Analytical Chem istry - DePauw University

92 Analytical Chemistry 2.0196 . × 7µ= 245 mg ± mg = 245 mg ± 6 mg5Thus, there is a 95% probability that the population’s mean is between 239to 251 mg of aspirin. As expected, the confidence interval when using themean of five samples is smaller than that for a single sample.Practice Exercise 4.6An analysis of seven aspirin tablets from a population known to havea standard deviation of 5, gives the following results in mg aspirin pertablet:246 249 255 251 251 247 250What is the 95% confidence interval for the population’s expectedmean?Click here when you are ready to review your answer.4D.4 Probability Distributions for SamplesIn working example 4.11–4.14 we assumed that the amount of aspirin inanalgesic tablets is normally distributed. Without analyzing every memberof the population, how can we justify this assumption? In situations wherewe can not study the whole population, or when we can not predict themathematical form of a population’s probability distribution, we must deducethe distribution from a limited sampling of its members.Sa m p l e Distributions a n d t h e Ce n t r a l Limit Th e o r e mLet’s return to the problem of determining a penny’s mass to explore furtherthe relationship between a population’s distribution and the distribution ofa sample drawn from that population. The two sets of data in Table 4.11are too small to provide a useful picture of a sample’s distribution. To gaina better picture of the distribution of pennies we need a larger sample, suchas that shown in Table 4.13. The mean and the standard deviation for thissample of 100 pennies are 3.095 g and 0.0346 g, respectively.A histogram (Figure 4.10) is a useful way to examine the data in Table4.13. To create the histogram, we divide the sample into mass intervalsand determine the percentage of pennies within each interval (Table 4.14).Note that the sample’s mean is the midpoint of the histogram.Figure 4.10 also includes a normal distribution curve for the populationof pennies, assuming that the mean and variance for the sample provide appropriateestimates for the mean and variance of the population. Althoughthe histogram is not perfectly symmetric, it provides a good approximationof the normal distribution curve, suggesting that the sample of 100 pennies