Confidence intervals for a single mean

We will first look at the way we calculate the confidence interval for a single mean p and then talk about its interpretation. Later in this chapter we will extend the methodology to deal with pj — p2 and other parameters of interest. 

D. Estimation (e.g., point, confidence intervals) for a single mean... 

The density of a liquid is measured by filling a 50 ml flask as close as possible to the index mark and weighing. In successive trials the weight of the liquid is found to be 45.736 g, 45.740 g, 45.705 g, and 45.720 g. For these weights calculate the average deviation, the standard deviation, the 95% confidence interval for a single value, and the 95% confidence interval for the mean. 

A radioactive sample shows the following counts for one-minute intervals 2642 2650 2649 2641 2641 2637 2651 2636. Find the average deviation, the standard deviation, and the 90% confidence interval for a single value and for the mean. 

By analogy with the confidence range of a single value or a mean, we can calculate a confidence interval for a single predicted value according to ... 

Table VI gives data obtained for RD 1333 lead azide. A statistical analysis using the 11 data points in the table gives a mean of AT of 15°C with a standard deviation of 2°. The 95% confidence interval is 1.3°. The importance of making more than one measurement is made clear because the 95% confidence interval for a single measurement using the same mean and standard deviation is 4°. If we compare data for lead azides (Table VII) for heating rates of 5° and 10°/min, dextrinated material is significantly more sensitive than the other products tested. The explosion temperature is also significantly lower. A higher heating...

Note that the standard error of prediction is not a constant for all values of xo. but reflects where xo is collected in relation to the mean. Observations removed from the mean of x will have larger standard errors of prediction than values close to the mean. Equation (2.34) is developed as the confidence interval for a single observation measured at x0. If more than one observation is made at xo, the term 1/n in Eqs. (2.33) and (2.34) is substituted with the term m/n, where m is the number of observations at x0. Note that m is contained within n. If the confidence interval is made for all points on the regression line, the result would be a confidence band. 

This SRM is intended for the verification of fracture toughness testing procedures and consists of a set of 5 hot pressed silicon nitride flexure specimens. .. The certified fracture toughness is 4.572 MPa y/m and the uncertainty at the 95% confidence interval for a single specimen is 0.228 MPa.y/m, and for the mean of five specimens, is 0.106MPay m. ... 

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 ... 

Additional measurements were made with the 17-)im sizing screen to obtain more information on the variability of our measurement techniques. Eight lint samples from a single source of cotton were analyzed by the procedures outlined previously. The dust levels obtained in this test were 11.7, 12.1, 13.5, 11.8, 10.8, 11.2, 10.9, and 9.7 mg, respectively, per 20 g of lint. The mean and standard deviation of these measurements were 11.5 and 1.1, respectively. The estimated standard error of the mean was 0.42, and the interval from 10.5 to 12.5 represented a 95% confidence interval for the lot mean. 

Calculate the 99% confidence interval for predicting a single new value of response at pH = 7.0 for the data of Equation 11.16 and the second-order model of Equation 11.39. Calculate the 99% confidence interval for predicting the mean of seven new values of response for these conditions. Calculate the 99% confidence interval for predicting the true mean for these conditions. What confidence interval would be used if it were necessary to predict the true mean at several points in factor space ... 

At the end of the previous chapter we saw how to extend the idea of a standard error for a single mean to a standard error for the difference between two means. The extension of the confidence interval is similarly straightforward. Consider the placebo controlled trial in cholesterol lowering described in Example 2.3 in Chapter 2. We had an observed difference in the sample means 3cj — 3c2 of 1.4 mmol/1 and a standard error of 0.29. The formula for the 95 per cent confidence interval for the difference between two means — p.2) i -... 

Note for any stated confidence level, the confidence interval about the mean is the narrowest interval, the prediction interval for a single future observation is wider, and the tolerance interval (to contain 95% of the population) is the widest.]... 

Meeting the foregoing criterion should not be interpreted to mean that an individual composite potency assay will meet the in-house limits with high assurance. If this is desired, a prediction interval for a single future observation, or better yet, a tolerance interval, should be used. The validation specialist should be cautioned that additional composite assays might need to be tested to meet either one of these criteria with high confidence. 

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. 

