D. Harvey - Modern Analytical Chemistry (794078), страница 25
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The shaded area showsthe percentage of tablets containingbetween 243 and 262 mg of aspirin.243 − 250= −1.45and the deviation for the upper limit isz up =262 − 250= +2.45Using the table in Appendix 1A, we find that the percentage of tablets with lessthan 243 mg of aspirin is 8.08%, and the percentage of tablets with more than262 mg of aspirin is 0.82%. The percentage of tablets containing between 243and 262 mg of aspirin is therefore100.00% – 8.08% – 0.82 % = 91.10%4D.3 Confidence Intervals for PopulationsIf we randomly select a single member from a population, what will be its most likely value? This is animportant question, and, in one form or another, it isthe fundamental problem for any analysis.
One of themost important features of a population’s probabilitydistribution is that it provides a way to answer thisquestion.Earlier we noted that 68.26% of a normally distributed population is found within the range of µ ± 1σ. Stating this another way, there is a 68.26% probability that amember selected at random from a normally distributedpopulation will have a value in the interval of µ ± 1σ. Ingeneral, we can writeXi = µ ± zσTable 4.11Confidence Intervals for NormalDistribution Curves Betweenthe Limits µ ± zσzConfidence Interval (%)0.501.001.501.962.002.503.003.5038.3068.2686.6495.0095.4498.7699.7399.954.9where the factor z accounts for the desired level of confidence. Values reportedin this fashion are called confidence intervals. Equation 4.9, for example, is theconfidence interval for a single member of a population.
Confidence intervalscan be quoted for any desired probability level, several examples of which areshown in Table 4.11. For reasons that will be discussed later in the chapter, a95% confidence interval frequently is reported.confidence intervalRange of results around a mean valuethat could be explained by random error.1400-CH04 9/8/99 3:54 PM Page 7676Modern Analytical ChemistryEXAMPLE 4.12What is the 95% confidence interval for the amount of aspirin in a singleanalgesic tablet drawn from a population where µ is 250 mg and σ2 is 25?SOLUTIONAccording to Table 4.11, the 95% confidence interval for a single member of anormally distributed population isXi = µ ± 1.96σ = 250 mg ± (1.96)(5) = 250 mg ± 10 mgThus, we expect that 95% of the tablets in the population contain between 240and 260 mg of aspirin.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 meanµ = Xi ± zσ4.10EXAMPLE 4.13The population standard deviation for the amount of aspirin in a batch ofanalgesic tablets is known to be 7 mg of aspirin. A single tablet is randomlyselected, analyzed, and found to contain 245 mg of aspirin. What is the 95%confidence interval for the population mean?SOLUTIONThe 95% confidence interval for the population mean is given asµ = Xi ± zσ = 245 ± (1.96)(7) = 245 mg ± 14 mgThere is, therefore, a 95% probability that the population’s mean, µ, lies withinthe range of 231–259 mg of aspirin.Confidence intervals also can be reported using the mean for a sample of size n,drawn from a population of known σ.
The standard deviation for the mean value,σ X–, which also is known as the standard error of the mean, isσX =σnThe confidence interval for the population’s mean, therefore, isµ = X ±zσn4.111400-CH04 9/8/99 3:54 PM Page 77Chapter 4 Evaluating Analytical DataEXAMPLE 4.14What is the 95% confidence interval for the analgesic tablets described inExample 4.13, if an analysis of five tablets yields a mean of 245 mg of aspirin?SOLUTIONIn this case the confidence interval is given asµ = 245 ±(1.96)(7)= 245 mg ± 6 mg5Thus, there is a 95% probability that the population’s mean is between 239 and251 mg of aspirin.
As expected, the confidence interval based on the mean offive members of the population is smaller than that based on a single member.4D.4 Probability Distributions for SamplesIn Section 4D.2 we introduced two probability distributions commonly encountered when studying populations. The construction of confidence intervals for anormally 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 analgesictablets is normally distributed. We are justified in asking how this can be determined without analyzing every member of the population. When we cannot studythe whole population, or when we cannot predict the mathematical form of a population’s probability distribution, we must deduce the distribution from a limitedsampling of its members.Sample Distributions and the Central Limit Theorem Let’s return to the problemof determining a penny’s mass to explore the relationship between a population’sdistribution and the distribution of samples drawn from that population.
