95% confidence interval

The basis for confidence intervals is that the error is normally distributed with the variance i.e. e iV(0, 7ct ). This also results in b being normally distributed and 

The variance, s, can be estimated by dividing the sum of square errors, SSE, by the degrees of freedom, v 

The number of degrees of freedom is the number of observations of the random variation. When there is only one source of errors, e.g. the chemical analysis, the degrees of freedom are 

180-188, 2006. Reproduced with permission from Elsevier. 

Thus there are no observations of die random error and consequently no degrees of freedom. Similar results can be obtained if a quadratic model (j = ho + b x + b2X ) were used to fit three experimental points. For this case, the number of degrees of freedom would be zero as weU (v = n — p = 3 — 3 = 0). 

F = F-value from the F distribution Tables (F-ratio Tables) at a required confidence level and at DOF 1 and error DOF 8 (Roy (1990)) 

Ne = Effective number of replications = (Total number of results (or number of S/N-ratios) / DOF of mean (=/, always) + DOF of all factors included in the estimate of the mean)or the total number of units in one level. 

F-value is sometimes refereed to as F-ratio, used to test the significance of factor effects. It is statistically analogue to Taguchi s signal-to-noise ratio for control factor effect vs. the experimental error. The F-ratio uses information based on sample variances (mean squares) to define the relationship between the power of the control factor effects (a type of signal) and the power of the experimental error (a type of noise) (Fowlkes and Creveling (1995)). 

If a random variable X is defined over a continuous domain Q in 91, the unknown mean p of a sample lies in a known two-sided confidence interval o = [x , x6] at 100(1 — a) percent, or, equivalently, is known at the a significance level, if 

The two-sided confidence interval is limited by the 100a/2 and 100(1 — a/2) percentiles. A commonly used confidence interval is 95 percent (a=0.05), although for the search of outliers (rogue values produced during data acquisition), larger confidence intervals are occasionally preferred. 

An application of the confidence interval concept central to most statistical assessment is the t2-test for small normal samples. Let us consider a normally distributed variable X with mean p and variance a2. It will be demonstrated below that for m observations with sample mean x and variance s2, the variable U defined as 

Multiplying the inequalities in parentheses by —1 changes the smaller than into greater than  

Since the f-distribution is even, t100a/2 m l = —t100(1-a/2),m i, and, finally 

Student s t is a statistical tool used most frequently to express confidence intervals and to compare results from different experiments. It is the tool you could use to evaluate the probability that your red blood cell count will be found in a certain range on normal days. 

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 

NOTE In calculating confidence intervals, o may be substituted for s in Equation 4-6 if you have a great deal of experience with a particular method and have therefore determined its true population standard deviation. If a is used instead of s, the value of t to use in Equation 4-6 comes from the bottom row of Table 4-2. 

50% chance that true value lies in this interval 

The standard deviation of the mean is a point estimate of//. However, a point estimate does not indicate the confidence that can be placed in such an estimate. When an objective measure of reliability is required, we report a range of values rather than a single value. These interval estimates are called confidence intervals (Cl) or confidence limits. 

The confidence interval of a mean, X, is calculated by using the following equation  

The confidence limit gives the range that the mean should be in a certain percentage of the time. 

the confidence interval at the 90% level is calculated by inserting the Z value 1.64 from Table 14.1 into the equation. 

The use of these statistical contents in ion chromatography is perhaps best illustrated by an example. 

The distribution parameters are estimated according to Eqs. (9.34) and (9.37) on the basis of samples. These are considered to be representative of the underlying population, for example all valves of a certain type which operate under certain well-defined conditions. If the entire population were known and the period of observation infinitely long, the exact values of the distribution parameters could be determined. However, this is not the case. This is why a so-called confidence interval is calculated based on the information from the sample. The exact value of the parameter lies within this interval with a predetermined level of confidence, y. This confidence interval is all the smaller the larger the sample and thus the accumulated time of observation and the lower the required level of confidence. In an extreme case we might ask for an interval, in which the parameter would be encountered with a level of confidence of one. However, such an interval comprises the entire domain of values of the parameter in question ([0, oo] in case of X, and [0, 1] for u) it is of no use. That is why normally confidence intervals for confidence levels of 90 or 95 % are calculated. 

