" I looked through some books, and they basically said no because its not a representative sample."

I think that these authors have a quite narrow understanding of "statistical analysis". I'd say: yes, sure you can analyze the data, and sure you can try to interpret the data and learn something from it.

But it is also correct that the estimates and confidence intervals do not have their presumed frequency-properties w.r.t. to the "popoulation of people living in the city". This is the major misconception I recognized in your post:

"The population was people that lived in the city."

You mean that this was your desired population, but it is not the one you can analyze or infer about with statistical means only. In a narrow sense your statistical population are those people that

a) were around (close to your place) in the two districts at the times you were there and

b) were willing to participate in your survey

It is part a) of these "limitations" that are related to a "stratification". I mean, if the whole city was your aimed population and if you have to sample locally, then stratification would mean to select some places of the city (randomly) and take a sub-sample at each of these places. So, having just two different places is not what I would call a good stratification in this sense. If you can argue that there selected places are just very representative for the city, then it is ok. But without this information you may have picked samples where the responses of the individuals are correlated so that you get a biased result (for the estimate and the precision). The problem with having data from "just" 2 districts is too little to sensibly judge whether or not the results will be biased (if you have data from many districts, and all the results were similar, then it would be sensible to argue that the variance is in fact small and that there is no [considerable] bias; if the results were quite differeing, you would not base a conclusion on one or two selected disticts - the whole data set shows you that there is a high variance -> impossible to give a very precise estimate without [possibly considerable] bias).

Further it is not clear if the participants were citicens of this city. If your locations, for instance, were shopping centers, I would expect a lot of people from other cities around. You should make clear if you asked these people if the really live in the city, better ask them where in the city they live.

Part b) is another very critical point. This makes your sampling much being a "convenience sampling". This will unevitably introduce some bias, since you will never get the answers from people that are not willing to participate. The problem is that this willingness is very likely correlated with either the experiences with X or with the general lifesytle and thereby indirectly with X again.

Any interpretation/discussion of your data/results should be made considering explicitly these limitations. Although this may then not be the ideal scenario it may well be the best possible one can do. The uncertainty of generalizations (from your sample to the city - and to people that were not willing to respond!) is neccesarily high (and clearly much higher than indicated by confidence intervals). You have no way to estimate this uncertainty from your data alone, but you may know a lot of the city and the people and their general behaviour and representativeness of your sample (based on common sense, experience, expertise). So you can argue about this in a sensible way. As long as you consider these additional uncertainties and others can follow your argumentation, everithing should be fine.

"there is one previous study somewhat similar to mines,"

If or how the results will be comparable depends on the precise meaning of this "somewhat". You have to judge if the differences are likely relevant to the survey. If they are relevant, a comparison is difficult.

[Edit: corrected some typos and added forgotten words. I seem to have consitently written "startification" instead of "stratification"... funny)]

" I looked through some books, and they basically said no because its not a representative sample."

I think that these authors have a quite narrow understanding of "statistical analysis". I'd say: yes, sure you can analyze the data, and sure you can try to interpret the data and learn something from it.

But it is also correct that the estimates and confidence intervals do not have their presumed frequency-properties w.r.t. to the "popoulation of people living in the city". This is the major misconception I recognized in your post:

"The population was people that lived in the city."

You mean that this was your desired population, but it is not the one you can analyze or infer about with statistical means only. In a narrow sense your statistical population are those people that

a) were around (close to your place) in the two districts at the times you were there and

b) were willing to participate in your survey

It is part a) of these "limitations" that are related to a "stratification". I mean, if the whole city was your aimed population and if you have to sample locally, then stratification would mean to select some places of the city (randomly) and take a sub-sample at each of these places. So, having just two different places is not what I would call a good stratification in this sense. If you can argue that there selected places are just very representative for the city, then it is ok. But without this information you may have picked samples where the responses of the individuals are correlated so that you get a biased result (for the estimate and the precision). The problem with having data from "just" 2 districts is too little to sensibly judge whether or not the results will be biased (if you have data from many districts, and all the results were similar, then it would be sensible to argue that the variance is in fact small and that there is no [considerable] bias; if the results were quite differeing, you would not base a conclusion on one or two selected disticts - the whole data set shows you that there is a high variance -> impossible to give a very precise estimate without [possibly considerable] bias).

Further it is not clear if the participants were citicens of this city. If your locations, for instance, were shopping centers, I would expect a lot of people from other cities around. You should make clear if you asked these people if the really live in the city, better ask them where in the city they live.

Part b) is another very critical point. This makes your sampling much being a "convenience sampling". This will unevitably introduce some bias, since you will never get the answers from people that are not willing to participate. The problem is that this willingness is very likely correlated with either the experiences with X or with the general lifesytle and thereby indirectly with X again.

