Introduction to Statistics and Data Analysis by Christian Heumann Michael Schomaker & Shalabh

Author:Christian Heumann, Michael Schomaker & Shalabh
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Confidence Interval for When is Unknown.

Let be an i.i.d. sample from where is assumed to be unknown and is being estimated by the sample variance . We know from Sect. 8.​3.​1 that

It can be shown that and are stochastically independent. Thus, we know that

follows a t-distribution with degrees of freedom. We can use this result to determine the confidence interval for as

or

The confidence interval for is thus obtained as

(9.16)

which is the confidence interval for or the confidence interval for with confidence coefficient .

The interval (9.16) is, in general, wider than the interval (9.15) for identical and identical sample size n, since the unknown parameter is estimated by which induces additional uncertainty. The quantiles for the t-distribution can be obtained using the R command qt or Table C.2.

Example 9.4.3

Consider Example 9.4.2 where we evaluated the weight of 10-year-old children. We have already calculated the point estimate of as . With = 2.093, obtained via qt(0.975,19) or Table C.2, the upper and lower limits of a 95 % confidence interval for are obtained as

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