Section 8.1 Summary

1. Using a single statistic only, such as $\overline{x}$, to estimate a population parameter is called point estimation. The value of the statistic is called the point estimate.

2. A confidence interval estimate of a parameter consists of an interval of numbers generated by a point estimate, together with an associated confidence level specifying the probability that the interval contains the parameter. The $100(1-\alpha )\%$ $Z$ confidence interval for $\mu$ is given by the interval

   $\begin{array}{l}\text{lower bound}=\overline{x}-Z_{\alpha /2}\left(\sigma /\sqrt{n}\right)\\ \text{upper bound}=\overline{x}+Z_{\alpha /2}\left(\sigma /\sqrt{n}\right)\end{array}$

   where $1-\alpha$ is the confidence level. If $\sigma$ is not known, then the $Z$ interval cannot be used. You may use the following generic interpretation for the confidence intervals that you construct: “We are 90% (or 95%, or 99%, and so on) confident that the population mean _____ lies between _____ (lower bound) and ______ (upper bound).”

3. The margin of error $E$ is a measure of the precision of the confidence interval estimate. For the $Z$ interval for the mean, the margin of error takes the form

   $E=Z_{\alpha /2}(\sigma /\sqrt{n})$

   Usually, our confidence intervals take the form

   $\text{point estimate}\pm \text{margin of error}$

4. To use a $Z$ interval to estimate the population mean $\mu$ to within a margin of error $E$ with confidence $100(1-\alpha )\%$, the required sample size is given by

   $n={\left[{\displaystyle \frac{\left(Z_{\alpha /2}\right)\sigma }{E}}\right]}^2$

   where $Z_{\alpha /2}$ is associated with the desired confidence level (Table 1), $E$ is the desired margin of error, and $\sigma$ is the population standard deviation. Round up to the next integer if there is a decimal.