11. In the distribution of heights in a group of a thousand people, height p is at the 42nd percentile, height r is at the 48th percentile, height s is at the 54th percentile, and height t is at the 60th percentile. Quantity A: r-p Quantity B: t-s
Answer: C
The two quantities are equal.
Since height r is at the 48th percentile and height p is at the 42nd percentile, the difference r - p represents the range of heights between these two percentiles. Similarly, height t at the 60th percentile and height s at the 54th percentile also provide a range of heights. Both ranges can be considered equal in terms of their percentile differences.
A) Quantity A is greater.
This option suggests that the difference between the heights at the 48th and 42nd percentiles (r - p) is greater than the difference between the heights at the 60th and 54th percentiles (t - s). However, since both differences are derived from the same distribution of height percentiles, this statement is incorrect.
B) Quantity B is greater.
This option implies that the difference between the heights at the 60th and 54th percentiles (t - s) is greater than the difference between the heights at the 48th and 42nd percentiles (r - p). Given the equal nature of the percentile differences, this statement is also incorrect.
C) The two quantities are equal.
This option accurately reflects that the difference between height r and height p is equal to the difference between height t and height s, as they are both derived from equal intervals in the percentile rankings of height.
D) The relationship cannot be determined from the information given.
This option suggests uncertainty in establishing a relationship between the two quantities. However, the defined percentiles allow for a clear comparison, making this choice incorrect.
Conclusion
The correct answer is that the two quantities are equal, as the differences in height between the respective percentiles are identical. All other options misinterpret the relationship between the differences, failing to recognize that both ranges in height represent equal intervals within the distribution.