Three Ways to Write a Confidence Interval

Assuming that you are familiar with how to calculate a confidence interval as well as how to interpret a confidence interval, you may be curious as to actually write down a final “answer” or result. The following lesson will show you some different ways that confidence intervals can be written.

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Example

We will use the following example to think about the different ways to write a confidence interval. For practice, you should make sure you know how to do the calculations needed to get the interval.

A college student wishes to estimate the percentage of students on his campus who can name the current president of the college. He chooses a random sample of 347 students and finds that 86 of them can in fact, name the current college president. Help the student estimate the percentage of all students who can name the current president by calculating a 95% confidence interval.

Using the formula for a confidence interval for the population proportion, The final answer for this is:

\(0.248 \pm 0.045\)

Let’s think about different ways this interval might be written.

Method 1 – point estimate +/- margin of error

All confidence intervals are of the form “point estimate” plus/minus the “margin of error”. If you are finding a confidence interval by hand using a formula (like above), your interval is in this form before you do your addition or subtraction. This is a common way to actually present your confidence interval.

Final Answer: \(0.248 \pm 0.045\)

Since this confidence interval is estimating a percentage, it might also be written as:

Final Answer: \(24.8\% \pm 4.5\%\)

This would be the method probably used on the news or in any report to a general audience. The student could say “about 24.8% of students at my college can name the college president.” He would then also note that his estimate has a margin of error of 4.5%. If you pay attention to the evning news or new articles, they typically show this number on the screen or mention it in the article while discussing similar polls.

Method 2 – as an interval

If you actually do the two calculations, 0.248 – 0.045 = 0.203 and 0.248 + 0.045 = 0.293, you will get two endpoints. These can be used to write the final answer as well.

Final Answer: \((0.203, 0.293)\)

Or, if using percentages:

Final Answer: \((20.3\%, 29.3\%)\)

This is typically how a confidence interval will look if you calculating it using technology such as a ti83 or ti84. It’s a little more “mathy” so you might not see it as often in reports to general audiences.

Method 3 – as an inequality

Finally, some textbooks like to write intervals in a way to help you remember that somewhere in that interval is your population value you are trying to estimate. In this case, you are estimating the population proportion p, so you would write.

Final Answer: \(0.203 < p < 0.293\) Note that if you were estimating the mean, you would place \(\mu\) within the inequality. [adsenseLargeRectangle]

Important

If you are taking a statistics course, it is of important to pay attention to how your professor or textbook prefers to present confidence intervals and generally stick to that method. If instead, you are using confidence intervals in your research, it is probably important to consider your audience. Most people have no trouble understanding the idea of adding and subtracting a margin of error, even if they haven’t had much formal training in statistics. This should be a consideration when you present your findings.

Finally, and most importantly, do take a moment to read about how to interpret confidence intervals. It is easy to get caught up in the math side but you still must know how this applies to real life situations.

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