IR Explained: Q13-18: Child Measurements

May 25, 2012

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This post is part of a series--IR Explained--that walks through the sample Integrated Reasoning questions provided in the latest edition of the GMAT Official Guide.

This set of Multi-Source Reasoning questions is very data-heavy. The first tab, "Height-for-age standards," provides a detailed table showing several heights for boys at specific ages between 2 and 5. The second tab, "Weight-for-height standards," provides a graph for the same age bracket, showing weights for boys at various heights.

Before attacking the questions, take your time to understand the information in the tabs. The table and graph are telling you quite a bit, but it is just as important to know what they are not telling you.

The height-for-age table has two variables: age (the independent variable) and height (the dependent variable). Thus, if we know a boy's age, we know the approximate likelihood that he is a certain height. Or if we know his height, we know approximately how tall he is relative to his age group.

But the process doesn't work as well in reverse. If we know, for instance, that a boy is 93cm tall, we don't know very much about the likelihood that he is a certain age. Sure, he would be a very tall 2:0 year old, a fairly tall 2:3 year old, a slightly above average 2:6 year old, and so on, but the level of precision in this direction is much lower.

The same is true with the weight-for-height graph. Here, height is the indepedent variable and weight is the dependent variable. If we know, for instance, that a boy in this age range is 120cm tall, we can determine from the graph the weights at various percentile levels. But if we know his weight, we cannot perform the same operation to find his height.

Most crucially, recognize that we have very little knowledge directly relating age to weight. If we know a boy's age, we can find the distribution of his probable height, and if we know his height, we can find the distribution of his probable weight, but if we know his age, we know next to nothing about his weight.

Question 13 jumps right in to these distinctions. We are given a boy's age, height, and weight. From there, we can find where his height falls in the distribution based on his age (a bit above the 85th percentile) and where his weight falls in the distribution based on his height (a bit above the 50th percentile).

But 13A tries to infer a distribution based on his weight, which is not found in either of the tabs. We cannot infer anything based on his weight, so 13A is No.

13B is more straightforward. The 85th percentile for a boy aged 4:3 is 109.5cm, so if this boy is 110cm tall, he is above the 85th percentile. Put another way, fewer than 15% of boys at his age are taller. 13B is Yes.

13C asks you compare this boy to boys of a different age group. In the 5:0 group, the 50th percentile is 110cm, the same as this boy. Thus, this boy's height is indeed greater than or equal to half of the boys in the older age group. 13C is Yes.

Question 14 is more general. The phrase "selected at random from a model population" isn't anything to worry about--it just means that the boy follows the rules of the model population described in the tabs.

14A asks you to consider several age groups. The 50th percentile height distribution for the 3:3 age group is 98cm. Thus, if a boy is aged 3:3, there is a 50% chance that he is at least 98cm. If he is older, the probability is higher than he is at least 98cm. 14A is Yes.

In 14B, you must look to the other tab. At 105cm tall, the 3rd percentile is approximately 14kg, meaning that there is a 3% chance that the boy weighs 14kg or less. If he is taller, the 3rd percentile is heavier, so the probability is less than 3%. 14B is Yes.

14C does not specify an age, but refers to the age-height table. Essentially, it claims that regardless of age, 114cm is the 85th percentile or higher. However, the 85th percentile of boys aged 5:0 is 114.8cm, so 114cm is below the 85th percentile. 14C is No.

Question 15 has many moving parts. We know that a certain boy is always at the 50th percentile in height based on his age (according to the age-height table) and at the 50th percentile in weight based on his height (according to the graph). Based on the discussion above about independent and dependent variables, we can quickly eliminate each of the first three choices. We don't have distributions of age based on height (choice (A))--it's the other way around. The same applies to (B) and (C).

The final two choices are testable. Choice (D) says that his weight at 5:0 is 150% of his weight at 2:0. First, we need his height at the two ages. At 2:0, his 50th percentile height is 87.1cm; at 5:0, it is 110cm. The corresponding 50th percentile weights are 12kg at 2:0 and 18kg at 5:0. 18kg is 150% of 12kg, so choice (D) is correct.

We return to multi-part problems with Question 16. The prompt is the same as in question 13, but the statements are different. 13A asks you to look at the height-weight graph. A boy 110cm who weighs 19kg is above the 50th percentile (18kg) in weight but well below the 85th percentile (21kg). Thus, at least 15% of boys at this height weigh less than B--in fact, more than half do. 16A is Yes.

16B takes us back to the age-height table. At age 4:3, 110cm falls just above the 85th percentile. 10% of 110 is 11, so any height between 99 (110-11) and 121 (110+11) is within 10% of B's height. 99cm is below the 15th percentile and 121cm is above the 97th percentile, so at least 82 percent of boys aged 4:3 have heights within 10% of B's height. 16B is Yes.

In 16C, we compare B's height to the distribution of a different age group. If he were 4:0, his height of 110cm would fall about 1cm below the 97th percentile. The statement is the same as "B's height is at or above the 97th percentile." Since B's height is lower, 16C is No.

Question 17 is another general one. 17A is too strong to be supported by the data. Percentile tables tell us the likelihood that a boy of a certain age is above or below a certain height, not the probability that he is at an exact height. 17A is No.

In 17B, we're asked to compare 81cm to the height distributions at every age. The lowest 3rd percentile height is 81.4cm, for boys aged 2:0. Thus, even at this lowest height distribution, 81cm is below the 3rd percentile, so any boy 81cm tall is shorter than at least 95% (97%, in fact) of boys his age. 17B is Yes.

Finally, consider 17C. This statement reverses the independent/dependent variable relationship of the graphs. We don't know the weight distribution of boys at a certain age, so we can't infer that a boy of any height weights more than that many boys of age 2:6. 17C is No.

We finish off this set with question 18, which hearkens back to question 15. In fact, the structure is identical, except instead of the 50th percentile for both height and weight, this boy is the 97th percentile in both. We can apply the same reasoning to the first three choices that we did with question 15: these choices confuse the dependent and independent variables of the table and graph. We have height distributions based on age, not age distributions based on height, which we would need to evaluate choice (A).

Choice (D) is one we can evaluate. At age 2:0, his 97th percentile height is 92.9cm, and his 97th percentile weight is approximately 16kg. At age 5:0, his 97th percentile height is 118.7, and his 97th percentile weight is approximately 26kg. 26/16 is approximately 1 and 5/8, or 1.63, so we've gotten very close to 166%. Choice (D) is correct.

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About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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