:Speaking of having problems, I seem to have developed a mental block on :being able to get the kids to understand applying concepts to :calculations in this context. Here's an example of a problem which has :been condensed a bit: :"If 113 of 130 people in an SRS respond favorably, construct a 95% :confidence interval for the population." For some stupid reason (I :hope), I'M STUCK!!!
I address two areas: first the question itself and second the application of concepts by the student.
I too have difficulty with the question. Assuming we are discussing a CI for the population PARAMETERS, several question arise. (1)Are the responses random? If one group can be expected to respond with a bias and that group is more likely to respond than others, then the results are biased. At this level, I feel most of us would allow/suggest the assumption of random responses; but, it does deserve discussion. (2) What is the population? In particular, how big is it? If it is 130, no confidence interval is needed--you have your parameter(assuming it is the proportion favorable). If it is relatively small, say less than 1300, or so, you may need a finite correction factor(hypergeometric model)to a standard binomial model. If larger,(or using the correction factor) one might use the standard normal approximation to the binomial proportion, p. (3) What model is appropriate? Under the assumption of random responses, I'd probably choose the binomial.
The more difficult question was deliberately held for last because I suspect each of us tries to discuss the ideas in the above paragraph. I believe part of the students problem is that we reveal so much to them that our caveats confuse them OR that they intuitively are aware of some of the real world problems and cannot believe a specific model applies. In the problem cited, there are valid reasons to believe the binomial model may not apply. The student, perhaps subconsciously, is aware of this--particularly if we have stressed that the responses must be random. Add to this the concept that we are APPROXIMATING one distribution with another and students already uncertain as to the appropriateness of the model, suddenly are faced with the intuitive feeling that APPROXIMATING an uncertain model is a fool's errand. Fear (perhaps too much information) of being incorrect makes them hesitate. I try to avoid this(only somewhat effectively) by stressing that we are choosing a model and we must always examine the reasonableness of our model's assumptions. I further stress that most of the models we study in THIS course are fairly forgiving of minor deviations from some of the assumptions. I frequently require that students provide me a discussion of the model's assumptions for one, or two, homework problems each night. They are asked to pick those which gave them the greatest intellectual difficulty. This discussion is to include whether they are certain each assumption of the model holds OR would they be willing to say that THEY expect the assumption to be "reasonable" for the given situation. In our case, we have discussed the discrete models: binomial, uniform, poisson, hypergeometric, and geometric(and, of course, general tabulated distributions).