Introduction to Biostatistics, Glover-Mitchell, Basic Review Exercises

Appendix A
Additional Problems

The problems in this document may be used as extra review or practice for quizzes and exams. They are organized by chapter and are very similar in spirit to those found in the text. When referring to problems in this document, we will use the chapter number followed by the problem number. For example, Problem 1.3 refers to the third problem in the problem set for Chapter 1. The answers to these problems may be found in a companion document at this website.

Problems for Chapter 7

Reconsider the cancer patient age data in Problem 1.1. For the control group _C = 48.5 yr and = 143.64 ( yr) ² while for the experimental group _E = 51.5 yards = 144.54 ( yr) ². Is there a significant difference in the mean ages of the two groups?
In a study of serum creatinine levels in thyroid cancer patients, patients were divided into two groups: Means of creatinine levels in the euthyroid (n = 56) and mild hypothyroid (n = 18) states were judged not significantly different (69.9 and 67.1 ľ mol/ l, respectively) . Assuming that (1) the variances were similar for the two populations and (2) the researchers carried out an unpaired t test with ∝ = 0.05, what is the smallest possible value of the pooled variance ?

The New York Public Library (NYPL) possesses a number of Dutch books. Some were deteriorated to such a extent, that it was decided to include them in a microfilming program during the 1960s. The original books were to be destroyed but, through contacts at the Koninklijke Bibliotheek (KB, National Library of the Netherlands) in The Hague, the Dutch books were returned to their land of origin. In this way the KB obtained possession of a few hundred Dutch books, which had been stored in New York.

At the end of the 1980s these books were rediscovered in the KB and submitted to some pilot tests. The study focused on the dfferences between six New York books and the identical editions that came from the KB collection. The NYPL books had been stored in depots without climate control before they were microfilmed. After the filming they were transferred to the former KB building (also no climate control). In 1982 the KB moved in to a new building, and the books were then stored in an office room with a temperature of about 20º C and 50% relative humidity (RH). The KB copies were situated in the KB depots until the moment of investigation, first until 1982 in a depot without air conditioning, and subsequently in a storage room at 18º C and 50% RH. The investigation showed remarkable differences between the NYPL and the KB copies: The paper in the NYPL copies was more acid ( especially at the margins), was more discolored and less strong. The much greater fluctuations in climate in New York campared to The Hague were suggested as a possible cause of the differences. A second possibility was the large difference in air pollution between the two locations: The higher concentrations of gasses like SO ₂ , NO, and NO ₂ in New York might lead to a greater absorption of these gasses into the books. This would explain the lower pH in the margins.

In 1994 the KB continued the investigation to characterize the differences between the NYPL and the KB copies of identical editions. Ninety-two identical titles (of the same edition) were located and compared. Among the data recorded were the differences in pH in the following table. (Derived from data reported in Sophia Pauk and Henk Porck, 1996, "A study on identical books, stored under two different conditions," www. kb. nl/ kb/ resources/ frameset kb. html? / kb/ sbo/ cons/ lol/ rappned. html. )

Differences in pH: KB-NYPL
Location	Margin	Center
n	92	92
_d	1.23	-0.16
s_d	0.37	0.26

Do these data support the results of the earlier pilot study? Test the hypothesis that the NYPL volumes have a lower pH (higher acidity) at the margins of the text. ( H ₀: ľ_d ≤ 0 versus H _a: _{ľ d}> 0. )
Test the hypothesis that the NYPL volumes have a lower pH at the center of the text.

The biological activity of insulin, like any protein, is affected if stored at extremes of temperature. Manufacturers recommend storing insulin under cool (15 - 30º C), shaded conditions. Neither condition is possible in the desert, where temperatures frequently exceed 40º C, thus constituting a major problem for diabetic patients. A small study was conducted to demonstrate the possible effect on the bioactivity of insulin following storage in a zeer, a semiporous clay pot containing water, in desert conditions. The small amounts of water that slowly seep through the walls of the zeer evaporate, using up heat, thereby cooling the remaining water. This principle is widely used in the Middle East to cool drinking water in such containers.

