Obtain mean, median, standard deviation, and so on, for each of the three In contrast to part a, notice that the difference between the individual medians of the Should the t test we did have been one-sided? In practice, we must make the decision We used

Text not available due to copyright restrictions Text not available due to copyright restrictions 360 CHAPTER 7 Inferences about Population Variances 7.1 Introduction and Abstract of Research Study 7.2 Estimation and Tests for a Population Variance 7.3 Estimation and Tests for Comparing Two Population Variances 7.4 Tests for Comparing t ⬎ 2 Population Variances 7.5 Research Study: Evaluation of Methods for Detecting E. coli 7.6 Summary and Key Formulas 7.7 Exercises

7.1 Introduction and Abstract of Research Study

When people think of statistical inference, they usually think of inferences con- cerning population means. However, the population parameter that answers an ex- perimenter’s practical questions will vary from one situation to another. In many situations, the variability of a population’s values is as important as the population mean. In the case of problems involving product improvement, product quality is defined as a product having mean value at the target value with low variability about the mean. For example, the producer of a drug product is certainly con- cerned with controlling the mean potency of tablets, but he or she must also worry about the variation in potency from one tablet to another. Excessive potency or an underdose could be very harmful to a patient. Hence, the manufacturer would like to produce tablets with the desired mean potency and with as little variation in potency as measured by or as possible. Another example is from the area of investment strategies. Investors search for a portfolio of stocks, bonds, real estate, and other investments having low risk. A measure used by investors to determine the uncertainty inherent in a particular portfolio is the variance in the value of the investments over a set period. At times, a portfolio with a high average value and a large standard deviation will have a value that is much lower than the average value. Investors thus need to examine the variability in the value of a portfolio along with its average value when determining its degree of risk. Abstract of Research Study: Evaluation of Methods for Detecting

E. coli

The outbreaks of bacterial disease in recent years due to the consumption of contaminated meat products have created a demand for new, rapid methods for detecting pathogens in meats that can be used in a meat surveillance program. Under s 2 s specific environmental conditions, certain strains of bacteria such as E. coli are capable of causing hemorrhagic colitis, hemolytic uremic syndrome, and even death. An effective pathogen surveillance program requires three main attributes: 1 a probability-based sampling plan as described in Chapter 2, 2 a method capable of efficiently removing viable organisms from the target surface of animals, and 3 a repeatable, accurate, and practical microbial test for the target pathogen. The paper “Repeatability of the Petrifilm HEC test and agreement with a hydrophobic grid membrane filtration method for the enumeration of Escherichia coli 0157:H7 on beef carcasses” 1998, Journal of Food Protection, Vol. 61, pp. 402 – 408, describes a formal comparison between a new microbial method for the detection of E. coli, the Petrifilm HEC testing, with an elaborate laboratory-based procedure, hydrophobic grid membrane filtration HGMF. The HEC test is easier to inoculate, more com- pact to incubate, and safer to handle than conventional procedures. However, it was necessary to compare the performance of the HEC test to that of the HGMF proce- dure in order to determine if HEC may be a viable method for detecting E. coli. What aspects of the E. coli counts obtained by HEC and HGMF should be of interest to the researchers? A comparison of just the mean concentrations ob- tained by the two procedures would indicate whether or not the two procedures were in agreement with respect to the average readings over a large number of de- terminations. However, we would not know if HEC was more variable in its deter- mination of E. coli than HGMF. For example, consider the two distributions in Figure 7.1. Suppose the distributions represent the population of E. coli concen- tration determinations from HEC and HGMF for a situation in which the true E. coli concentration is 7 log 10 CFUml. The distributions would indicate that the HEC evaluation of a given meat sample may yield a reading very different from the true E. coli concentration, whereas the individual readings from HGMF are more likely to be near the true concentration. In this type of situation it is crucial to compare both the means and standard deviations of the two procedures. In fact, we need to examine other aspects of the relationship between HEC and HGMF determinations in order to evaluate the comparability of the two procedures. FIGURE 7.1 Hypothetical distribution of E. coli concentrations from HEC and HGMF HGMF HEC 4.0 3.5 3.0 2.5 2.0 1.5 1.0 .5 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 E. coli concentration PDFs from tw o procedures