In order to fix the endpoints of the confidence interval for a confidence level y we search for the value of X, for which fewer than the actually observed k failures would have occurred with a probability of (1 — y)H. This gives the lower endpoint, which is denoted here by X. Analogously, the upper endpoint is that value of X for which more than the actually observed failures would have occurred with a probability of (1 - - y)/2. This value is denoted by X. The calculation of the endpoints leads to sums over the Poisson distribution of Eq. (9.30). These sums may be expressed in terms of the distribution [24]. 

Example 9.13 Determination of confidence interval endpoints for failure rates The lower and the upper endpoints of the confidence interval for a confidence level of 90 % are to be determined for the multiport valves of Example 9.11. How would these endpoints change, if twice the number of failures were observed after doubling the period of observation with all other conditions remaining unchanged  

2 X 37 X 8,278.2 h 612,586.8 h After doubling the time of observation the following result is obtained 

The confidence interval becomes narrower with the expected value remaining unchanged. The reason is the more reliable sample size for the longer observation.  

Statisticians have introduced the notion of fractile (quantile), which is a special argument of the density function. The quantile, xq, of order q (0 q 1) is determined by the equation q = P(x xq), which means that the probability of finding a value of x below xq is equal to q (Fig. 2-3). 

In this respect xq is equal to a certain value k which replaces infinity as the upper integral limit in Eq. 2-2. So we can realize that k values depend on probability P. Again the GAUSSian (normal distribution) function is the simplest model function, because of its sole dependence on P which makes k = k(P). 

Using these lvalues we are now able to compute (to predict) so-called confidence intervals, cnf, of measured or calculated (estimated) numerical values xt. 

the true parameter cr (standard deviation) of the distribution of a variable x is known, one is able to calculate 

If we report an analytical result in such a way, we are confident that, with a defined probability P, the true result of, e.g., the mean will lie within the range defined by Ax around the reported mean value. 

Comparison of equation (9.194) to (9.148) shows that the approximate Hessian matrix approach (since we are using and not H ) leads to the same equation 

The procedure that we use to develop confidence intervals is based on making some assumptions about the variables we measure (concentration of A as a function of time, e.g.) and the parameters in the model equation. One assump- 

The measured dependent variable is equal to its true value plus measurement error and is also equal to an estimated value plus a residual error, which is often referred to simply as a residual. That is, the measured value of the dependent variable, Y, is equal to its true value, Y-, which we will never know for sure, plus a measurement error, V-, which we also do not know  

the measured dependent variable Y- is equal to its estimated value, plus a residual, C, at condition, i  

An alternate way to concepmalize this point is to note that the ultimate purpose of the results from a specific clinical trial is not to tell us precisely what happened in that trial, but, in combination with results from other trials in the drug s clinical development program, to gain insight into likely drug responses in patients who may be prescribed the drug should it subsequently be approved for marketing. 

A commonly used confidence interval is the two-sided 95 % confidence interval, and so this confidence interval is used here, and discussions continue to use the hypothetical example presented earlier in this chapter. The treatment effect used in the example, which is now referred to as the treatment effect point estimate, was 8.00 mmHg. The calculations performed yield a value that is subtracted from the treatment effect point estimate to give the lower limit of the confidence interval and added to the treatment effect point estimate to give the upper limit. Therefore, in this setting, the lower and upper limits of the confidence interval lie symmetrically around the treatment effect point estimate. Imagine that the two-sided 95% confidence interval has a lower limit of 6.5 mmHg and, therefore, an upper limit of 

The lower and upper limits define a range of values that we are 95 % confident will cover the true but unknown population treatment effect and allow the following statement to be made  

5 mmHg The clinicians involved in the drug s development program may again decide that the answer is yes. If so, the drug s efficacy is deemed to be clinically significant. 

