8 Comparing Population Proportions

Where We’ve Been

  • Presented both parametric and nonparametric methods for comparing two or more population means

Where We’re Going

  • Discuss methods for comparing two population proportions. (8.18.2)

  • Present a chi-square hypothesis test for comparing the category proportions associated with a single qualitative variable—called a one-way analysis (8.3)

  • Present a chi-square hypothesis test for relating two qualitative variables—called a two-way analysis. (8.4)

Statistics in Action The Case of the Ghoulish Transplant Tissue

In the United States, tissue implants are routinely performed to aid patients in various types of surgery, including joint replacements, spinal surgery, sports-related surgeries (tendons and ligaments), and others. The process of obtaining a tissue transplant involves several parties. First, of course, is the donor, who has agreed to have tissue removed upon death and whose family has approved the donation. The tissue is then “harvested” by an approved tissue bank. Next, the harvested tissue is sent to a processor that sterilizes the tissue. Finally, the processor sends it either directly to the hospital/surgeon doing the implant or to a distributor that inventories the tissue and ultimately sends it on to the hospital/surgeon. The entire process is highly regulated by the Federal Trade Commission (FTC), particularly the harvesting and processing aspects.

Given this background, we consider an actual case that began in the early 2000s when the owner of a tissue bank—Biomedical Tissue Services (BTS)—became a ringleader of a group of funeral home directors that harvested tissue illegally and without permission of donors or their families. In some cases, the cadavers were cancerous or infected with HIV or hepatitis, all of which would, of course, disqualify them as donors. BTS then sent the tissue to processors without divulging that it had obtained the tissue illegally. (Note: The owner is currently serving 18 to 24 years in a New York prison.) The unsuspecting processors sterilized the tissue and sent it on for use as surgical implants. When the news story broke about how the tissue had been obtained, the processors and their distributors were required to send recall notices using an FTC recall letter to the hospital/surgeons who had received the tissue. Some of the BTS tissue was recovered; however, much of the tissue had already been implanted, and hospitals and surgeons were required to inform patients receiving implants of the potentially infectious tissue. Although few patients subsequently became infected, a number filed suit against the distributors and processors (and BTS), asking for monetary damages.

After the bulk of the lawsuits had been either tried or settled, a dispute arose between a processor and one of its distributors regarding ultimate responsibility for payment of damages to litigating patients. In particular, the processor claimed that the distributor should be held more responsible for the damages because in its recall package, the distributor had of its own volition included some salacious, inflammatory newspaper articles describing in graphic detail the “ghoulish” acts that had been committed. None of the patients who received implants that had been sterilized by this processor ever became infected, but many still filed suit.

To establish its case against the distributor, the processor collected data on the patients who had received implants of BTS tissue it had processed and the number of those patients who subsequently filed suit: The data revealed that of a total of 7,914 patients, 708 filed suit. A consulting statistician subdivided this information according to whether the recall notice had been sent to the patient’s surgeon by the processor or one of its distributors that had sent only the notice or by the distributor that had included the newspaper articles. The breakdown was as shown in Table SIA8.1.

Table SIA8.1 Data for the Tainted Tissue Case*

Recall Notice Sender Number of Patients Number of Lawsuits
Processor/Other Distributor 1,751 51
Distributor in Question 6,163 657
Totals 7,914 708

Do these data provide evidence of a difference in the probability that a patient would file a lawsuit depending on which party sent the recall notice? If so, and if the probability is significantly higher for the distributor in question, then the processor can argue in court that the distributor that sent the inflammatory newspaper articles is more responsible for the damages.

We apply the statistical methodology presented in this chapter to solve the case of the ghoulish transplant tissue in the following Statistics in Action Revisited example.

Statistics in Action Revisited

  • Testing whether Likelihood of a Lawsuit Is Related to Recall Notice Sender (p. 477)

Many experiments are conducted in the biological, physical, and social sciences to compare two or more proportions. Those conducted in business and the social sciences to sample the opinions of people are called sample surveys. For example, a state government might wish to estimate the difference between the proportions of people in two regions of the state who would qualify for a new welfare program. Or, after an innovative process change, an engineer might wish to determine whether the proportion of defective items produced by a manufacturing process was less than the proportion of defectives produced before the change. Or, a sports psychologist might want to determine if the proportions of professional hockey players differ depending on their highest level of education attained (some high school, high school diploma, some college, college undergraduate degree, college graduate degree). In Section 8.1 we show you how to test hypotheses about the difference between two population proportions based on independent random sampling. We will also show how to find a confidence interval for the difference. Then, in Section 8.3 we will compare more than two population proportions, and in Section 8.4 we will present a related problem.

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