Any interpretation/discussion of your data/results should be made considering explicitly these limitations. Although this may then not be the ideal scenario it may well be the best possible one can do. The uncertainty of generalizations (from your sample to the city - and to people that were not willing to respond!) is neccesarily high (and clearly much higher than indicated by confidence intervals). You have no way to estimate this uncertainty from your data alone, but you may know a lot of the city and the people and their general behaviour and representativeness of your sample (based on common sense, experience, expertise). So you can argue about this in a sensible way. As long as you consider these additional uncertainties and others can follow your argumentation, everithing should be fine.

"there is one previous study somewhat similar to mines,"

If or how the results will be comparable depends on the precise meaning of this "somewhat". You have to judge if the differences are likely relevant to the survey. If they are relevant, a comparison is difficult.

[Edit: corrected some typos and added forgotten words. I seem to have consitently written "startification" instead of "stratification"... funny)]

Thank you very much.  To clarify, I did ask respondents if they lived in the city before administering the survey.  One of the questions on the survey also asked for their home postal codes.  I disqualified answers with postal codes outside of the range for the city.

It looks like I can run descriptive statistics, tests of association, cross-tabulation on the results.  As I understand it now, I need to interpret all of these things together for each question as best as I can with context in mind.  Then, I must make clear that the sample may be biased.  I definitely can't make the argument that the sample is representative of the city based on common sense alone.  You're right in that I am very hamstrung by who is willing to take the survey.

I agree with Wilhem, but I think convenient sampling does not represent the population you are interested in. only to select randomly can not provide true representatives of the population therefore mostly statistical and social research methods' books do not even include it into probability sampling.

You can apply statistical test but it might not provide you the better or somehow logical answers. 

Wilhem is ok. To me though convenient is not representative but some times it is necessary and having no other alternative. for example, if you are conducting a survey on street beggers, convenience sampling is appropriate.

You did not say why you chose the particular places to sample.  Was it convenience or was there another reason?  That is was there a purpose related to your study for choosing those particular locations. 

How large is the city?  If the city is very large then two locations is totally inadequate to even approach representing the city in any way.  On the other hand, if the city is small two locations might give you a sense of the people who live in the city.

You cannot do inferential statistics since you do not have a probability sample.  You can do descriptive statistics to describe your sample.  What this means is that you cannot generalize to the city residents; you can only make statements about your sample.  If the locations were purposively chosen and the city is small, you can make statements that it is possible that this is the way that city residents behave/believe.  If the city is large and/or your choice of locations was not purposive, you cannot make any statements about city residents.

You can combine the data from the two locations and talk about your respondents.  If it makes sense, you can compare the respondents from the two locations (e.g., respondents who grocery shop versus those who attend a sports game).

You can also look to see how the characteristics of your sample compare to the the same characteristics of the population.  This would be useful if your choice of locations was purposive and the city is small.

Hi,

I feel that all statistical theory of inference is based on probability sampling and in my opinion, no statistical inference can be drawn on convenient or non-random sampling.

Farah Yasmeen

Non-probability based convenience sampling won't be as good as probability sampling, but sometimes you have no choice, or is cheaper (in behavioural terms and economic terms). Books says lots of things, most of them really useful... but you can't let them narrow absolutely everything. I think the most important thing of all is the kind of inferences you make about the phenomena and the data. At the end, the statistical software can gives you an average of the gender... but is up to you to put in your paper that everyone in the sample are half male/half female.

You are allowed to analyse convenience sample data for descriptives and statistical significance using the desired tests but you need to state clearly that your sample is a biased one in your report or paper under the heading of limitations. It is acceptable.

Hi

The issue here should be what informed

You can't do inferential statistics but, yes, you can calculate descriptive statistics, e.g., percentages for categorical variables and the mean, median, and mode for ratio scale variables . . . so long as you make clear that they represent values in a convenience sample. You should also describe how you implemented the convenience sampling, making known any foreseeable biases that may have resulted from your procedures. Convenience sampling is useful in the conduct of exploratory research in preparation for more rigorous hypothesis testing in a randomized representative sample of a known population universe.

A lot of good points have been made.  As several others have pointed out, you can do statistics (after all, statistic programs don't know the nature of your data, it's all just numbers in the formulas).  Your  limitations are in the type you can run, and most importantly, in the inferences or conclusions you make.  Happens all the time in science; you simply need to be up front about your limitations and explain them.   And it is always good to consult with the previous research in your field.  If your survey methodology is standard practice in your field, then others face the same limitations as you and thus, are limited in their conclusions as well.