You can estimate the population proportion P of part-time students using the calculated sample proportion p. The true proportion P is not known.  Its (1-alpha)% confidence interval can be calculated exclusively as a function of p. There is no basis for the determination of the degree of representativeness of the population by the sample by comparing them. 

However randomness of sample might be tested using nonparametric tests like the sign test. 

You can estimate the population proportion P of part-time students using the calculated sample proportion p. The true proportion P is not known.  Its (1-alpha)% confidence interval can be calculated exclusively as a function of p. There is no basis for the determination of the degree of representativeness of the population by the sample by comparing them. 

However randomness of sample might be tested using nonparametric tests like the sign test. 

Magda -

Transport attitudes?  That does not sound like something for which you have quantitative modeling data, but if you did, you might be able to test how well the same model might, or might not, predict for test data in each segment of the population which you might imagine in a partitioning. You might also graph y on the y-axis of a scatterplot, and predicted-y on the x-axis for each separate part of the population under study.  The estimated variances of the prediction errors should be low enough to meet your needs.

But it sounds like you have a situation for which you have random sampling and wish to know if every part of the population is adequately represented.  If you have different ways to stratify the population, then you should draw data from each stratum, but whether stratified random sampling is adequate depends on whether or not you only need good overall results, or do you need to know the individual strata well, in which case you generally need a bigger sample, and really just have a set of simple random samples from different populations, not one stratified random sample. 

You might find this interesting:

Brewer, K.R.W. (2014), “Three controversies in the history of survey sampling,” Survey Methodology,

(December 2013/January 2014), Vol 39, No 2, pp. 249-262. Statistics Canada, Catalogue No. 12-001-X.

http://www.statcan.gc.ca/pub/12-001-x/2013002/article/11883-eng.htm

If you have a qualitative study, again I suggest that you break your population into groups/strata, based on whatever you know about the population, and perhaps here you could draw from each until you don't find more 'surprises.' 

These were just a few thoughts on this. 

Best wishes - Jim 

PS  -  If you have no other characteristics to use for grouping other than full-time/part-time, and you are comparing them, then you may be over- or under-representing or even not representing some category or categories of which you may be unaware.  That could always be a problem, and is a good reason to know as much about your population as you can, and document your methodology to aid in perhaps comparing results to other such studies. 

You may want to research "Simpson's Paradox," as that could apply to your results if there are factors of which you are unaware. 

Magda,

To answer your question, what you want is a goodness-of-fit test for nominal data. I would use a binomial test, but you could also use a chi-square goodness-of-fit test or g-test goodness of fit.

Some examples using R are here: http://rcompanion.org/handbook/H_03.html

Because you have a relatively large sample size, you are likely to find a significant result even for a small difference. Be sure to consider the "practical significance" of the difference between your sample and the population.

You can run the R code below by downloading the software or using http://www.r-fiddle.org/#/

Let's assume n=1914, your full-time students = 1500, the proportion of full-time students in the population is 0.80

### Binomial test

binom.test(1500, 1914, 0.80)

### Chi-square goodness-of-fit

chisq.test(x = c(1500, 414), p = c(0.80, 0.20))

BTW, the confidence interval approach also makes sense.  You'll note that the binom.test output in R includes the confidence interval for the proportion.  It uses a method appropriate for proportions, I think the Clopper and Pearson method.

Hello, Thank you very much all for your time and interesting answers. I am studying all the suggestions.

Dear James Knaub, you are right is a qualitative study (using 4 point- Likert scales). Yes, the sample can be break into other socio-demographic characteristics. I am checking the paper you suggest. Thank you

Ette Etuk, is there any suggestion of nonparametric test? I checked the sign test, but this when one wants to compare two dependent samples. And I just have one independent sample. Thank you

Salvatore S. Mangiafico, thank you for your suggestion, I was reviewing the binomial test and it seems to be very helpful, but for variables with more levels lets say age, which test would you suggest me? 

Kind regards, 

Magda

A multinomial test is like a binomial test but with more levels in the variable.  It is less common, and can get computationally intensive if counts are high. 

You can also use a chi-square goodness of fit or g-test goodness of fit with multiple levels.

If it's an option, I recommend the binomial / multinomial approach, since there are fewer assumptions.  But you may have to tell your audience, "This is like a chi-square goodness-of-fit test", since that's the most common test in this situation.

Dear Magda

The following remarks need to be studied carefully.

1. Testing statistical significance of the difference between the means of two sets of scores or more than two sets, and not between the sample and the population. How we can observe the population in our routine problems/researches ? It can be observed in Census, and not possible elsewhere.

2. When you want to draw a sample, the characteristic of the population should be studied and represented in your sample carefully. In addition, research studies are usually carried out on sample of subjects rather than whole populations. The most challenging aspect of fieldwork is drawing a random sample from the target population to which the results of the study would be generalized. The key to a good sample is that it has to be typical of the population from which it is drawn.

3.When the information from a sample is not typical of that in the population in a systematic way, we say that error has occurred. In actual practice, the task is so difficult that several types of errors, i.e. sampling error, non-sampling error, Response error, Processing error,… In addition, the most important error is the Sampling error, which is statistically defined as the error caused by observing a sample instead of the whole population. The underlying principle that must be followed if we are to have any hope of making inferences from a sample to a population is that the sample be representative of that population. A key way of achieving this is through the use of “randomization”.

4. There several types of random samples, Some of which are: Simple Random Sampling, Stratified Random Sampling, Double-stage Random Sampling... Moreover, the most important sample is the simple random sample which is a sample selected in such a way that every possible sample of the same size is equally likely to be chosen.

5. In order to reduce the sampling error, the simple random sample technique and a large sample size have to be developed.

Regards,

Zuhair

In addition to the confidence interval around mean, you may use the one Sample t-test against to a known value only if the data is continuous and parametric.

the question might be old but all the same i have some thoughts to add - and a question! :-)

the way i understand the original problem, magda has collected a sample of n_f full time students and n_p part time students (with n = n_ft + n_pt = 1914 ). on the other hand, she knows, probably from census data, the proportions p_ft and p_pt of full time and part time students in the whole population (e.g. all students nationwide). obviously, she could perform a chi2-test to check whether the proportions in her sample differ significantly from the known proportions in the whole population. in R, she would do that with chisq.test(c(n_ft, n_pt), p=c(p_ft, p_pt)).

now, the problem is that both her sample and also the reference population are rather large, and that means that her chi2-test will almost certainly lead to a very small p-value, implying that her sample proportions are significantly different from the corresponding proportions in the whole population. the reason is that however small (but > 0) the deviation between sample and true proportions is, a test will always reveal this difference if only the sample is large enough. however small an effect is, you can always "prove" it with a test and a really large sample!

and that leads me to my follow up question: is a chi2-test really the appropriate answer to the question of representativity if you have 2000 observations? shouldn't one use a different criterion than just statistical significance? something like maybe the sample proportions just being "large enough" in some sense...?!

i'd be glad if somebody experienced in representative surveys would share his knowledge! thank you very much in advance.