Sampling confidence interval

Hyslop T, Hsuan F, Holder DJ. A small sample confidence interval approach to assess individual bioequivalence. Stat Med 2000 19 2885-2897. 

Detailed testing plan test length, number of samples, confidence intervals, acceleration factors, test environment... 

There will be incidences when the foregoing assumptions for a two-tailed test will not be true. Perhaps some physical situation prevents p from ever being less than the hypothesized value it can only be equal or greater. No results would ever fall below the low end of the confidence interval only the upper end of the distribution is operative. Now random samples will exceed the upper bound only 2.5% of the time, not the 5% specified in two-tail testing. Thus, where the possible values are restricted, what was supposed to be a hypothesis test at the 95% confidence level is actually being performed at a 97.5% confidence level. Stated in another way, 95% of the population data lie within the interval below p + 1.65cr and 5% lie above. Of course, the opposite situation might also occur and only the lower end of the distribution is operative. 

Confidence intervals also can be reported using the mean for a sample of size n, drawn from a population of known O. The standard deviation for the mean value. Ox, which also is known as the standard error of the mean, is... 

In Section 4D.2 we introduced two probability distributions commonly encountered when studying populations. The construction of confidence intervals for a normally distributed population was the subject of Section 4D.3. We have yet to address, however, how we can identify the probability distribution for a given population. In Examples 4.11-4.14 we assumed that the amount of aspirin in analgesic tablets is normally distributed. We are justified in asking how this can be determined without analyzing every member of the population. When we cannot study the whole population, or when we cannot predict the mathematical form of a population s probability distribution, we must deduce the distribution from a limited sampling of its members. 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

There is a temptation when analyzing data to plug numbers into an equation, carry out the calculation, and report the result. This is never a good idea, and you should develop the habit of constantly reviewing and evaluating your data. For example, if analyzing five samples gives an analyte s mean concentration as 0.67 ppm with a standard deviation of 0.64 ppm, then the 95% confidence interval is... 

In the previous section we noted that the result of an analysis is best expressed as a confidence interval. For example, a 95% confidence interval for the mean of five results gives the range in which we expect to find the mean for 95% of all samples of equal size, drawn from the same population. Alternatively, and in the absence of determinate errors, the 95% confidence interval indicates the range of values in which we expect to find the population s true mean. 

The probabilistic nature of a confidence interval provides an opportunity to ask and answer questions comparing a sample s mean or variance to either the accepted values for its population or similar values obtained for other samples. For example, confidence intervals can be used to answer questions such as Does a newly developed method for the analysis of cholesterol in blood give results that are significantly different from those obtained when using a standard method or Is there a significant variation in the chemical composition of rainwater collected at different sites downwind from a coalburning utility plant In this section we introduce a general approach to the statistical analysis of data. Specific statistical methods of analysis are covered in Section 4F. 

Relationship between confidence intervals and results of a significance test, (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on fexp. The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a,v). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. 

Unpaired Data Consider two samples, A and B, for which mean values, Xa and Ab, and standard deviations, sa and sb, have been measured. Confidence intervals for Pa and Pb can be written for both samples... 

Three replicate determinations are made of the signal for a sample containing an unknown concentration of analyte, yielding values of 29.32, 29.16, and 29.51. Using the regression line from Examples 5.10 and 5.11, determine the analyte s concentration, Ca, and its 95% confidence interval. 

Construct an appropriate standard additions calibration curve, and use a linear regression analysis to determine the concentration of analyte in the original sample and its 95% confidence interval. 

In the previous section we considered the amount of sample needed to minimize the sampling variance. Another important consideration is the number of samples required to achieve a desired maximum sampling error. If samples drawn from the target population are normally distributed, then the following equation describes the confidence interval for the sampling error... 

Determine the %CO for each sample, and report the mean value and the 95% confidence interval. 

Confidence Interval for a Mean For the daily sample tensile-strength data with 4 df it is known that P[—2.132 samples exactly 16 do fall witmn the specified hmits (note that the binomial with n = 20 and p =. 90 would describe the likelihood of exactly none through 20 falling within the prescribed hmits—the sample of 20 is only a sample). 

Example For the composite sample of 100 tensile strengths, what is the 90 percent confidence interval for i ... 

Example Compute the 95 percent confidence interval based on the original 100-point sample and the subsequent 5-point sample ... 

Example For the first week of tensile-strength samples compute the 90 percent confidence interval for <3 (df = 24, corresponding to n = 25, using 5 percent and 95 percent in Table. 3-6) ... 

Increa.se the number of mea.surements included in the mea.sure-ment. set by using mea.surements from repeated. sampling. Including repeated measurements at the same operating conditions reduces the impact of the measurement error on the parameter estimates. The result is a tighter confidence interval on the estimates. 

The confidence interval for a given sample mean indicates the range of values within which the true population value can be expected to be found and the probability that this will occur. For example, the 95% confidence limits for a given mean are given by... 

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