Statistics for Business: Decision Making and Analysis, 3rd Edition

15.2 Confidence Interval for the Mean

A similar procedure produces a confidence interval for μ, the mean of a population. The similarity comes from again using a normal model for the sampling distribution of the statistic,

$$\overline{X}\sim N\left(\mu , \frac{{\sigma }^2}{n}\right)$$

This sampling distribution implies that

$$P\left(\overline{X}-1.96 \sigma /\sqrt{n}\le \mu \le \overline{X}+1.96 \sigma /\sqrt{n}\right)=0.95$$

The average of 95% of samples lies within $1.96\sigma /\sqrt{n}$ of μ. Once again, the sample statistic lies within about two standard errors of the corresponding population parameter in 95% of samples. As in Chapters 13 and 14, $\overline{X}$ is a random variable that represents the mean of a randomly chosen sample, and $\overline{x}$ in lowercase is the mean of the observed sample.

Because σ is unknown, we do not know the standard error of $\overline{X}$. The solution is to plug in a sample ...