Sample proportions confidence intervals

The frequency interpretation of the interval estimates on the unknown amounts is given by the following ( 27 ) With at least 1- a confidence, based on the sampling characteristics of the observations on the standards, at least P proportion of the interval estimates made from a particular calibration will contain the true amounts. The Bonferroni inequality insures the 1-a confidence since the confidence interval about the regression line and the upper bound on cr are each performed using a 1- a/2 confidence coefficient. Hence, the frequency interpretation states that at least (1-a) proportion of the standard calibrations are such that at least P proportion of the intervals produced by the method cover the true unknown amounts. For the remaining a proportion of standard calibrations the proportion of intervals which cover the true unknown values may be less than P. 

Note that for binary data and proportions the multiplying constant is 1.96, the value used previously when we first introduced the confidence interval idea. Again this provides an approximation, but in this case the approximation works well except in the case of very small sample sizes. 

In the example quoted earlier, we found that 42 out of a sample of 50 patients (84 per cent) showed a successful response to treatment, but, what would happen if we were to adopt this treatment and record the outcomes for thousands of patients over the next few years The proportion of successful outcomes would (hopefully) settle down to a figure in the region of 84 per cent, but it would be most surprising if our original sample provided an exact match to the long-term figure. To deal with this, we quote 95 per cent confidence intervals for the proportion in the population based upon a sample proportion. 

Figure 15.2 The 95 percent confidence intervals for the proportion of successful outcomes to treatment with varying sample sizes...

A measure of variability of the estimate can be gained from the standard error, but it can be seen from Equations 12.4 and 12.5 that the magnitude of the standard error is inversely proportional to n that is, the larger the sample size, the smaller will be the standard error. Therefore, without prior knowledge of the sample size, a reported standard error cannot be evaluated. A standard error value of 0.2 indicates a great deal more variability in the estimate if n = 100 than if n = 3. One way around this shortcoming is to report n for every estimate of mean standard error. Another, and better, method is to report confidence intervals of the mean. 

This is also a confidence interval for the parameter p, probability of success, of the binomial distribution. The use of the Z distribution for this interval is made possible because of the Central Limit Theorem. Consider the random variable X taking on values of 0 or 1, such that the sampling distribution of the sample mean (the proportion) is approximately normally distributed. A table of the most commonly encountered values of the standard normal distribution is provided in Table 8.3 for quick reference. Others are provided in Appendix 1. 

The statistical analysis approach is to calculate 95% confidence intervals for the proportion of participants in each group (placebo and combined active) reporting a headache. This analysis approach is reasonable because the sample size is sufficiently large (that is, the values, pn, in each group are at least five). Satisfying this assumption enables us to use the Z distribution for the reliability factor. 

So the reliability factor for interval estimates will come from the Z distribution. Then a two-sided (1 - a)% confidence interval for the difference in sample proportions, p - p is ... 

Fitting a subjective belief form requires that the questions be posed in terms of statistical parameters. That is, decision makers could be asked to first consider their uncertainty regarding the true value of a given probability and then estimate their mean, mode, or median belief. This approach can be further extended by asking decision makers to describe how certeiin they are of their estimate. For example, a worker might subjectively estimate the mean and veuiance of the proportion of defective circuit boards before inspecting a small sample of circuit boards. If the best estimate corresponds to a mean, mode, or median, and the estimate of certainty to a confidence interval or stemdard deviation, a functional form such as the Beta-1 probability density function (pdf) can then be used to fit a subjective probability distribution (Clemen 1996 Buck 1989). 

Application of equation (5.17) to the data in the example above shows that the test samples with absorbance of 0.100 and 0.600 have confidence limits for the calculated concentrations of 1.23 0.12 and 8.01 0.72 pgmH respectively. The widths of these confidence intervals are proportional to the observed absorbances of the two solutions. In addition the confidence interval for the less concentrated... 

It is clear from Equation 2.9 that the length of the confidence interval is linearly proportional to the population standard deviation, and inversely related to the square root of the sample size. If o were known. Equation 2.9 could be used to determine the minimum sample size required to obtain a confidence interval which will contain the unknown mean p, with a (1-cc) probability. An expression for the minimum sample size will, therefore, be... 

The width of the confidence interval is inversely proportional to Vn, which implies that if the sample size increases, the sample mean is a more precise, but not necessarily a more accurate estimate of the population mean, see Fig. 20.4 Confidence intervals (95 % confidence level). 

X is not usually available because most experimental samples are counted only once. As a result of this limitation it has become customary to indicate a proportional error of a given count. At a confidence level of 95% (i.e., there is a 95% chance the observed value will fall within the interval x - l-96o-) the proportional error may be calculated from the relationship... 

A prediction interval is fundamentally different from a confidence statement. It is appropriate when trying to find the bounds for future outcomes, given that some knowledge of past performance exists. A prediction interval is a confidence statement about future individual samples. In both instances, the method for interval estimates assumes that the data are normally distributed. The tolerance interval is used when one is interested in finding the proportion of future samples falling within limits, with a stated confidence level. 

Another approach to setting specifications involves the use of tolerance intervals mentioned in Section II.B. Again, a tolerance interval provides the bounds or confidence limits to contain a stated proportion of future samples. The equation for the tolerance limits looks very similar to that for the prediction interval it is given by Eq. (10). 

A tolerance interval and range for each component calcnlated for the conventional commercial varieties that are grown in the same field trials. The tolerance interval is calcnlated to inclnde a chosen proportion (e.g. 99%) of all samples from the popnlation with a specified degree of confidence 100(1 - a)%, where, e.g. a = 0.05... 

---