1. Consider an individual boy from a model population. Suppose that from grades 1 through 5, this boy's height is at the 50th percentile for his running time and his running time is at the 50th percentile for his grade level. Which one of the following statements must be true of the boy at Grade 5?This problem requires the use of both graphs.
We can't use the first statement right away, because we don't have the boy's height or running time to start with.
But we can use the second statement: The boy's running time is the median (50th percentile) for 5th graders. We can find this in the table (first datagraphic): 8 minutes 21 seconds. That's 8·60 + 21 = 501 seconds.
Now go back to the second datagraphic. 501 seconds is approximately 500 seconds, which is the left-hand extreme of the graph. The boy's height is the y-coordinate of the 50th percentile there, which is 120 cm. (Lol that's only about 4 feet tall... awfully short for a fifth-grader!)
Let's see whether we can decide any of the answer choices:
Quote:
A. His grade level is at the 50th percentile for his running time.
This is impossible to decide, because we don't have distributions of the numbers of students by running time or by grade level. (Among other things, the value cited here could vary if the classes at different grade levels have different numbers of boys in them.)
Quote:
B. His height is at the 50th percentile for his grade level.
We don't have any information about heights by grade level, only heights by running time. This can't be decided.
Quote:
C. His running time is at the 50th percentile for his height.
We can't use this unless we have some data for which height is the explanatory (x axis) variable. We have no such data here, so this is unresolvable as well.
For D and E, we need the boy's 2nd grade height.
This just takes another jaunt through the same two steps. First, find the 50th percentile time for second grade (11 min 15 sec, or 675 sec). Then go to the graph with the curves and read off the boy's height: about 65 cm.
(LOL!! that's barely more than 2 feet! Literal newborns are almost 2 feet long, so this is an absolutely insane figure. It's very uncharacteristic of GMAC to create a problem with numbers that are absurd in the real world—I can only guess that this problem didn't go through as much editing as most of them do.)
Twice 65cm is 130cm, so the fifth-grade height of 120cm is
just barely less than twice the second-grade height.
Quote:
D. His height is approximately 130% of his height in Grade 2.
Nah.
Quote:
E. His height is approximately 185% of his height in Grade 2.
Yeah.
It's E.(The listed OA isn't correct here. Not sure what happened there, but the datum in choice A can't be found from the data we're given.)
2. For each of the following statements, select Yes if the statement must be true of a boy selected at random from a model population. Otherwise, select No.Some of the wordings in these statements are a bit dodgy (see details below). In the upcoming Data Insights section—essentially, "IR goes to play in the big leagues"—I trust that the real exam problems will be more extensively edited, and thus more meticulously written for clarity, than the below.
Quote:
If he is in the third grade or above, the probability that his speed is faster than 10:50 is greater than 50%.
"Probability" is singular here, so the only possible interpretation of this statement is that
the boy is chosen at random from a pool containing all third-, fourth-, AND fifth-graders. We're talking about a SINGLE probability, for that whole pool of boys from three grade levels combined.
What do we know about the 3 individual grades that combine to make the pool?
•
3rd grade: 10m50s is exactly the 50th percentile running time. So, according to the given info,
EXACTLY HALF of 3rd graders run faster times than this.
•
4th grade: 10m50s is beyond 95th percentile running time. So
MORE THAN 95% of boys in 4th grade run faster than this time.
•
5th grade: 10m50s is exactly the 50th percentile running time. So
MORE THAN 95% of boys in 5th grade run faster than this time.
The OVERALL probability here, therefore, is some weighted average of 50%, something greater than 95%, and something else greater than 95%. This definitely has to be greater than 50% overall, so the first statement
IS satisfied.
Quote:
If he runs at least a 10 minute kilometer run (no slower), the probability that his height is up to 73cm is no greater than 5%.
This statement doesn't involve grade level, so we can disregard the table this time. We're only going to use the graph.
10 minutes is equal to 600 seconds."No slower" means we need to look at the part of the graph to the LEFT of 600 seconds.
Throughout that region of the graph, all of the curves are above 73cm the whole time (see previous comments about the utter ridiculousness of this height in the real world...)
The bottom curve is the 5th percentile, so the data points for 95% of the boys lie above that curve at any given x value. Since this bottom curve sits entirely above 73cm in the desired region,
this statement is ALSO TRUE.
Quote:
If he runs an 8 minute kilometer run, he is faster than at least 95% of boys of his same grade level.
This time our statement isn't about probability, and is instead about the boy's
INDIVIDUAL GRADE LEVEL. Therefore, this statement is only "True" if it works for ALL FIVE GRADE LEVELS individually.
But if we look at the 5th graders, we see that 15% of them run times faster than 7min 53sec. This statement is therefore false if our boy happens to be in fifth grade, so
it is not supported overall.
Yes, yes, no.
3. A boy, John, selected at random from a model population runs a 10:50 kilometer. Another randomly selected boy, Peter, runs a 10:00 kilometer. Both boys measured 80cm tall. The difference in their percentiles in height among boys who run their respective speeds is most closest toThis part is more concrete than the preceding ones, because we can actually find the specific values for both halves of the comparison (quite the contrast with Data Sufficiency, where, if you're asked for a difference or other relative value, you probably
won't be able to find all of the individual component values separately).
John's time is 650 seconds. We go to x = 650 sec and locate a height of y = 80 cm—which lands pretty much right on the 85th percentile curve.
JOHN = 85th percentilePeter's time is 600 seconds. We go to x = 600 sec and locate a height of y = 80 cm—which lands slightly UNDER the 50th percentile curve.
PETER = ≈40th percentile (we can't read this one with much accuracy; all we know is "somewhere between 15th and 50th percentiles").
But with these choices...
Quote:
A. 15%
B. 25%
C. 30%
D. 35%
E. 40%
... that's enough to give us E, because we'd get D by landing ON BOTH the 85th and 50th percentile curves (respectively for John and Peter). so we know it's more than D, and only E does that.
"Most closest"? Does the problem actually... say that? •_______•
Other 3 problems will follow in another post, just to keep a lid on the total volume in each post. Ron