Can anyone Explain why this statistics statement is incorrect and give an alternatively, correct statement?

Question

In an examination students were asked to say what is meant by:

“The interval (11.1, 16.8) is 100(1-α) % confidence interval for the parameter θ^n. One student wrote that:

“the interval (11.1,16.8) is 100(1-α)% confidence interval for θ if θ lies within the interval with probability (1-α)”

Explain why this statement is incorrect and give an alternatively, correct statement.

Answer 1

I think student is saying that that is the interval if the probability of being in the interval is 1-a

should be "there is 1-a probability that theta is in the interval"

Answer 2

There isn't much wrong with it, but it could be seen as incomplete or insufficient.  I assume you were marked down for providing this answer on a test?

A 95% confidence interval for a population parameter (theta) provides the range that the average of a  sample (of size n) will fall within with probability 1-alpha.

The only thing thats wrong with your answer is that you confuse the population with a sample of the population. Remember a parameter refers to the population and a statistic (such as a confidence interval) refers to samples.

Its a somewhat odd question because it asks for a 100% confidence interval.  With parametric statistics, there is no such thing as a 100% confidence interval.  An assumption of parametric statistics is that extreme results are always possible, if extremely unlikely.  The 100% confidence interval of 11.1-16.8 implies that all possible data points in the population fall within this range.  You could never come up with that number as an estimate; no statistical analysis would come up with that answer.  You could only come up with that interval by counting every single datapoint in the population, at which point statistical descriptions are meaningless because you have data on the entire population.