For the study, after intravenous injection of a standardized dose of insulin, blood was taken every 5 minutes for a period of 30 minutes. The slope of the fall in plasma glucose level (in mmol/ l/ min) was determined for each subject. (Based on data reported in Khiriah Al Shaibi, et al., "Storing insulin in a clay pot in the desert causes no loss of activity: A preliminary report", Annals of Saudi Medicine, 1999, 19 (6); see www.kfshrc.edu.sa/annals/196/98-243.html. )

In one phase of the experiment (subjects 1 - 4) , the efficacy of insulin that had yet to be stored was compared to insulin which was stored in a zeer over a six week period. Given the concern that insulin stored at higher temperatures may be less effective in lowering glucose levels, carry out an appropriate test on the first group to determine whether there is evidence for such differences.

Subject	Before storage	After storage in refrigerator	After storage in zeer
1	0.258		0.254
2	0.216		0.180
3	0.176		0.129
4	0.139		0.176
5		0.300	0.250
6		0.198	0.179
7		0.186	0.146
8		0.230	0.220

In a second phase of the experiment (subjects 5 - 8), the efficacy of insulin that was stored in a refrigerator (4º C) was compared to insulin which was stored in a zeer (mean temperature 26.7º C). Carry out an appropriate test to determine whether there is evidence for a decrease in efficacy of zeer-stored insulin.

Mulberry, Morus spp., is increasingly being used as forage plant because of its high protein content and high digestibility. However, its cultivation management is not yet defined when destined to be used for domestic farm animals. In Brazil, mulberry is known as a forage plant for the silkworm, Bombyx mori L., where it is cut close to the ground (low-cut) at intervals of 13 weeks. The management of mulberry like a shrub or a tree (high-trunk) has been encouraged by some in Europe and Japan with the advantages of the system being: greater production, better exploitation of soil, better leaf quality, and greater longevity. In a field trial, clones of 12 different mulberry species that are typically grown using the low-cut technique were grown using two different high-trunk methods: cutting every 9 weeks and cutting every 13 weeks. (Based on data reported in Jose Eduardo de Almeida and Tamara Canto Fonseca.
"A contribution to the introduction of the high-trunk mulberry system in tropical climates," www. fao. org/ WAICENT/ FAOINFO/ AGRICULT/ AGA/ AGAP/ FRG/ Mulberry/ Posters/ HTML/ Almeida2. htm )

		Cutting frequency
Clone	9 weeks(4 cuts per year)		13 weeks (3cuts per year)
	1997/98 kg/plant	1998/99 kg/plant	1997/98 kg/plant	1998/99 kg/plant
IZ 1/16	7.19	6.67	6.44	9.06
IZ 3/2	5.58	3.91	5.33	7.45
IZ 6/7	5.15	5.55	4.32	6.05
IZ 10/1	5.49	4.73	5.98	9.20
IZ 10/4	5.97	5.03	7.01	8.61
IZ 10/8	6.46	5.72	5.55	9.32
IZ 11/9	4.59	3.26	5.90	4.92
IZ 13/6	6.48	6.42	6.59	9.03
IZ 56/4	7.01	4.73	4.72	8.00
IZ 57/2	5.54	3.57	6.03	6.67
IZ 40	6.62	7.26	6.80	9.33
KORIN	6.81	5.36	6.62	9.43

The researchers used a sign test to determine whether there were significant differences in the production levels for the two high-trunk methods for the initial year (1997/ 98) of the field trial. What did they find?
Having given the plants a year to adapt to the new cultivation methods, the researchers tested the hypotheses H ₀: There is no increase in production from the first year to the second year, versus H _a: There is an increase in production during the second year. Carry out two paired sign tests, one for each cultivation method and interpret the results.
Finally carry out a sign test to compare the production results for the two cultivation methods during the second year of the trial. Interpret your results.