91Ó°ÊÓ

An interval is a set of numbers that includes all the real numbers between the two endpoints of the interval.

\({\rm{100(1 - \alpha )\% confidence interval}}\)

the mean is calculated using,

\(\left( {{\rm{\bar x - }}{{\rm{z}}_{{\rm{\alpha /2}}}}{\rm{ \times }}\frac{{\rm{\sigma }}}{{\sqrt {\rm{n}} }}{\rm{,\bar x + }}{{\rm{z}}_{{\rm{\alpha /2}}}}{\rm{ \times }}\frac{{\rm{\sigma }}}{{\sqrt {\rm{n}} }}} \right)\)

when it is known what the value\({{\rm{\sigma }}^{\rm{2}}}\)is.

The broader the interval, the higher the confidence level.

As a result, a \({\rm{90\% }}\) confidence interval is narrower than a \({\rm{95\% }}\) confidence interval. Because \({{\rm{z}}_{{\rm{\alpha /2}}}}\) for \({\rm{90\% }}\) confidence level is smaller than \({{\rm{z}}_{{\rm{\alpha /2}}}}\) for \({\rm{95\% }}\) confidence level \({\rm{(1}}{\rm{.645 < 1}}{\rm{.96)}}\), the confidence interval will be shorter.

In general, there is \({\rm{95\% }}\) confidence in the technique when the confidence interval is formed from a sample, whether the mean \({\rm{\mu }}\) belongs to it or not.

The population mean \({\rm{\mu }}\), not the population values, is the subject of the confidence interval.

It isn't required. The sample mean \({\rm{\mu }}\) is predicted to fall within \({\rm{95}}\) and \({\rm{100}}\) percent confidence intervals. In theory, if you repeated the process infinitely instead of \({\rm{100}}\) times, the sample mean \({\rm{\mu }}\) would fall into \({\rm{95\% }}\) of those confidence ranges.