Official Explanation
1. 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.
Using the table in Height-for-age standards, the 50th percentile height for boys aged 3 years 3 months is 98.0 cm, so 50% of boys this age are taller than this. For boys aged 3 years 6 months and older, the 50th percentile height is greater than 98.0 cm, so at least 50% of boys this age are taller than this. Moreover, for boys 4 years 0 months and older, the 15th percentile height exceeds 98.0 cm, so at these ages, at least 85% of boys are taller than 98.0 cm. Thus, for a boy selected at random from a model population whose age is greater than 3 years 3 months, the probability that his height is at least 98.0 cm is greater than 50%.
The correct answer is Yes.
Using Weight-for-height standards, the 3rd percentile weight for height 105 cm is just over 14 kg, and for heights greater than 105 cm, the 3rd percentile weight is greater than 14 kg. Thus, at most, 3% of the population of boys at least 105 cm tall has a weight of 14.0 kg or less. In particular, at most 3% of the population of boys at least 105 cm tall has a weight of exactly 14.0 kg.
The correct answer is Yes.
Using Height-for-age standards, the 85th percentile in height for boys 5 years 0 months is 114.8 cm. This means that a boy 114.8 cm tall is as tall or taller than 85% of boys aged 5 years 0 months. It does not follow that a boy 114.8 cm tall is necessarily taller than 85% of boys aged 5 years 0 months, so it is not necessarily true of a boy 114 cm tall, either.
The correct answer is No.
2. Consider an individual boy from a model population. Suppose that from age 2 through age 5, this boy's weight is at the 97th percentile for his height and his height is at the 97th percentile for his age. Which one of the following statements must be true of the boy at age 5 years 0 months?
Neither Height-for-age standards nor Weight-for-height standards provide a distribution of age on the basis of weight, height, or any other variable, so it is impossible to determine where the boy's age falls in the distribution of age for height (choice A).
The boy's weight is less than that of 97% of boys his height, and his height is less than that of 97% of boys his age, but it is impossible to tell where his weight falls among the weights of boys his age (choice B).
Likewise, knowing height-for-weight percentiles does not provide information about weight-for-height percentiles (choice C).
However, it is possible to approximate his height at each age and his weight at each height; thus it is possible to determine his weight at each age. Height-for-age standards indicates that at age 2 years 0 months, the boy was 92.9 cm tall, and at age 5 years 0 months, he was 118.7 cm tall. Weight-for-height standards indicates that the 97th percentile weight for boys 92.9 cm tall is approximately 16 kg, while the 97th percentile weight for boys 118.7 cm tall is between 26 kg and 27 kg. Thus his weight at age 5 years 0 months is roughly between 26/16 = 162 .5% and 27/16 = 168 .75%, or approximately 166%, of his weight at age 2 years 0 months.
The correct answer is D.
3. B is a boy aged 4 years 3 months whose height is 110 cm and whose weight is 19 kg. For each of the following statements, select Yes if, based on the given information, it must be true of B relative to a model population. Otherwise, select No.
Neither Height-for-age standards nor Weight-for-height standards provide a distribution of height on the basis of weight, so it is impossible to determine the proportion of boys weighing 19 kg whose height is less than 110 cm.
The correct answer is No.
Using the table in Height-for-age standards, the 85th percentile in height for boys aged 4 years 3 months is 109.5 cm, so (100 − 85)%, or 15%, of boys aged 4 years 3 months are taller than 109.5 cm. Thus, at most 15% of boys of this age are taller than B's height of 110 cm.
The correct answer is Yes.
Using the table in Height-for-age standards, the 50th percentile in height for boys aged 5 years 0 months is 110.0 cm, which is the same height as B. Therefore, B's height is greater than or equal to that of (100 − 50)%, or 50%, of boys aged 5 years 0 months.
The correct answer is Yes.
4. B is a boy aged 4 years 3 months whose height is 110 cm and whose weight is 19 kg. For each of the following statements, select Yes if, based on the given information, it must be true of B relative to a model population. Otherwise, select No.
