0
$\begingroup$

My professor was trying to explain something to me about confidence interval and I haven't been able to understand it.

There is a statement I think is true that she says is false. I can't understand why it is false.

The situation is that the (107.8, 116.2) is a 95% confidence interval for a mean statistic.

The statement is:

There is a 95% probability that the interval from 107.8 to 116.2 contains μ

My statistics professor says that the statement is false because the probability is either 0 or 1.

However, I am fairly sure that the statement is true considering that the definition of probability is:

the extent to which an event is likely to occur, measured by the ratio of the favorable cases to the whole number of cases possible.

My statistics professor has tried to explain to me her point but I have not understood it yet.

We did both agree that the following statement is true

This interval was constructed using a method that produces intervals that capture the true mean in 95% of all possible samples

I am fairly sure this statement says exactly the same thing as the first statement does. What am I missing?

  • 0
    That your professor is a frequentist. – 2017-01-06
  • 0
    Can you please explain that further? – 2017-01-06

1 Answers 1

1

The typical interpretation of confidence intervals goes something like this:

Let's say I have a statistic $\xi$ and I have decided upon an experiment to create a confidence interval. A 95% CI means that if I repeated the experiment over and over and over (infinitely many times) 95% of the time the CI I generated would contain $\xi$. Now, $\xi$ is a fixed number. So for any fixed interval, the interval either contains or does not contain $\xi$. This is what your teacher means.

Does this make sense, or should I give it another go?

----EDIT-----

Additionally, if we have a 95% confidence interval, and $\xi$ is outside the interval, we can interpret the sampling event (the collection of data used to make the CI) to have had a probability of 5% (or less). In this sense, it is the sampling event that was "rare."

  • 0
    Why does that invalidate saying there is a 95% probability. I agree it either is or isnt but lets say we set out a physical number line on the ground and dropped a penny randomly there would be a non 0 or 1 chance that the penny would land on the line on the ground right? – 2017-01-06
  • 1
    Because given the fixed interval there isn't a 95% probability that the value is contained in the interval. Before the sampling (collection of data) has occurred, we know that we will generate an interval that contains the value with a probability of 0.95. The issue is you shouldn't think about the mean as the random variable, but rather the interval is random (or more precisely, probabilistic). Therefore, once we have fixed the interval, we have already realized a value of the random quantity. – 2017-01-06
  • 0
    Thankyou! That makes sense! – 2017-01-06