Interpreting confidence levels and confidence intervals

When we create a confidence interval, it's important to be able to interpret the meaning of the confidence level we used and the interval that was obtained.

The confidence level refers to the long-term success rate of the method, that is, how often this type of interval will capture the parameter of interest.

A specific confidence interval gives a range of plausible values for the parameter of interest.

Let's look at a few examples that demonstrate how to interpret confidence levels and confidence intervals.

Example 1: Interpreting a confidence level

A political pollster plans to ask a random sample of

500

voters whether or not they support the incumbent candidate. The pollster will take the results of the sample and construct a

90 %

confidence interval for the true proportion of all voters who support the candidate.

Which of the following is a correct interpretation of the $90 %$ ‍ confidence level?

(Choice A)
If the pollster repeats this process and constructs $20$ ‍ intervals from separate independent samples, we can expect about $18$ ‍ of those intervals to contain the true proportion of voters who support the candidate.
(Choice B)
About $90 %$ ‍ of people who support the candidate will respond to the poll.
(Choice C)
If the pollster repeats this process many times, then about $90 %$ ‍ of the intervals produced will capture the true proportion of voters who support the candidate.

Example 2: Interpreting a confidence interval

A baseball coach was curious about the true mean speed of fastball pitches in his league. The coach recorded the speed in kilometers per hour of each fastball in a random sample of

100

pitches and constructed a

95 %

confidence interval for the mean speed. The resulting interval was

(110, 120)

Which of the following is a correct interpretation of the interval $(110, 120)$ ‍?

(Choice A)
If the coach took another sample of $100$ ‍ pitches, there's a $95 %$ ‍ chance the sample mean would be between $110$ ‍ and $120 km/hr$ ‍.
(Choice B)
About $95 %$ ‍ of pitches in the sample were between $110$ ‍ and $120 km/hr$ ‍.
(Choice C)
We're $95 %$ ‍ confident that the interval $(110, 120)$ ‍ captured the true mean pitch speed.

We shouldn't say there is a

95 %

chance that this specific interval contains the true mean, because it implies that the mean may be within this interval, or it may be somewhere else. This phrasing makes it seem as if the population mean is variable, but it's not. This interval either captured the mean or didn't. Intervals change from sample to sample, but the population parameter we're trying to capture does not.

It's safer to say we're

95 %

confident that this interval captured the mean, since this phrasing more closely agrees with the long-term capture rate of confidence levels.

Example 3: Effect of changing confidence level

Suppose that the coach from the previous example decides they want to be more confident. The coach uses the same sample data as before, but recalculates the confidence interval using a

99 %

confidence level.

How will increasing the confidence level from $95 %$ ‍ to $99 %$ ‍ affect the confidence interval?

(Choice A)
It's impossible to say without seeing the sample data.
(Choice B)
Increasing the confidence will increase the margin of error resulting in a wider interval.
(Choice C)
Increasing the confidence will decrease the margin of error resulting in a narrower interval.

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