Researchers asked volunteers to imagine they were running a five-kilometer race against 50 people and then against 500 people, races in each of which the top 10 percent would receive a $1,000 prize. Asked about the effort they would apply in the respective cases, the volunteers indicated, on average, that they would run slower in the race against the greater number of people. A likely explanation of this result is that those of the volunteers who were most comparatively inclined—those who most tended to compare themselves with others in the social environment—determined (perhaps unconsciously) that extreme effort would not be worthwhile in the 500-competitor race.  

Which of the following would, if known to be true, most help justify the explanation offered above?

Criminologists studying public perceptions of crime versus actual crimes reported by police found that,in counties worldwide, most people think violent crime accounts for a much larger percentage of the total crimes than it actually does.The criminologists hypothesize that   these estimates are caused by media misrepresentations,nothing that violent crime is depicted prominently in many counties' most popular media,like television shows and books,while other types of crime are underrepresented or not depicted at all.

Which of the following would it be most useful to determine in evaluating the criminologist hypothesis?

To increase business this year, a travel agency changed the prices of all of its tour packages in a way that created a linear relationship between this year’s prices and last year’s prices. What is the price this year of a tour package that had a price of $5,500 last year?

(1) A tour package with a price of $2,000 last year has a price of $1,700 this year.

(2) A tour package with a price of $3,000 last year has a price of $2,300 this year.

For positive integers m and n, is m² - n² divisible by 3?

(1) m - n is divisible by 3

(2) When m² + n² is divided by 3, the remainder is 2

If x > y > z, which of the following could be true?

I. x³>y³>z³

II. z>y²>x³

III. x³>z³>y³

Sophia is buying a smartphone and a phone charger at an electronics store for a total price of $630. This price reflects a discount from the original prices of both devices due to a sale the store is offering. Assuming no taxes or other fees are involved, what was the original price of the phone charger alone?

(1) The sale that the store is offering is 15 percent off a total purchase of $500 or more or 20 percent off a total purchase of $800 or more.

(2) The original price of the smartphone alone was $720.

In how many ways can the letters of the word MACHINE be arranged so that the vowels may occupy only odd position?

United Energy recently invested in a series of large windmills which are able to produce renewable energy with minimal negative effect to the environment. The company has not drilled oil wells in the same area, even though greater revenues and profits could be generated from oil wells. Because any drilling would disrupt the native habitat of certain marine species in the area, some environmentalists assert that, by foregoing this drilling, United Energy has established that it places environmental impact over financial returns. However, United Energy may be acting in a manner consistent with its financial goals. Recent patterns of increasing annual hurricane activity have some experts questioning the long-term viability and profitability of oil wells in the area.

The two boldfaced portions play which of the following roles?

Historian:  Newton developed mathematical concepts and techniques that are fundamental to modern calculus.  Leibniz developed closely analogous concepts and techniques.  It has traditionally been thought that these discoveries were independent.  Researchers have, however, recently discovered notes of Leibniz’ that discuss one of Newton’s books on mathematics.  Several scholars have argued that since the book includes a presentation of Newton’s calculus concepts and techniques, and since the notes were written before Leibniz’ own development of calculus concepts and techniques, it is virtually certain that the traditional view is false.  A more cautious conclusion than this is called for, however.  Leibniz’ notes are limited to early sections of Newton’s book, sections that precede the ones in which Newton’s calculus concepts and techniques are presented.

In the historian’s reasoning, the two boldfaced portions play which of the following roles?

If ab ≠ 0 and a³b=ab³, then which of the following must be true?

I. a = b
II. a = –b
III. ab = 1