In Chapter 7 Wild and Seber call these ``two standard error intervals,'' but in Chapter 8 we find out they are really called confidence intervals.
The thingies discussed in this section are called interval estimates. For contrast, the estimates previously discussed, like and are called point estimates.
For any point estimate having an approximately normal sampling distribution
Generically,
And
Plugging in the formulas for the standard errors,
For confidence levels other than 95% see Section 1.4.4 below.
Nothing in the preceeding section is useful for small samples. For proportions there is no small sample theory. But for means there is. In Chapter 7 we only do the one-sample case (the two-sample case will come later).
The confidence interval for in the preceeding section, was derived from the fact that
The difference between the two theories is that
The relation between (1) and the confidence interval is that the two equations
Hence in order to get exact (not approximate) confidence intervals assuming a normal population distribution we only need to substitute for 2 the such that
The critical values for 95% confidence are given in the column headed 0.025 of Appendix 6 in Wild and Seber or by either of the R commands
qnorm(0.975, n - 1) - qnorm(0.025, n - 1)where n is the sample size (so n - 1 is the degrees of freedom).
For example, if , then
For confidence levels other than 95% see Section 1.4.4 below.
For confidence levels other than 95%, just change the 0.95 in (3) to some other number.
To get
confidencethe critical value is
the quantile of the Student distributionor
minus the quantile of the Student distribution.Thus
confidence level | column of Appendix 6 headed |
90% | 0.05 |
95% | 0.025 |
99% | 0.005 |
The same trick works for large-sample intervals based on the approximate normality of the sampling distribution of a point estimate. Just use the Normal distribution instead of the Student distribution. This the Student's -distribution with ``infinity degrees of freedom'' in the bottom row of Appendix 6 in Wild and Seber. Hence
confidence level | critical value |
90% | 1.645 |
95% | 1.960 |
99% | 2.576 |
Note also that a finicky person also uses 1.96 s. e. intervals rather than 2 s. e. intervals for 95% confidence (not that it really matters, it's only approximate anyway).