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Ranking Sci-Fi Films

GMAT-XRZGG4

Two friends decide to rank ten science fiction films. Each person gives a rank of 10 to their favorite film and a rank of 1 to their least favorite film. Each rank (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) is used only once. Here are the ranks assigned by the two friends:

Film Rank of Person A Rank of Person B Difference in Ranks
A.I. Artificial Intelligence 7 5
Interstellar 9 10
Dark City 10 8
Solaris 5 4
Star Trek II: The Wrath of Khan 8 3
Star Wars Episode III 3 1
Avatar 1 2
Blade Runner 6 9
2001: A Space of Odyssey 4 7
District 9 2 6

The two friends plan to use a statistical procedure known as the Spearman Rank Correlation to see if there is a significant tendency for them to agree on rank preferences. If evidence suggests a relationship of rank agreement, then it offers good reason for the friends to believe that the opinion of one of them about a movie is likely to be similar to the opinion of the other.

In the Spearman Rank Correlation test, the difference between the ranks of each film is calculated and then squared. The squares are added together. Call this sum of squares $D$.

The correlation statistic r is given by the formula:

$$r=1-\cfrac { 6D }{ n(n^{ 2 }-1) }$$

...where $n$ stands for the number of films ranked.

The chart below contains the value of n in the left column. In the row with the appropriate value of $n$, the values given are those of the correlation statistic $r$. The column headings are probabilities; smaller probabilities indicate stronger evidence of a relationship of similar ranking systems. Probabilities below 0.01 are small enough that the two friends would be convinced that they rank science fiction movies in a similar way.

α

$n$ .10 .05 .025 .01 .005 .0025 .001 .0050
4 1.000 1.000
5 .800 .900 1.000 1.000
6 .657 .829 .886 .943 1.000 1.000
7 .571 .714 .786 .893 .929 .964 1.000 1.000
8 .524 .643 .738 .833 .881 .905 .952 .976
9 .483 .600 .700 .783 .833 .867 .917 .933
10 .455 .564 .648 .745 .794 .830 .879 .903
11 .427 .536 .618 .709 .755 .800 .845 .873
12 .406 .503 .587 .678 .727 .769 .818 .846
13 .385 .484 .560 .648 .703 .747 .791 .824
14 .367 .464 .538 .626 .679 .723 .771 .802
15 .354 .446 .521 .604 .654 .700 .750 .779
16 .341 .429 .503 .582 .635 .679 .729 .762
17 .328 .414 .485 .566 .615 .662 .713 .748
18 .317 .401 .472 .550 .600 .643 .695 .728
19 .309 .391 .460 .535 .584 .628 .677 .712
20 .299 .380 .447 .520 .570 .612 .662 .696
21 .292 .370 .435 .508 .556 .599 .648 .681
22 .284 .361 .425 .496 .544 .586 .634 .667
23 .278 .353 .415 .486 .532 .573 .622 .654
24 .271 .344 .406 .476 .521 .562 .610 .642
25 .265 .337 .398 .466 .511 .551 .598 .630

Decide whether the following statements are true or false about this scenario.

True

False

True

False

About one-third of the value of D comes from the ranks for Star Trek II.

True

False

The friends have a statistically significant ranking relationship in which they have a tendency to rank movies similarly.

True

False

If the two friends had a probability of close to 0.005 and 12 movies had been ranked, then the value of D for their ranking system would have been at least 100.

True

False

With 10 movies that were ranked in the precisely opposite order, the value of D would have been greater than 300.