Calculating the Confidence Interval

In the box below, please enter the two endpoints of the 95% confidence interval for the population mean, then click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.

95% Confidence Bounds: (, )

Make sure the lower limit is less than the upper limit.

You got the correct endpoints to the 95% confidence: (55.3648, 61.7018). Congratulations!

Unfortunately, your answer was not correct. Either try again or click on “Show Solution” below to see how to obtain the correct answer.

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Here is the formula you can use to calculate the confidence bounds.

In this formula, there are a few symbols to know:

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Thus, we are 95% confident that the mean height for the population is between 55.3648 and 61.7018.

Note that the value following the plus/minus sign ± is known as the margin of error, \( E = 3.1685 \), and is always half of the width of the confidence interval. The margin of error is affected by the sample size, the level of confidence, and the variability of the population. If n increases, the margin of error shrinks. If the level of confidence increases, the margin of error expands. If the variability in the population increases, the margin of error expands.

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The z-procedures are sensitive to knowing the population variance. Logic dictates that if we do not know the population mean, then we will not know the population variance. As such, the z-procedures are rarely used now. As such, there is no z-test in the base R package.

The following code will perform the above steps, however. Copy and paste the following code into your R script window, then run it from there.

In the R output, the limits to the 95% confidence interval are the last numbers output by the script.

For practical reasons, the base R installation does not contain the z-procedures. However, I have created one for pedagogical purposes. If you are connected to the Internet, the following will also calculate the test statistic:

The limits of the 95% confidence interval follow the line “95 percent confidence interval:” near the middle of the output.

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The z-procedures are sensitive to knowing the population variance. Logic dictates that if we do not know the population mean, then we will not know the population variance. As such, the z-procedures are rarely used now. As such, there is no z-test in the base Excel program.

Copy and paste the following code into your Excel spreadsheet window, making sure the value x ends up in A1 after pasting.

The limits for the 95% confidence interval are the numbers calculated in cells C5 and C6.

Again, when you paste this code into Excel, make sure that you start the pasting in call A1. To help with that, you may want to also copy this notice. It seems to help.