Confidence Interval

 

The confidence interval is the range where you expect something to be. By saying "expect" you live open the possibility of being wrong. The degree of confidence measures the probability of that expectation to be true.

 

The degree of confidence is linked with the width of the confidence interval. It's easy to be very confident that something will be within a very wide range, and vice versa. Also, the amount of information (typically related with the sample size) has an influence on the degree of confidence and the width of the confidence interval. With more information you will be more confident that "the thing" will be within a given interval. Also, with more information, and keeping a given degree of confidence, you can narrow the interval.

 

Then you can finish with an example from an Internet forum:

 

In a given city a survey is made. The question is: "Do you prefer Coke or Pepsi?" 60% answer Coke, and 40% answer Pepsi. So an estimation is that, in this city, 60% prefer Coke. Does it mean that 60% of the population in this city prefer Coke? No unless the survey had been answered by all the population. However, you can be somehow confident that the actual proportion of people choosing Coke will be within some interval around the 60% found in the sample. How confident? How wide is the interval?

 

If the survey is based on a sample of 100 persons, you can be 90% confident that the actual proportion of Coke will be between 52% and 68%. Also, you can be 99% confident that the actual proportion will be between 48% and 72% (for the same sample size, more confidence, and wider interval).

 

If the survey had been on a sample of 1000 persons instead of 100, you could be 90% confident that the actual proportion is between 57.5% and 62.5% (compare with 52% and 68% for the same confidence with a sample of 100: larger sample, narrower interval for the same degree of confidence). And you could be 99.99998% (let's say 100%?) confident that the actual proportion will be between 52% and 68% (compare with a degree of confidence of 90% for the same interval with a sample of 100: larger sample, better degree of confidence for the same interval).[1]

 

In the example you gave, would you explain where the "52% and 68%" figures come from.

An excellent explanation, however a couple of other things come in to play here, and as a teacher, one needs to be prepared for further questions.  The confidence level is a measure of power (the degree to which I'm confident about these results).  But how precise do we want to be (the degree to which the sample results would actually reflect the population results)?  This is +/- number reported in Gallop Polls (e.g., 56% (+/- 3%).   Power and precision have an inverse relationship for a given sample size:  more power means less precision, and vice versa.   The end result is a not-very-intuitive statement like, "I'm 95% confident that 63% (+/- 3%) of the population favors Coke over Pepsi", or "I'm 90% confident that 63% (+/- 2%) of the population favors Coke over Pepsi" (for a given sample size).