Question

An elementary school class ran one mile with a mean of 12 minutes and a standard deviation of three minutes. Rachel, a student in the class, ran one mile in nine minutes. A junior high school class ran one mile with a mean of nine minutes and a standard deviation of two minutes. Kenji, a student in the class, ran 1 mile in 7.5 minutes. A high school class ran one mile with a mean of eight minutes and a standard deviation of four minutes. Nedda, a student in the class, ran one mile in ten minutes.

(a) Why is Kenji considered a better runner than Nedda, even though Nedda ran faster than he? (Round your standard deviations to two decimal places.)
A) Kenji's time for one mile was standard deviations smaller than the mean of his class, and Nedda's time was standard deviations larger than her class.
B) Kenji's time for one mile was standard deviations larger than the mean of his class, and Nedda's time was standard deviations smaller than her class.
C) Kenji's time for one mile was standard deviations smaller than the mean of his class, and Nedda's time was standard deviations smaller than her class.
D) Kenji's time for one mile was standard deviations larger than the mean of his class, and Nedda's time was standard deviations larger than her class.

(b) Who is the fastest runner with respect to his or her class? Explain why. (Round your standard deviation to two decimal places.)
A) Rachel in the elementary school class; her time was standard deviations smaller than the mean of her class.
B) Kenji in the junior high school class; his time was standard deviations smaller than the mean of his class.
C) Nedda in the high school class; her time was standard deviations smaller than the mean of her class.
D) Kenji in the junior high school class; his time was standard deviations larger than the mean of his class.

Answer 1

Kenji is considered a better runner due to his relative performance within his class's distribution, and Rachel is the fastest relative to her class, based on how her time compares to her classmates using standard deviations.

Step-by-step explanation:

Comparing the running times of students in different classes requires understanding the concept of the standard deviation and its role in measuring how individual values relate to the mean of a dataset.

For part (a), Kenji is considered a better runner than Nedda even though Nedda ran faster than Kenji because his time is closer to the mean of his class when considering the standard deviation. Kenji's time was 1.25 standard deviations below his class mean (9 minutes - 7.5 minutes = 1.5 minutes; 1.5 minutes / 2 minutes = 0.75 standard deviations), while Nedda's time was 0.5 standard deviations above the mean of her class (10 minutes - 8 minutes = 2 minutes; 2 minutes / 4 minutes = 0.5 standard deviations).

For part (b), Rachel is the fastest runner with respect to her class because her time was the furthest below her class mean in terms of standard deviations. Rachel's time was 1 standard deviation below her class mean (11 minutes - 8 minutes = 3 minutes; 3 minutes / 3 minutes = 1 standard deviation). This concept is crucial in statistics, as it allows us to compare individual performance across different datasets.