Prof. David Draper
Department of
Applied Mathematics and Statistics
University of California, Santa Cruz
AMS 7: Discussion Section 1
1. A central theme of this course is quantification: how to measure things that interest you,
and what to do with the measurements once you have them. This problem poses some
questions on how to measure variables that arise in biology, environmental science, health,
and other fields.
(a) The number of deaths from cancer in the United States has risen steadily over time.
In 1985, for example, about 462,000 people died of cancer, up from 331,000 deaths in
1970. In a speech on the floor of the House of Representatives in 1986, a member of
Congress said that “these numbers show that no progress has been made in treating
cancer.” Explain how the number of people dying of cancer could increase even if
treatment of the disease were improving, and describe at least one variable that
would be a more appropriate measure of the effectiveness of medical treatment for a
potentially fatal disease.
(b) Cost is often an important factor in deciding which variable to measure in a study.
Sometimes the cost can be determined directly (in dollars, for example), while in other
cases it may be expressed in terms of the time required to take the measurement.
A crude but inexpensive measure is sometimes preferable to a more precise measure
that is too expensive. Blood tests, for example, are not as accurate as exploratory
surgery in diagnosing some diseases, but they’re faster, cheaper, and pose a lower risk
of side-effects.
Foresters need quick measurements of the size of a tree, since they must measure
many trees in a woodlot in order to assess its overall condition. List three variables
that measure the size of a tree, and arrange them in order from most to least costly.
(c) Various studies have attempted to rank cities in terms of how desirable it is to live
and work in each city. Describe five variables that you would measure for each city
if you were designing such a study, and briefly explain your choices.
(d) Scientists who study human growth use different measures of the size of an individual;
weight, height, and weight divided by height are three of the most common measures.
If you were interested in studying the short-term effects of a digestive illness, which
(if any) of these three variables would you use? Explain briefly.
(e) The usual method of determining heart rate in humans is to take the person’s pulse
by counting the number of beats in a given time period. The results are generally
reported as beats per minute; for example, if the time period is 15 seconds, the
observed count is multiplied by 4. You can either decide ahead of time to specify
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