Population confidence interval

A-basis Also called A-allowable. It is the value above which at least 99% of the population of values is expected to fall with a confidence of 95%. See B-basis population confidence interval S-basis typical basis. 

Population confidence interval The limits on either side of a mean value of a group of observations which will, in a stated fraction or percent of the cases, include the... 

Typical-basis The typical property value is an average value. No statistical assurance is associated with this basis. See A-basis B-basis C-basis population confidence interval. 

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. 

What is the 95% confidence interval for the amount of aspirin in a single analgesic tablet drawn from a population where p, is 250 mg and is 25 ... 

According to Table 4.11, the 95% confidence interval for a single member of a normally distributed population is... 

Alternatively, a confidence interval can be expressed in terms of the population s standard deviation and the value of a single member drawn from the population. Thus, equation 4.9 can be rewritten as a confidence interval for the population mean... 

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... 

Thus, there is a 95% probability that the population s mean is between 239 and 251 mg of aspirin. As expected, the confidence interval based on the mean of five members of the population is smaller than that based on a single member. 

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. 

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. 

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... 

Confidence Interval for the Difference in Two Population Means The confidence intei val for a mean can be extended to include the difference between two population means. This intei val is based on the assumption that the respective populations have the same variance <7 ... 

Confidence Interval for a Variance The chi-square distribution can be used to derive a confidence interval for a population variance <7 when the parent population is normally distributed. For a 100(1 — Ot) percent confidence intei val... 

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... 

Figure 1.18. The Student s ( resp. t/Vn for various confidence levels are plotted the curves for p = 0.05 are enhanced. The other curves are for p = 0.5 (bottom), 0.2, 0.1, 0.02, 0.01, 0.002, 0.001, and 0.0001 (top). By plotting a horizontal, the number of measurements necessary to obtain the same confidence intervals for different confidence levels can be estimated. While it takes n - 9 measurements (/ = 8) for a t-value of 7.12 and p = 0.0001, just n = 3 f - 2) will give the same limits on the population for p = 0.02 (line A - C). For the CL on the mean, in order to obtain the same t/ /n for p = 0.02 as for p = 0.0001, it will take n = 4 measurements (line B ) note the difference between points D and ...

Figure 2.9. The confidence interval for an individual result CI( 3 ) and that of the regression line s CLj A are compared (schematic, left). The information can be combined as per Eq. (2.25), which yields curves B (and S, not shown). In the right panel curves A and B are depicted relative to the linear regression line. If e > 0 or d > 0, the probability of the point belonging to the population of the calibration measurements is smaller than alpha cf. Section 1.5.5. The distance e is the difference between a measurement y (error bars indicate 95% CL) and the appropriate tolerance limit B this is easy to calculate because the error is calculated using the calibration data set. The distance d is used for the same purpose, but the calculation is more difficult because both a CL(regression line) A and an estimate for the CL( y) have to be provided.

The target number of commodity samples to be obtained in the OPMBS was 500, as determined using statistical techniques. A sample size of 500 provided at least 95% confidence that the 99th percentile of the population of residues was less than the maximum residue value observed in the survey. In other words, a sample size of 500 was necessary to estimate the upper limit of the 95% confidence interval around the 99th percentile of the population of residues. 

Confidence Interval for the Difference in Two Population Means The confidence interval for a mean can be extended to... 

The 206Pb/204Pb ratios of four samples from a Polynesian island have been determined to be 18.999, 19.091, 19.216, and 19.222. Assuming that these measurements represent a sample from a normal population, find a 95 percent confidence interval for the mean and the standard deviation of the population. 

A 95% confidence interval is 1.96SD and is the most frequently quoted. There is a 95% certainty that this range of values around the mean will contain the population mean. 

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