Population mean confidence interval

Means The confidence interval for a mean can be extended to include the difference between two population means. This interval is based on the assumption that the respective populations have the same variance... 

Often we have data from several populations that we believe follow the same parametric distribution (such as the normal distribution), but may have different values of the parameter (such as the mean). The classical frequentist approach would be to analyze each population separately. The maximum likelihood estimate of the parameter for each population would be estimated from the sample from that population. Simultaneous confidence intervals such as Bonferroni, Tiikey, or Scheff6 intervals would be used for the difference between different population parameter values. These wider intervals would control the overall confidence level, and the overall significance level for testing the hypothesis that the differences between all the population parameters are zero. However, these intervals don t do anything about the parameter estimates themselves. 

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... 

The population standard deviation for the amount of aspirin in a batch of analgesic tablets is known to be 7 mg of aspirin. A single tablet is randomly selected, analyzed, and found to contain 245 mg of aspirin. What is the 95% 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. 

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. 

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... 

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... 

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 ...

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. 

In some textbooks, a confidence interval is described as the interval within which there is a certain probability of finding the true value of the estimated quantity. Does the term true used in this sense indicate the statistical population value (e.g., p if one is estimating a mean) or the bias-free value (e.g., 6.21% iron in a mineral) Could these two interpretations of true value be a source of misunderstanding in conversations between a statistician and a geologist ... 

The difference between these two concepts, the population standard deviation and the standard error of the mean, is important and we will return to it when considering confidence intervals. 

When estimating a population mean, the 95% confidence interval is approximately given by... 

One question that is often asked of statisticians is in what sense can we be 95% confident that the population mean lies within the limits 3.84 and 4.13 To answer the question we can again conduct a sampling experiment as follows. Suppose that the 40 blood glucose measurements in Figure 8.3 comprised the total population of values. For random sample of size 10 from the populations of blood glucose values determine the sample mean, standard error and the corresponding 95% confidence interval. Repeat the process 100 times. The results of such an experiment are shown in Figure 8.6. 

In this figure, each individual confidence interval has been drawn as a vertical straight line joining the lower and upper limits. The horizontal line is positioned at the value 4.055 mmol/L- the population mean. This gives us 40 sample means that are not equal to one another, so they on their own like the original measurement show random... 

Exploration of the scope of NPS in electrochemical science and engineering has so far been rather limited. The estimation of confidence intervals of population mean and median, permutation-based approaches and elementary explorations of trends and association involving metal deposition, corrosion inhibition, transition time in electrolytic metal deposition processes, current efficiency, etc.[8] provides a general framework for basic applications. Two-by-two contingency tables [9], and the analysis of variance via the NPS approach [10] illustrate two specific areas of potential interest to electrochemical process analysts. 

Confidence interval The numerical interval constructed around a point estimate of a population parameter. It is combined with a probability statement linking it to the populations true parameter value, for example, a 90% confidence interval. If the same confidence interval construction technique and assumptions are used to calculate future intervals, they will include the unknown population parameter with the same specified probability. For example a 90% confidence interval around an arithmetic mean implies that 90% of the intervals calculated from repeated sampling of a population will include the unknown (true) arithmetic mean. 

Mostly the 0.95 confidence interval of mean is given, i.e., you can be 95% sure that your randomly selected sample of a population is included in the population mean. 

We have seen in the previous chapter that it is not possible to make a precise statement about the exact value of a population parameter, based on sample data, and that this is a consequence of the inherent sampling variation in the sampling process. The confidence interval provides us with a compromise rather than trying to pin down precisely the value of the mean p or the difference between two means — p2> for example, we give a range of values, within which we are fairly certain that the true value lies. 

Now look at all 100 samples taken from the normal population with p = 80 mmHg. Figure 3.1 shows the 95 per cent confidence intervals plotted for each of the 100 simulations. A horizontal line has also been placed at 80 mmHg to allow the confidence intervals to be judged in terms of capturing the true mean. 

Most of the 95 per cent confidence intervals do contain the true mean of 80 mmHg, but not all. Sample number 4 gave a mean value 3c = 81.58 mmHg with a 95 per cent confidence interval (80.33, 82.83), which has missed the true mean at the lower end. Similarly samples 35, 46, 66, 98 and 99 have given confidence intervals that do not contain p = 80 mmHg. So we have a method that seems to work most of the time, but not all of the time. For this simulation we have a 94 per cent (94/100) success rate. If we were to extend the simulation and take many thousands of samples from this population, constructing 95 per cent confidence intervals each time, we would in fact see a success rate of 95 per cent exactly 95 per cent of those intervals would contain the true (population) mean value. This provides us with the interpretation of a 95 per cent confidence interval in... 

The interpretation of this interval is essentially as before we can be 95 per cent confident that the true difference in the (population) means, — p.2> is between 0.8 and 2.0. In other words the data are telling us that the mean reduction p. in... 

Figure 5. Mean EAG responses of three male and three female Colorado beetles from different populations to tta.ns-2-hexen-l-ol (t-2-H-l-ol), cis-3-hexen-l-ol (c-3-H-l-ol), ds-3-hexenyl acetate (c-3-H-ace), and tra.ns-2-hexeruil (t-2-H-al) at two dilutions in paraffin oil, 10 and 10 (v/v). Key A, laboratory stock culture B, field population Wageningen C, field population Utah vertical lines indicate 95% confidence intervals and, significant a/ P < O.OI (Mann-lVhitney V test).

A particular use of the t-statistic is calculating confidence intervals (Cl). When we calculate the mean of a sample we do not expect that it will be exactly equal to the mean of the population from which the sample was drawn. Nonetheless, we can expect that it will be reasonably close to the population mean. A confidence interval provides an estimate as to how close. The 95% confidence interval is a random interval such that, in 95% of hypothetical replications of the sampling process, the confidence intervals obtained will include the true value of p The confidence interval for p is of the form x multiples of s.e.m. The multiple used is tl-a/2 (n-1), which is the 100(l-o/2) percentage point of the t-distribution with n-1 degrees of freedom. Thus, the 95% Cl (o=0.05) is given by ... 

However, the GUM [Guide to the Expression of Uncertainty of Measurement approach (ISO 1993a), which leads to the verbose statement concerning expanded uncertainty quoted above, might not have been followed, and all the analyst wants to to do is say something about the standard deviation of replicates. The best that can be done is to say what fraction of the confidence intervals of repeated experiments will contain the population mean. The confidence interval in terms of the population parameters is calculated as... 

When n becomes large the t value tends toward the standardized normal value of 1.96 (z = 1.96), which was approximated to 2 above. The 95% confidence interval, calculated by equation 2.13, is sometimes explained much like the expanded uncertainty, as a range in which the true value lies with 95% confidence. In fact, the situation is more complicated. The correct statistical statement is if the experiment of n measurements were repeated under identical conditions a large number of times, 95% of the 95% confidence intervals would contain the population mean. ... 

Having repeated a measurement, how can the result be compared with a given value We have already encountered this scenario with tests against a regulatory limit, and the problem has been solved by looking at the result and its 95% confidence interval in relation to the limit (see figure 2.5). This procedure can be turned into a test by some simple algebra on the equation for the 95% confidence interval (equation 2.13). If the population mean, p is to be found in the 95% confidence interval, in 95% of repeated measurements. 

From a limited number of measurements, we cannot find the true population mean, p, or the hue standard deviation, a. What we can determine are x and. v, the sample mean and the sample standard deviation. The confidence interval is an expression stating that the true mean, p, is likely to lie within a certain distance from the measured mean, x. The confidence interval of p is given by... 

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