Confidence interval for mean

Figure 5.1 How wide a confidence interval for mean imipramine content in tablets ...

Figure 5.5 The 95 per cent confidence intervals for mean imipramine content. Population mean = 25 mg. Sample size = 9. The SD varies between 0.2 and 2 mg...

Figure 5.12 The 95 per cent confidence interval for mean pesticide content calculated (a) directly or (b) via Log transformation...

The calculated values from any sample are considered as point estimates. Any such estimate may be close to the true value v>f the population (/c, a or other) or it may vary substantially from the true value. An indication of the interval around this point estimate, within which the true value is expected to fall with some stated probability, is called a coiifich iHc interval, and the lower and upper boundary values are called the confidence limits. The probability used to set the interval is called the level of eonfidenee. This level is given by (1 - ), where a is the probability as discussed above for rejecting a null hypothesis when it is true. In most circumstances, means are the most important point estimates, and confidence intervals for means are evaluated at some probability / — (I - a) that the true population mean is within the stated confidence limits. This can be expressed for a population with a known standard deviation a as given in Eq. 21. 

Figure 5. Course of the broken function with 95% confidence interval for mean value.

Using the preceding information, we construct a 95% confidence interval for Mean as follows ... 

Listing 8.14. Monte Carlo analysis of confidence intervals for mean and standard deviation. 

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 ... 

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. 

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. 

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 ... 

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... 

It is apparent that the confidence interval for the mean rapidly converges toward very small values for increasing n, because both (/) and 1/Vn become smaller. 

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 ...

In all the above cases we presented confidence intervals for the mean expected response rather than a future observation (future measurement) of the response variable, y0. In this case, besides the uncertainty in the estimated parameters, we must include the uncertainty due to the measurement error (so). 

Having determined the uncertainty in the parameter estimates, we can proceed and obtain confidence intervals for the expected mean response. Let us first consider models described by a set of nonlinear algebraic equations, y=f(x,k). The 100(1 -a)% confidence interval of the expected mean response of the variable y at x0 is given by... 

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

For the data set given in Table 6.2, calculate the mean, sample standard deviation, relative standard deviation, degrees of freedom and the 95% confidence interval for the mean. 

Fig. 4 Left the mean 1961-1990 monthly temperature for the Ebro catchment. Part (a) shows the annual cycle, each line representing a different RCM simulation and the bold line representing the CRU observed series. The shading represents the 95% confidence interval for the estimate of the observed 30-year sample mean. Part (b) represents the individual monthly model means as an anomaly from the CRU mean with 95% confidence interval superimposed. Part (c) represents the mean absolute annual error for each of the RCMs. Right-, as for left column but for mean precipitation (d) for the Gallego catchment. Model anomalies in parts (e) and (f) are expressed as a percentage relative to the CRU monthly mean. Model numbers correspond to experiments shown in Table 1. Figure from [35]...

All in vivo data, including the human and rat absorption data used by both Egan and Zhao et al., have considerable variability. Zhao et al. comment that measurements of percent absorbed for the same molecule may vary by 30%, and that the 95% confidence interval for a prediction is approximately 30% given a model RMSE of 15%. This is approximately the same as the normal experimental error for absorption values. This means that models predicting percent absorbed have to be carefully interpreted, i.e., a prediction of 30% absorbed really means the molecule is predicted to have absorption from 15 to 45%, and that classification models should work nearly as well as regression models. 

As can be seen from Table 1, the estimated coefficients b[0] are not equal to zero for different samples, whereas the estimated coefficients b[l] are close to 1 within confidence interval. That means that coefficients b[0] estimated for different points of the territory are generalized relative characteristics of elements abundance at the chosen sampling points. Statistical analysis has confirmed that hypotheses Hi and H2 are true with 95% confidence level for the data obtained by any of the analytical groups involved. This conclusion allowed us to verify hypothesis H3 considering that the estimated average variances of the correlation equation (1) are homogeneous for all snow samples in each analytical group. Hypothesis H3... 

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