According to the graph in Weight-for-height standards, the 15th percentile in weight for a boy of height 110 cm is approximately 17 kg. Therefore, 17 kg is greater than or equal to the weight of 15% of boys of that height. But 17 kg is less than B's weight of 19 kg, so the percentage of boys whose weight is less than or equal to B's weight may be somewhat larger than 15%. Consequently, at least 15% of boys at B's height have a weight that is less than or equal to that of B.
The correct answer is Yes.
B's height is 110 cm, and 10% of 110 is equal to 11, so a boy is within 10% of B's height provided that he is between 99 cm and 121 cm tall. According to Height-for-age standards, for boys aged 4 years 3 months, 100.5 cm is the 15th percentile in height and 113.1 cm is the 97th percentile in height. Therefore, (97 − 15)%, or 82% of boys B's age have heights between 100.5 cm and 113.1 cm. Since 99 < 100.5 and 113.1 < 121, the heights of all these boys are within 10% of B's height, so at least 82%—and hence at least 80%—of boys B's age have heights within 10% of B's height.
The correct answer is Yes.
According to Height-for-age standards, for boys aged 4 years 0 months, the 97th percentile in height is 111.2 cm, so B's height of 110 cm is less than that of at least 3% of boys this age. If B's height were also less than that of at most 3% of boys this age, then 110 cm would have to be the 97th percentile in height for boys aged 4 years 0 months, which it is not.
The correct answer is No
5. 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.
Using Height-for-age standards for boys aged 4 years 0 months, the 15th percentile in height is 99.0 cm. Thus, 15% of boys of this age are 99.0 cm tall or shorter. Note that the 3rd percentile in height for boys of this age is 95.4 cm, so 3% of boys of this age are at this height or shorter. Therefore, (15 − 3)%, or 12% of boys aged 4 years 0 months have heights greater than 95.4 cm and less than or equal to 99.0 cm. The proportion of boys of this age whose height is exactly 99.0 cm must therefore be less than or equal to 12%, so the indicated probability is also less than or equal to 12%.
The correct answer is No.
Using Height-for-age standards, the 3rd percentile in height for ages from 2 years 0 months through 5 years 0 months is at least 81.4 cm. This means that a boy from a model population whose height is 81.4 cm is shorter than at least 97% of boys his age. Thus, a boy who is 81 cm tall is shorter than at least 97% of boys his age and thus shorter than at least 95% of boys his age.
The correct answer is Yes.
Weight-for-height standards provide a weight distribution for boys 120 cm tall; in particular, 97% of boys at this height weigh more than 18 kg. But a randomly selected boy at this height could conceivably weigh much less than 18 kg. Moreover, even if the boy's weight were known, a percentile distribution of weight for boys age 2 years 6 months is lacking, so it is impossible to answer the question based on the information provided.
The correct answer is No.
6. Consider an individual boy from a model population. Suppose that from age 2 through age 5, this boy’s weight is at the 50th percentile for his height and his height is at the 50th percentile for his age. Which one of the following statements must be true of the boy at age 5 years 0 months?
Neither Height-for-age standards nor Weight-for-height standards provide a distribution of age on the basis of weight, height, or any other variable, so it is impossible to determine where the boy's age falls in the distribution of age for height (option A). The boy's weight is less than that of 50% of boys his height, and his height is less than that of 50% of boys his age, but it is impossible to tell where his weight falls among the weights of boys his age (option B). Likewise, knowing height-for-weight percentiles does not provide information about weight-for-height percentiles (option C).
What can be approximated, however, is his height at each age and his weight at each height; thus, it is possible to know his weight at each age. Height-for-age standards indicates that at age 2 years 0 months, the boy was 87.1 cm tall, and at age 5 years 0 months, he was 110.0 cm tall. Weight-for-height standards indicates that the 50th percentile weight for boys 87.1 cm tall is approximately 12 kg, while the 50th percentile weight for boys 110.0 cm tall is approximately 18 kg. Thus, his weight at age 5 years 0 months is approximately 18/12 (100)%, or 150%, of his weight at age 2 years 0 months.
The correct answer is D.