5.4 - Inference for the Population Mean

Overview Section

In this section, we discuss how to find confidence intervals for the population mean. The idea and interpretation of the confidence interval will be similar to that of the population proportion only applied to the population mean, \(\mu\).

We start with the case where the population standard deviation, \(\sigma\), is known. We continue to the more realistic case where \(\sigma\) is not known. For the latter case, we need to recall the \(t\)-distribution. We end this section by presenting how to determine a sample size for a desired margin of error and confidence.

Point Estimates for a Population Mean

The point estimate of the population mean, \(\mu\) is:

Point Estimate of the Population Mean
\(\bar{x}=\) sample mean

If one wants to know how accurate the sample mean is to estimate the population mean, we need some probability statement. We will want to know the sampling distribution of \(\bar{x}\). From this distribution, we can get a confidence interval. Such an interval provides a range of values for which the parameter value is believed to fall. An interval is more likely to be "correct" than a point estimate.