i.e. strata) N1, N2, N3, ... Ni, such that N1 + N2 + N3 + ... + Ni = N. Then do a simple random sample of f = n/N in each strata.

Advantages and disadvantages

1. It assures that you will be able to represent...

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