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 500500 voters whether or not they support the incumbent candidate. The pollster will take the results of the sample and construct a 90%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%90\% confidence level?
Choose all answers that apply:
Choose all answers that apply:

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 100100 pitches and constructed a 95%95\% confidence interval for the mean speed. The resulting interval was (110,120)(110, 120).
Which of the following is a correct interpretation of the interval (110,120)(110, 120)?
Choose all answers that apply:
Choose all answers that apply:

We shouldn't say there is a 95%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%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%99\% confidence level.
How will increasing the confidence level from 95%95\% to 99%99\% affect the confidence interval?
Choose 1 answer:
Choose 1 answer:

Try it for yourself

Here's a simulation that lets you explore these concepts. When you click "draw sample," it takes a sample from the population shown, calculates the sample mean, and constructs a confidence interval based on that sample data. Click "draw sample" a few times (or many times). Notice that some intervals contain the population mean μ\mu, while others don't.
Try changing the confidence level, draw many samples, and pay attention to "percent hit"—this shows what percentage of the intervals successfully captured the population mean μ\mu. Click "reset" and try again for a different confidence level. What do you notice about "percent hit" now? Try changing only the sample size to see what effect that has.
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