- Interval Estimates and Point Estimates
- Large Sample (Approximate) Intervals
- Small Sample (Exact) Intervals
- Different Confidence Levels

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

point estimatepoint estimate

is an approximate 95% confidence interval for the parameter that the
point estimate estimates.
Generically,

And

- The sample size is large (both sample sizes are large in the two-sample cases).
- In the two-sample cases, the samples are independent.

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

where the ``double wiggle'' sign means ``approximately distributed as.'' An exact (small sample) confidence interval for can be derived from the analogous fact that

where the ``single wiggle'' sign means ``exactly distributed as'' and Student means the Student's -distribution with degrees of freedom.

The difference between the two theories is that

- (1) holds (approximately) regardless of the population distribution for sufficiently large sample size .
- (2) holds (exactly) for a normal population distribution regardless of the sample size .

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

where Student. This is called the critical value for 95% confidence and is different for each sample size .

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

For example, if , then

- These intervals are exact only if the population distribution is exactly normal.
- If the population distribution is close to but not exactly normal, then the these intervals are approximate (their actual coverage probability is near their nominal 95% coverage probability).
- If the population distribution is nowhere near normal, then these intervals are totally bogus.

For confidence levels other than 95% see Section 1.4.4 below.

Different Confidence Levels

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).