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 95% chance that the population proportion of college students who have not had a nap in the last 7 days is between 0.3565 and 0.3661, or 35.65% and 36.61%." Rather than measuring the likelihood that a single confidence interval contains the population proportion, the confidence level instead tells us the percentage of off 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. Let's explore this idea a bit more. Suppose we knew that the population proportion of college students who did not take a nap in the last week was 0.40, or 409%. Of course, in reality we wouldn't know the population proportion,- which is why we're creating a confidence interval to begin with-but let's assume that we did. Now suppose we took 10 random samples of 100 college students and constructed a 95% confidence interval for each one. We can run this simulation using the DCMP Explore Coverage and Confidence Intervals tool available at https://domathpathways.shinyapps.io/ExploreCoverage/ @. The following picture illustrates a simulated result of this situation using the tool (i.e., 10 confidence intervals constructed from our 10 random samples). 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Population Proportion pPart A: The red lines in the previous picture represent the confidence intervals that did not contain the population proportion 0.40. Out of the 10 confidence intervals from this simulated result, how many of them did not contain the population proportion 0.40? Hint The red lines are those that did not contain the population proportion--how many red lines are in the previous picture? Part B: The green lines in the previous picture represent the confidence intervals that did contain the population proportion that we assumed (0.40). Out of the 10 confidence intervals from this simulated result, how many of them did contain the population proportion 0.40? Hint The green lines are those that actually did contain the population proportion--how many green lines are in the previous picture? Part C Given your answers to Parts A and B, what proportion of the 10 confidence intervals did contain the population proportion 0.40? What is this value as a percentage? Hint In Part B, you calculated the number of intervals containing the proportion 0.40. Divide this value by 10 to calculate the proportion. 96 of the intervals