• Calculate a confidence interval and explain what it means.
• Recognize common misinterpretations of confidence intervals.

Did you take a nap?

Let’s explore data from a national survey of college students conducted by the American College Health Association.

One of the questions posed to the participants was, “On how many of the last 7 days did you take a nap?” Out of the [latex]38,440[/latex] participants, [latex]13,888[/latex] stated they had not taken a nap in the last 7 days.

Let’s use the statistical tool below to construct a confidence interval for the scenario above.

Step 1: Change the Enter Data box to Number of Successes.

Step 2: Input the sample size ([latex]n[/latex]) and the number of successes ([latex]x[/latex]), which in this case is the number of students who stated they had not taken a nap in the last [latex]7[/latex] days.

Step 3: (Optional) Specify appropriate labels for success/failure by checking the appropriate boxes and typing in labels.

Step 4: Leave the confidence level at the default [latex]95\%[/latex].

Step 5: View the [latex]z[/latex] critical value by selecting the box Show z-score for Margin of Error.

Step 6: The confidence interval will appear to the right along with the point estimate, standard error, margin of error, and [latex]z[/latex] critical value (labeled as [latex]z[/latex]-score).

[Trouble viewing? Click to open in a new tab.]

Accurately interpreting a confidence interval is just as important as ensuring our calculations are correct. One common but incorrect interpretation is that the confidence level is the probability (expressed as a percentage) that the population proportion is contained within the bounds of our confidence interval.

For example, using the confidence interval calculated previously, a student might incorrectly state: “There is a [latex]95\%[/latex] chance that the population proportion of college students who have not had a nap in the last [latex]7[/latex] days is between [latex]0.3565[/latex] and [latex]0.3661[/latex], or [latex]35.65\%[/latex] and [latex]36.61\%[/latex].”

Rather than measuring the likelihood that a single confidence interval contains the population proportion, the confidence level instead tells us the percentage of all confidence intervals that we’d expect to contain the population proportion, if we were to repeatedly take random samples and construct confidence intervals around our point estimates.