Wiki Page, Math 13 - Perez Sandbox : Module 23 - Wrap Up "Estimating a Population Mean"

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2026-01-17T21:15:40-08:00

Let's Summarize

A confidence interval approximates a population mean by giving us a range of values that likely contains the population mean, . The general form of the confidence interval is .

To say that we are “95% confident that the population mean falls within our confidence interval” really means that “about 95% of all confidence intervals computed in this way will capture the true population mean.”

We can use a sample mean to build a confidence interval as an estimate for . There are two possible cases:

Suppose the population standard deviation, σ, is known. We check the conditions for use of the normal model. Conditions: The variable must be normally distributed in the population, or the sample size is large enough (n > 30). In this case, the confidence interval has the form
.

Suppose the population standard deviation, σ, is not known. Then we use the sample standard deviation, s, as an approximation for σ. We check the conditions for use of the T-model. Conditions are the same: The variable must be normally distributed in the population, or the sample size is large enough (n > 30). In this case, the confidence interval has the form
.
Also, when using the T-model to find the critical value, we need to select an appropriate number of degrees of freedom (df). The number of degrees of freedom is 1 less than the sample size
(df = n - 1).

As we have seen with other confidence intervals, the width of a confidence interval is twice the margin of error. The smaller the margin of error, the more narrow the confidence interval and the more precise the estimate of .

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