Analytical Chem istry - DePauw University

90 Analytical Chemistry 2.04D.3 Confidence Intervals for PopulationsIf we randomly select a single member from a population, what is its mostlikely value? This is an important question, and, in one form or another, itis at the heart of any analysis in which we wish to extrapolate from a sampleto the sample’s parent population. One of the most important features ofa population’s probability distribution is that it provides a way to answerthis question.Figure 4.8 shows that for a normal distribution, 68.26% of the population’smembers are found within the range of m ± 1s. Stating this anotherway, there is a 68.26% probability that the result for a single sample drawnfrom a normally distributed population is in the interval m ± 1s. In general,if we select a single sample we expect its value, X i to be in the rangeWhen z = 1, we call this the 68.26% confidenceinterval.X = µ ± z σ 4.9iwhere the value of z is how confident we are in assigning this range. Valuesreported in this fashion are called confidence intervals. Equation 4.9,for example, is the confidence interval for a single member of a population.Table 4.12 gives the confidence intervals for several values of z. For reasonswe will discuss later in the chapter, a 95% confidence level is a commonchoice in analytical chemistry.Example 4.12What is the 95% confidence interval for the amount of aspirin in a singleanalgesic tablet drawn from a population for which μ is 250 mg and σ 2is 25?So l u t i o nUsing Table 4.12, we find that z is 1.96 for a 95% confidence interval.Substituting this into equation 4.9, gives the confidence interval for asingle tablet asX i = μ ± 1.96σ = 250 mg ± (1.96 × 5) = 250 mg ± 10 mgTable 4.12 Confidence Intervals for aNormal Distribution (μ ± zσ)z Confidence Interval (%)0.50 38.301.00 68.261.50 86.641.96 95.002.00 95.442.50 98.763.00 99.733.50 99.95