Consider now a different scenario. Imagine a different trial of a similar design in which the treatment effect point estimate was also 8.00 mmHg, but the two-sided 95 % confidence interval has a lower limit of 2.5 mmHg and, therefore, an upper limit of 13.5 mmHg. The result would be written as follows  

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The distribution of the /-statistic (x — /ji)s is symmetrical about zero and is a function of the degrees of freedom. Limits assigned to the distance on either side of /x are called confidence limits. The percentage probability that /x lies within this interval is called the confidence level. The level of significance or error probability (100 — confidence level or 100 — a) is the percent probability that /X will lie outside the confidence interval, and represents the chances of being incorrect in stating that /X lies within the confidence interval. Values of t are in Table 2.27 for any desired degrees of freedom and various confidence levels. 

From Equation 4, x is 77.11 and from Equation 5, is 0.24 for 4 degrees of freedom. Because cr is not known, the Student 975 (2.78 for 4 degrees of freedom) is used to calculate the confidence interval at the 95% probability level. 

We used a two-tailed test. Upon rereading the problem, we realize that this was pure FeO whose iron content was 77.60% so that p = 77.60 and the confidence interval does not include the known value. Since the FeO was a standard, a one-tailed test should have been used since only random values would be expected to exceed 77.60%. Now the Student t value of 2.13 (for —to05) should have been used, and now the confidence interval becomes 77.11 0.23. A systematic error is presumed to exist. 

There will be incidences when the foregoing assumptions for a two-tailed test will not be true. Perhaps some physical situation prevents p from ever being less than the hypothesized value it can only be equal or greater. No results would ever fall below the low end of the confidence interval only the upper end of the distribution is operative. Now random samples will exceed the upper bound only 2.5% of the time, not the 5% specified in two-tail testing. Thus, where the possible values are restricted, what was supposed to be a hypothesis test at the 95% confidence level is actually being performed at a 97.5% confidence level. Stated in another way, 95% of the population data lie within the interval below p + 1.65cr and 5% lie above. Of course, the opposite situation might also occur and only the lower end of the distribution is operative. 

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. 

The confidence limits for the slope are given by fc where the r-value is taken at the desired confidence level and (A — 2) degrees of freedom. Similarly, the confidence limits for the intercept are given by a ts. The closeness of x to X is answered in terms of a confidence interval for that extends from an upper confidence (UCL) to a lower confidence (LCL) level. Let us choose 95% for the confidence interval. Then, remembering that this is a two-tailed test (UCL and LCL), we obtain from a table of Student s t distribution the critical value of L (U975) the appropriate number of degrees of freedom. 

Example 14 For the best-fit line found in Example 13, express the result in terms of confidence intervals for the slope and intercept. We will choose 95% for the confidence interval. 

Confidence Intervals for Normal Distribution Curves Between the Limits p zo... 

What is the 95% confidence interval for the amount of aspirin in a single analgesic tablet drawn from a population where p, is 250 mg and is 25 ... 

According to Table 4.11, the 95% confidence interval for a single member of a normally distributed population is... 

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... 

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... 

What is the 95% confidence interval for the analgesic tablets described in Example 4.13, if an analysis of five tablets yields a mean of 245 mg of aspirin ... 

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. 

In Section 4D.2 we introduced two probability distributions commonly encountered when studying populations. The construction of confidence intervals for a normally distributed population was the subject of Section 4D.3. We have yet to address, however, how we can identify the probability distribution for a given population. In Examples 4.11-4.14 we assumed that the amount of aspirin in analgesic tablets is normally distributed. We are justified in asking how this can be determined without analyzing every member of the population. When we cannot study the whole population, or when we cannot predict the mathematical form of a population s probability distribution, we must deduce the distribution from a limited sampling of its members. 

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... 

This confidence interval states that the analyte s true concentration lies within the range of -0.16 ppm to 1.44 ppm. Including a negative concentration within the confidence interval should lead you to reevaluate your data or conclusions. On further investigation your data may show that the standard deviation is larger than expected. 

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. 

If the significance test is conducted at the 95% confidence level (a = 0.05), then the null hypothesis will be retained if a 95% confidence interval around X contains p,. If the alternative hypothesis is... 

The equation for the test (experimental) statistic, fexp, is derived from the confidence interval for p,... 

Relationship between confidence intervals and results of a significance test, (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on fexp. The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a,v). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. 

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. 

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

These standard deviations can be used to establish confidence intervals for the true slope and the true y-intercept... 

Calculate the 95% confidence intervals for the slope and y-intercept determined in Example 5.10. 

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