Confidence interval meaning

Calculate the mean, the standard deviation, and the 95% confidence interval about the mean. What does this confidence interval mean ... 

Element Mean Std Dev (%) Confidence Interval Mean Std Dev (%) Confidence Interval Mean Std Dev (%) Confidence Interval... 

Parametric methods, i.e. those based on normal distribution theory, are recommended for the analysis of log-transformed bioequivaience measures. The general approach is to construct a 90% confidence interval for the quantity pT-pR and to reach a conclusion of pharmacokinetic equivalence if this confidence interval is within the stated limits. The nature of parametric confidence intervals means that this is equivalent to carrying out two one-sided tests of the hypothesis at the 5% level of significance 10, 77). The antilogs of the confidence limits obtained constitute the 90% confidence interval for the ratio of the geometric means between the multisource and comparator products. 

A 95-percent confidence interval means there is a 5 percent chance the true mean will lie outside the interval. 

Site Field 95% confidence interval Mean Lower Upper ... 

The statistical reliability of a result is expressed by the scatter range T and the confidence interval. The confidence interval means that P% of all individual measurements, whose mean value x is stated as a result, can be expected to be within the range x+T and x—T. Using details about T, one judges the quality... 

Figure 5.5Qq shows the 95% confidence interval which brackets (or sandwiches) the curve. From a statistics point of view, the 95% confidence interval means that out of 100 samples being measured fory, at the given x 95 of them will have a value ofy that lies within the range ofy, (i.e., between lower f(x,) and upper / (x,)) at the given x,. For the new look of the MATLAB Curve Fitting Toolbox (see Fig. 5.25bi. click on the Tools menu, followed by the Prediction Bounds submenu, and pick up, for example, a 95% confidence interval so that the 95% confidence envelope will sandwich the curve, as shown in Fig. 5.30b. 

The F statistic, along with the z, t, and statistics, constitute the group that are thought of as fundamental statistics. Collectively they describe all the relationships that can exist between means and standard deviations. To perform an F test, we must first verify the randomness and independence of the errors. If erf = cr, then s ls2 will be distributed properly as the F statistic. If the calculated F is outside the confidence interval chosen for that statistic, then this is evidence that a F 2. 

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. 

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. 

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... 

Determine the density at least five times, (a) Report the mean, the standard deviation, and the 95% confidence interval for your results, (b) Eind the accepted value for the density of your metal, and determine the absolute and relative error for your experimentally determined density, (c) Use the propagation of uncertainty to determine the uncertainty for your chosen method. Are the results of this calculation consistent with your experimental results ff not, suggest some possible reasons for this disagreement. 

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

We begin by determining the confidence interval for the response at the center of the factorial design. The mean response is 0.335, with a standard deviation of 0.0094. The 90% confidence interval, therefore, is... 

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). 

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 ... 

First, the parameter estimate may be representative of the mean operation for that time period or it may be representative of an extreme, depending upon the set of measurements upon which it is based. This arises because of the normal fluc tuations in unit measurements. Second, the statistical uncertainty, typically unknown, in the parameter estimate casts a confidence interv around the parameter estimate. Apparently, large differences in mean parameter values for two different periods may be statistically insignificant. 

Fig. 25. Relationship between the measured interfacial strength and the (negative) Gibbs free energy of mixing, (-AG )o5, for glass beads treated with various silane coupling agents embedded in a PVB matrix. Error bars correspond to 95% mean confidence intervals. Redrawn from ref. [165].

Confidence interval That portion of a distribution which is expected to contain the mean value a certain percentage of time. 

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... 

When a small number of observations is made, the value of the standard deviation s, does not by itself give a measure of how close the sample mean x might be to the true mean. It is, however, possible to calculate a confidence interval to estimate the range within which the true mean may be found. The limits of this confidence interval, known as the confidence limits, are given by the expression ... 

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... 

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