Why the marbles?

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Why the marbles?

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I have introduced the marbles binomial example as it mirrors the testing process, at least in a fair number of cases.

 

The marbles may be look at as test items.

 

An item pool is the bag of marbles.

 

In the pool are a certain number of items on which a given person knows the right answer.

 

In the case of a scale, we might say that the pool has in it a certain number of items on which a given person strongly agrees.

 

When we give a test or survey, we may be seen as drawing random samples of items from the pool.

 

We calculate the proportion of correct answers in the sample, or the proportion of strongly-agrees, and then use the binomial to get a confidence interval, a range within which we believe the true proportion lies.

 

Drawing bigger samples of marbles or items gives us shorter confidence intervals, greater precision, more confidence.

 

But there's at least one thing to ponder here. Drawing a sample of items from a pool and presenting them to a student is not quite the same as drawing marbles from a bag.

 

When presented with a red marble, we can be sure it's a red marble. But when a student gives a correct answer to a question, s/he may have just made a good guess. Or, the student may in fact know the right answer but fail to give it because of, for example, personal problems (headache, worries about home, test anxiety, and so on), or, perhaps, environmental issues (the room is too hot, there are noises coming from the street, the lighting is bad, and so on).

 

Testing students is not quite the same as looking at the color of a marble. Nonetheless, the binomial confidence interval is frequently used in testing in order to have a working confidence interval for a student's "true" score (the true proportion of items in the pool which the student knows).