Chapter 19 Page 2

Samples
and
populations

Probability
theory and statistical inferrence

Inferring a population
mean

Tests of
statistical significance

Tests of
differences between
means

Coefficient
of
correlation

Caution
with
association

Chi square

Other
tests of
significance

Caution in
using
statistical
test results

Aids

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Our illustration is based on 25 workers in an office.Table 19.1 shows the number of days each worker was absent from work over the past two years.

Table 19.1. IIllustrative distribution of days absent for a population of 25 workers

Worker number	Days absent	Worker number	Days absent	Worker number	Days absent
01	10	10	11	18	12
02	8	11	13	19	10
03	10	12	9	20	9
04	12	13	12	21	14
05	9	14	13	22	8
06	11	15	10	23	13
07	10	16	11	24	10
08	14	17	7	25	11
09	6

The total number of days absent for all 25 workers was 263 and the mean was 10.5 days. The mean of 10.5 is the parameter. It is the mean of the population. It is sometimes represented by a Greek letter µ (mu). In most studies, we do not know the population mean or any other parameter. We have only the mean for the sample we are analyzing. In this illustration, we start by showing the population mean so that you can see how sample variation occurs and its effect on any statistic based on sample data. We show this effect by calculating a series of means based on independently selected random samples from the population of 25 workers.

To illustrate sample variation, let's draw three random samples of five workers each and calculate means for each sample. Using a table of random numbers, we got the three samples listed in Table 19.2. (See Chapter 8 for guidance in using a table of random numbers.)

Table 19.2. Means based on three randomly selected samples

Sample 1 Worker Number	Days absent	Sample 2 Worker number	Days absent	Sample 3 Worker number	Days absent
06	11	20	9	11	13
24	10	09	6	24	10
19	10	12	9	16	11
10	11	19	10	08	14
17	7	05	9	01	10
∑X	49		43		58
M	9.8		8.6		11.6

None of the sample means equals the population mean. Two are less than and one is greater than the population mean. Why? The variation in sample means was caused solely by the operation of chance in the random selection of workers who comprised each sample. In each sample, each worker had 1 chance in 25 or a probability of .04 of being selected. In the first sample we happened to select by chance 5 workers whose scores added to 49; for the second sample, workers whose scores totaled 43; and for the third sample, workers whose scores totaled 58. If we selected additional random samples we would find further variation in sums and means, solely due to the operation of chance.

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