Excursions in Modern Mathematics (9th Edition)

9th Edition

ISBN: 9780134468372

Author: Peter Tannenbaum

Publisher: PEARSON

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Chapter 14, Problem 38E

Exercises 37 through 40 refer to a clinical study conducted at the Houston Veterans Administration Medical Centre on the effectiveness of knee surgery to cure degenerative arthritis (osteoarthritis) of the knee. Of the 324 individuals who met the inclusion criteria for the study, 144 declined to participate. The researchers randomly divided the remaining 180 subjects into three groups: One group received a type of arthroscopic knee surgery called debridement; a second group received a type of arthroscopic knee surgery called lavage; and a third group received skin incisions to make it look like they had had arthroscopic knee surgery, but no actual surgery was performed. The patients in the study did not know which group they were in and in particular did not know if they were receiving the real surgery or simulated surgery. All the patients who participated in the study were evaluated for two years after the procedure. In the two-year follow-up, all three groups said that they had slightly less pain and better knee movement, but the “fake” surgery group often reported the best results. [Source: New England Journal of Medicine, 347 , no. 2 (July 11, 2002 ): 81 – 88 .]

a. Was the sample chosen by random sampling? Explain.

b. Was this study a controlled placebo experiment? Explain.

c. Describe the treatment group(s) in this study.

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Chapter 14, Problem 37E

Chapter 14, Problem 39E

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For each real-valued nonprincipal character x mod k, let A(n) = x(d) and F(x) = Σ : dn * Prove that F(x) = L(1,x) log x + O(1). n

By considering appropriate series expansions, e². e²²/2. e²³/3. .... = = 1 + x + x² + · ... when |x| < 1. By expanding each individual exponential term on the left-hand side the coefficient of x- 19 has the form and multiplying out, 1/19!1/19+r/s, where 19 does not divide s. Deduce that 18! 1 (mod 19).

Proof: LN⎯⎯⎯⎯⎯LN¯ divides quadrilateral KLMN into two triangles. The sum of the angle measures in each triangle is ˚, so the sum of the angle measures for both triangles is ˚. So, m∠K+m∠L+m∠M+m∠N=m∠K+m∠L+m∠M+m∠N=˚. Because ∠K≅∠M∠K≅∠M and ∠N≅∠L, m∠K=m∠M∠N≅∠L, m∠K=m∠M and m∠N=m∠Lm∠N=m∠L by the definition of congruence. By the Substitution Property of Equality, m∠K+m∠L+m∠K+m∠L=m∠K+m∠L+m∠K+m∠L=°,°, so (m∠K)+ m∠K+ (m∠L)= m∠L= ˚. Dividing each side by gives m∠K+m∠L=m∠K+m∠L= °.°. The consecutive angles are supplementary, so KN⎯⎯⎯⎯⎯⎯∥LM⎯⎯⎯⎯⎯⎯KN¯∥LM¯ by the Converse of the Consecutive Interior Angles Theorem. Likewise, (m∠K)+m∠K+ (m∠N)=m∠N= ˚, or m∠K+m∠N=m∠K+m∠N= ˚. So these consecutive angles are supplementary and KL⎯⎯⎯⎯⎯∥NM⎯⎯⎯⎯⎯⎯KL¯∥NM¯ by the Converse of the Consecutive Interior Angles Theorem. Opposite sides are parallel, so quadrilateral KLMN is a parallelogram.

Chapter 14 Solutions

Excursions in Modern Mathematics (9th Edition)

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Solution Summary: The author explains that a clinical study on effectiveness of knee surgery to cure degenerative arthritis is given.

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