Is this sample representative of the population?


Hello,

I've received the following problem (no, it's not homework), and I'm trying
to figure out what would be the best way to answer it.

A client (in marketing, you can probably guess) sent us this table,
containing counts:

Profession Population Sample

Executive 10041 1164
Commercial 1734 162
Employee/labourer 3591 309
Farmer 414 29
CEO or upper executive 5330 797
Self-employed 3621 410
Retired 5699 927
Student 5051 294
blank 21 21
Other 8536 403
Total 44038 4495

"Population" implies the true population from which the sample ("Sample")
was taken.

The question asked by the client is : "Is this sample representative of the
population?"

I can't figure out how to answer this. I've tried as follows:

I wondered if the idea was that the proportions should be compared, so I did
a line-by-line parametric comparison of two percentages. I got the p-values
(for each category/row respectively) as:

100,00%
100,00%
86,42%
99,87%
97,62%
100,00%
98,12%
100,00%
100,00%
100,00%

....implying that the differences were not significant.
This obviously doesn't make sense for "other", which compares the two
percentages 19.38% and 8.97%.

I then had the idea that since the first column is the true count value of
the population, I should treat each line as comparing a theoretical
percentage (the "true" percentage) to an observed one (the sample). This got
me the results (for each category/row respectively):

0,00% (significantly different)
0,00% (significantly different)
12,51% (not significantly different)
0,09% (significantly different)
2,02% (significantly different)
0,00% (significantly different)
1,41% (significantly different)
0,00% (significantly different)
0,00% (significantly different)
0,00% (significantly different)

The last line was however not considered valid, since the value of "count x
theoretical %" was 2.14, ie <10.

I'm still not sure what I'm doing, as you can guess, and this sort of
problem is likely to come up again. Can anyone please advise me as to what
procedure I should use? and especially as to how to justify it? Even a
helpful hint would be great as to which direction to follow.

Thank you in advance.

Z


.

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