Confidence intervals single mean

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 ... 

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. 

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. 

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. 

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... 

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 ... 

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. 

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 -... 

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 ... 

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. 

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.]... 

This interval estimate is really based on the two-sided test of the third set of hypotheses previously given. Although it is possible to define one-sided confidence intervals based on the other two sets of hypotheses (1.59) and (1.60), such one-sided intervals are rarely used. By one-sided, we mean an interval estimate that extends from plus or minus infinity to a single random confidence limit. The one-sided confidence interval may be understood as the range one limit of which is the probability level a and the other one °°. 

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 ... 

We have already established that a mean derived from a sample is unlikely to be a perfect estimate of the population mean. Since it is not possible to produce a single reliable value, a commonly used way forward is to quote a range within which we are reasonably confident the true population mean lies. Such a range is referred to as a confidence interval . 

Fig. 13. Plasma concentration vs. time profiles of XYZ1234 following a single oral 100 mg dose given to poor CYP 2C19 metabolizing male and female subjects (straight lines), to extensive CYP 2C19 metabolizing females (dashed lines), and to extensive CYP 2C19 metabolizing males (dotted lines) respectively Mean values (bold) and 90% Confidence Interval...

It is because of these unavoidable, random, and indeterminate errors that replicates are run. The mean of a series of replicates should be more accurate than any single value because the indeterminate errors will tend to cancel (average themselves out) in the runs. It is statistically seldom worth running more than four replicates because the number of trials in the standard deviation, cr, and confidence interval calculations is a square-root term in the denominator. Figure 6-13 illustrates the term accuracy and precision. 

Applying the first method, four different criteria, namely Dixon s test, Grubbs test, the coefficient of dewness test and the coefficient of kurtosis test are used at a significance level of a = 0.05. If a laboratory mean for each element as single unweighted value was declared to be an outlier by any criterion, it is rejected and the whole procedure repeated until no more outliers could be identified. The remaining laboratory means are then combined in the usual way to provide estimates of the overall mean (consensus value) and its associated standard deviation, standard error and 95% confidence interval. 

Eighteen experimental measurements of product yields from the acid catalyzed dehydration of 1-propanol and 2-propanol in water (at 375 C and 34.5 MPa) are displayed in Table I. These data span a wide range of reactant (0.05 to 2.0 M) and catalyst (0.001 to 0.025 M) concentrations, and residence times (1.15 to 68.4 s). The error bars displayed in Table I reflect 95% confidence intervals based on yield variations detected during a single run. A typical sample standard deviation is almost 2% of the mean yield value for these data. Using these measurements, optimal values of the kinetic parameters associated with each trial mechanism were calculated. 

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,... 

Often, limitations in time or in the amount of available sample prevent us from making enough measurements to assume i is a good estimate of cr. In this case, a single set of replicate measurements must provide not only a mean but also an estimate of precision. As indicated earlier, calculated from a small set of data may be quite uncertain. Thus, confidence intervals are necessarily broader when we must use a small sample value of as our estimate of cr. 

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