The datashown in Tables 4.1 and 4.10 are insufficient for our purpose because they are notlarge enough to give a useful picture of their respective probability distributions. Abetter picture of the probability distribution requires a larger sample, such as that–shown in Table 4.12, for which X is 3.095 and s2 is 0.0012.The data in Table 4.12 are best displayed as a histogram, in which the frequency of occurrence for equal intervals of data is plotted versus the midpoint ofeach interval.
Table 4.13 and Figure 4.8 show a frequency table and histogram forthe data in Table 4.12. Note that the histogram was constructed such that the meanvalue for the data set is centered within its interval. In addition, a normal distribu–tion curve using X and s2 to estimate µ and σ2 is superimposed on the histogram.It is noteworthy that the histogram in Figure 4.8 approximates the normal distribution curve. Although the histogram for the mass of pennies is not perfectlysymmetrical, it is roughly symmetrical about the interval containing the greatestnumber of pennies. In addition, we know from Table 4.11 that 68.26%, 95.44%,and 99.73% of the members of a normally distributed population are within, respectively, ±1σ, ±2σ, and ±3σ.
If we assume that the mean value, 3.095 g, and thesample variance, 0.0012, are good approximations for µ and σ2, we find that 73%,histogramA plot showing the number of times anobservation occurs as a function of therange of observed values.771400-CH04 9/8/99 3:54 PM Page 7878Modern Analytical ChemistryTable 4.12Penny Weight(g)12345678910111213141516171819202122232425aPennies3.1263.1403.0923.0953.0803.0653.1173.0343.1263.0573.0533.0993.0653.0593.0683.0603.0783.1253.0903.1003.0553.1053.0633.0833.065Individual Masses for a Large Sample of U.S. Penniesin CirculationaPennyWeight(g)PennyWeight(g)PennyWeight(g)262728293031323334353637383940414243444546474849503.0733.0843.1483.0473.1213.1163.0053.1153.1033.0863.1033.0492.9983.0633.0553.1813.1083.1143.1213.1053.0783.1473.1043.1463.095515253545556575859606162636465666768697071727374753.1013.0493.0823.1423.0823.0663.1283.1123.0853.0863.0843.1043.1073.0933.1263.1383.1313.1203.1003.0993.0973.0913.0773.1783.0547677787980818283848586878889909192939495969798991003.0863.1233.1153.0553.0573.0973.0663.1133.1023.0333.1123.1033.1983.1033.1263.1113.1263.0523.1133.0853.1173.1423.0313.0833.104are identified in the order in which they were sampled and weighed.Table 4.13Frequency Distributionfor the Data in Table 4.12IntervalFrequency2.991–3.0093.010–3.0283.029–3.0473.048–3.0663.067–3.0853.086–3.1043.105–3.1233.124–3.1423.143–3.1613.162–3.1803.181–3.199204191523191213121400-CH04 9/8/99 3:54 PM Page 79Chapter 4 Evaluating Analytical Data792520Frequency1510502.991to3.0093.010to3.0283.029to3.0473.048 3.067 3.086 3.105 3.124tototototo3.066 3.085 3.104 3.123 3.142Weight of pennies (g)3.143to3.1613.162to3.1803.181to3.19995%, and 100% of the pennies fall within these limits.
It is easy to imagine that increasing the number of pennies in the sample will result in a histogram that evenmore closely approximates a normal distribution.We will not offer a formal proof that the sample of pennies in Table 4.12and the population from which they were drawn are normally distributed; however, the evidence we have seen strongly suggests that this is true. Although wecannot claim that the results for all analytical experiments are normally distributed, in most cases the data we collect in the laboratory are, in fact, drawn froma normally distributed population.
That this is generally true is a consequence ofthe central limit theorem.6 According to this theorem, in systems subject to avariety of indeterminate errors, the distribution of results will be approximatelynormal. Furthermore, as the number of contributing sources of indeterminateerror increases, the results come even closer to approximating a normal distribution. The central limit theorem holds true even if the individual sources of indeterminate error are not normally distributed.
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