The confidence intervals defined for a single random variable become confidence regions for jointly distributed random variables. In the case of a multivariate normal distribution, the equation of the surface limiting the confidence region of the mean vector will now be shown to be an n-dimensional ellipsoid. Let us assume that X is a vector of n normally distributed variables with mean n-column vector p and covariance matrix Ex. A sample of m observations has a mean vector x and an n x n covariance matrix S. 

A related confidence interval is used for estimating a single mean of several new values of response at a given point in factor space. It can be shown that the estimated variance of predicting the mean of m new values of response at a given point in factor space, s y,o, is... 

In general, the calculation of the confidence interval for any statistic, be it a single mean, the difference between two means, a median, a proportion, the difference between two proportions and so on, always has the same structure ... 

In general, oral dose forms are usually considered to be bioequivalent when confidence intervals for the ratios of their geometric mean C and area-under-the-curve (AUC from time zero to infinity for single doses or within a dosing interval at steady state) are within the range 0.8-1.25 and any difference between their Tma s is within clinically acceptable limits. This range may be tightened for medicines that have ... 

If we make a single measurement x from a distribution of known a, we can say that the true mean should lie in the interval x zcr with a probability dependent on z. This probability is 90% forz = 1.64, 95% fore = 1.96, and 99% forz = 2.58, as shown in Figure 7-lc, d, and e. We find a general expression for the confidence interval of the true mean based on measuring a single value x by rearranging Equation 6-2. (Remember that z can take positive or negative values.) Thus,... 

Describe in your own words why the confidence interval for the mean of five measurements is smaller than that for a single result. 

An atomic absorption method for determination of copper content in fuels yielded a pooled standard deviation of Spooled = 0.32 p-g Cu/mL (s a). The analysis of oil from a reciprocating aircraft engine showed a copper content of 8.53 p.g Cu/mL. Calculate the 90% and 99% confidence intervals for the result if it was based on (a) a single analysis, (b) the mean of four analyses, (c) the mean of 16 analyses. 

The assessment of bioequivalence is based on 90% confidence intervals for the ratio of the population geometric means (test/reference) for the parameters under consideration. This method is equivalent to two one-sided tests with the null hypothesis of bio-inequivalence at the 5% significance level. Two products are declared bioequivalent if upper and lower limits of the confidence interval of the mean (median) of log-transformed AUC and Cmax each fall within the a priori bioequivalence intervals 0.80-1.25. It is then assumed that both rate (represented by Cmax) and extent (represented by AUC) of absorption are essentially similar. Cmax is less robust than AUC, as it is a single-point estimate. Moreover, Cmax is determined by the elimination as well as the absorption rate (Table 2.1). Because the variability (inter- and intra-animal) of Cmax is commonly greater than that of AUC, some authorities have allowed wider confidence intervals (e.g., 0.70-1.43) for log-transformed Cmax, provided this is specified and justified in the study protocol. 

The prediction of a single mean from several y values at a given factor combination is feasible with modification of Eq. (6.25). For m new y values, we obtain for prediction of the mean and its confidence interval... 

Equation [8] is more appropriate than [7], as we usually have a single experimental mean x and wish to use it to provide a range for n, the true value. The range fi = x+1.96a/ /n is known as the confidence interval for /c Similarly, 99.7% confidence interval and limits are obtained from n = x+ial /n. 

Table 7-5 shows results for two of seven compounds sent to many labs to compare their performance in combustion analysis. For each compound, the first row gives the theoretical wt% for each element and the second row shows the measured wt%. Accuracy is excellent Mean wt% C, H, N, and S are usually within 0.1 wt% of theoretical values. The 95% confidence intervals for uncertainty for C for the first compound is 0.63 wt% and the uncertainty for the second compound is 0.33 wt%. The mean uncertainty for C listed in the bottom row of the table for all seven compounds in the study was 0.47 wt%. Mean 95% confidence intervals for H, N, and S are 0.24, 0.31, and 0.76 wt%, respectively. Chemists consider a result within 0.3 wt% of theoretical to be good evidence that the compound has the expected formula. This criterion can be difficult to meet for C and S with a single analysis because the 95% confidence intervals are larger than 0.3. 

Results from the study showed that the upper bound of the 90% confidence interval (Cl) of the mean predicted placebo-adjusted QTc change from baseline at geometric Cmax with aU five QT-positive drugs exceeded 10 msec and that the slope of the exposure-response model was positive for all of these five drugs. In contrast, the upper bound for levocetirizine was less than 10 msec even when a single dose comprising six times the therapeutic dose was administered. 

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