Introduction to a variety of useful statistical ideas and techniques - Improving the User Experience through Practical Data Analytics - page 30

SIDEBAR: DOES ACTUAL POPULATION SIZE MATTER? IT MIGHT! CONSIDER
THE NEED FOR THE FINITE POPULATION MULTIPLIER (FPM.)—cont'd
There are two basic ways of sampling from a population: sampling with replacement and sampling
without replacement . These two terms have the obvious meanings - with replacement means that after
we sample one data point, we replace it into the population before selecting the second data point; the
key consequence is that we might possibly sample the same data point (e.g., person) again!!
Sampling without replacement , on the other hand, means that once we sample a data point, he/
she/it is no longer eligible to be selected again - we have not replaced that data point back into the
population. The vast, vast majority of sampling that takes place is sampling without replacement .
In fact, in the rare case of sampling with replacement, this entire sidebar is moot! The reader should
assume that in the entire text, all sampling is sampling without replacement .
Now, we've provided in this chapter the formula for a conidence interval for the mean as
X-bar ± e
Of course, X-bar and “e” are determined by the software you use.
Technically, when sampling without replacement (which is usually the case), the formula you
should is:
X-bar ± e*SQRT{(N−n)/(N−1)}
The portion of the formula that multiplies the “e” − SQRT{(N − n)/(N − 1)} − is called the “inite
population multiplier (FPM).” Some texts refer to it as the “inite population correction factor.”
Thus, in our example, N = 20 (a total of 20 major clients) and n = 6 of these 20, without replace-
ment; the application of the FPM will actually make a material difference! Using this scenario,
suppose on a typical 5-point Likert scale, we have data of: 5, 4, 4, 4, 3, 4. We have X-bar = 4.0, and
e = .664 without the FPM. Thus, we have a 95% conidence interval of:
3.336 ---- 4.664
With the FPM, we have .664*SQRT{(20 − 6)/(20 − 1) = .664*(.8584) = .544, and a correspond-
ing 95% conidence interval of:
3.456 ---- 4.544
This is an interval that is about 18% narrower—i.e., 18% more precise! How important this 18%
is depends on the speciic context, but few would refer to it as totally negligible.
Having said this, the rule-of-thumb generally used is that we ignore the FPM unless we have
the somewhat rare case that n/N exceeds .05 (i.e., we are sampling more than 5% of the total
population.) The logic behind this rule-of-thumb is that under these conditions, the FPM would not
make a material difference to the answer.
By the way, you may be wondering what Excel and SPSS do about the FPM. The answer is:
Nothing!! They don't compute a conidence interval incorporating the FPM. The reason follows the
above logic that nearly always, including the FPM has a negligible impact.
For example, suppose that N = 10,000 and n = 10. Then the FPM equals SQRT{(10,000 − 10)/
(10,000 − 1)} = SQRT(9990/9999) = SQRT(.9991) = .9995. Well, multiplying “e” by .9995 is surely
negligible, and will have not even a remotely material effect on “e” (especially when “e” is likely to be
rounded to 2 or 3 digits at most). So, even though we theoretically need the FPM, we hardly ever use it.
Bottom line: If you have a case which does require the FPM, irst calculate the conidence
interval provided by the software to determine “e.” Then, multiply “e” by the FPM and then add and
subtract the “adjusted e” to the X-bar.
1.4 HYPOTHESIS TESTING
One topic, and perhaps the only topic, that is with us in a major way in every chapter
of this topic is what is called “hypothesis testing.” It has several variations in terms
of its goals and the speciic questions it answers, but, ultimately, it has to do with
deciding whether something is true or not , based on analyzing the data.