Statistical Process Control and Data Analytics, 8th Edition

Appendix GConfidence limits and tests of significance

Confidence limits

When an estimate of the mean of a parameter has been made it is desirable to know not only the estimated mean value, which should be the most likely value, but also how precise the estimate is.

If, for example, 80 results on weights of tablets give a mean $\overline{X}$ = 250.5 mg and standard deviation σ = 4.5 mg, have these values come from a process with mean µ = 250.0 mg? If the process has a mean µ = 250.0, 99.7 per cent of all sample means $(\overline{X})$ should have a value between:

$\mu \pm 3\sigma /\sqrt{n,}$

that is

$\mu -3\sigma /\sqrt{n}<\overline{X}<\mu +3\sigma /\sqrt{n},$

therefore:

$\overline{X}-3\sigma /\sqrt{n}<\mu <\overline{X}+3\sigma /\sqrt{n},$

that is, µ will lie between:

$\overline{X}\pm 3\sigma \sqrt{n},$

this is the confidence interval at the confidence coefficient of 99.7 per cent.

Hence, for the tablet example, ...