21.2 Interpretation of a CI

Interpreting CIs correctly is tricky. The correct interpretation (Definition 20.3) of a 95% CI is the following:

If samples were repeatedly taken many times, and the 95% confidence interval computed for each sample, 95% of these confidence intervals formed would contain the population parameter.

This is the idea shown in the animation in Sect. 20.4. In practice, this definition is unsatisfying, since we almost always have only one sample. And since the value of the parameter is unknown (after all, we went to the bother of taking a sample so we could estimate the value of the parameter), we don’t know if our CI includes the population parameter or not.

A reasonable alternative interpretation is:

The interval gives a range of values of the parameter that could plausibly (with 95% confidence) have given rise to our observed value of the statistic.

Or we might say that:

There is a 95% chance that our computed CI straddles the value of the population parameter.

These alternatives are not absolutely correct, but are reasonable interpretations.

Many people will write—and you will see it written in many places—that the CI means that there is a 95% chance that the CI contains the population parameter. This is not strictly correct, but is common (probably because it is easier to understand).

I use this analogy: Most people say the sun rises in the east. This is incorrect: the sun doesn’t rise at all. It appears to rise in the east because the earth rotates on its axis. But almost everyone says that the ‘sun rises in the east,’ and for most circumstances this is fine and serviceable, even though technically incorrect.

Similarly, most people use the final interpretation above for a CI in practice, even though it is technically incorrect.

Example 21.1 (Energy drinks in Canadian youth) In Example 20.1, the approximate 95% CI was from 0.192 to 0.236 The correct interpretation is:

If we took many samples of 1516 Canadian youth, and computed the approximate 95% CI for each one, about 95% of those CIs would contain the population proportion.

We don’t know if our CI includes the value of \(p\), however. We might say:

This 95% CI is likely to straddle the actual value of \(p\).

The range of values of \(p\) that could plausibly (with 95% confidence) have produced \(\hat{p} = 0.241\) is between 0.192 and 0.236.

In practice, the CI is usually interpreted as saying:

There is a 95% chance that the population proportion of Canadian youth who have experienced sleeping difficulties after consuming energy drinks is between 0.192 to 0.236.

This is not strictly correct, but is commonly used.

Think 21.1 (Interpretation of a CI) In Example 20.2 about koalas crossing roads, the approximate 95% CI was from 0.130 to 0.209.

What is the correct interpretation of this CI?