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Question 340:



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So it turns out that the reason for the difference is that when there are either all successes or all failures, as is the case in the example I gave (5/5), you cannot have a 2-sided confidence interval (since you can't go above 100%). In these situations a 95% CI ends up being a 97.5% CI. With a 95% interval, there is a 2.5% likelihood of the population completion rate being above this upper-bound or a 2.5% chance of it being below the level. When all pass or fail a task you only have 2.5% below, so you need to use the critical value for a 90% Confidence Level so there is 5% above and 5% below the interval, with the top again getting excluded, leaving you with just 5% below the interval.

There is already a note on the confidence interval calculator page, but that doesn't help while you're reading the Restoring Confidence article. You will notice when you change the Confidence Level to 90% on the calculator or spreadsheet, you'll get 55%.

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