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What is a confidence interval?

Mario Banuelos, PhD
Associate Professor of Mathematics
Mario Banuelos is an Associate Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels.

Mario Banuelos, PhD

Associate Professor of Mathematics

Expert Answer

You can think of a confidence interval as a kind of a net that captures the potential region where a parameter lies. For example, you might want to calculate the average number of hours students spend online. To do that, you might ask a sample of 100 students how many average hours they spend online, then add or subtract the margin of error.

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Question

How can I find the z value of 95% on the table?

Community Answer

On your table, look to the larger (inner) box, find the closest to .9500 (it will probably be .9495, or .9505). These translate to 1.64 and 1.65 respectively.

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Given a sample of 100 projector bulbs from a company has a mean length of life of 20.5 hours with a standard deviation of 1.6 hours, how do I find a 95% confidence interval for the average length of life of those bulbs and then interpret the results?

Community Answer

20.6 is the upper limit and 20.4 is the lower limit. There is 95% confidence that the constructed interval includes the population mean.

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Question

A poll of a sample of Minnesotans indicates that 60% favor this plan. We also know that the standard error is 1.2. How do I construct a 95% confidence interval for this measurement?

Community Answer

By looking up the Z table, you can find that the confidence coefficient Z_0.475 is equal to 1.96. We then multiply this value by the standard error, which is 1.2, and we get 2.352. Therefore, the 95% confidence interval for this measurement is: 60% ± 2.352%.

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How can I decrease the width of the confidence interval?

Community Answer

Make the confidence lower! If you have a 99% confidence level, it means that almost all the intervals have to capture the true population mean/proportion (and the critical value is 2.576). However, if you use 95%, its critical value is 1.96, and because fewer of the intervals need to capture the true mean/proportion, the interval is less wide.

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