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 seven 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 8.5 minutes. A high school class ran one mile with a mean of seven minutes and a standard deviation of four minutes. Nedda, a student in the class, ran one mile in eight minutes. Required:
a. Why is Kenji considered a better runner than Nedda, even though Nedda ran faster than he?
b. Who is the fastest runner with respect to his or her class? Explain why.

a) Since Kenji's time has a lower Z-score's than Nedda, he is considered a better runner relative to his competition.

b) Since her time has the lowest Z-score, Rachel is the fastest runner with respect to her class.

Step-by-step explanation:

Z-score:

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

a. Why is Kenji considered a better runner than Nedda, even though Nedda ran faster than he?

We have to see their z-scores.

Whoever has the lower z-score is in the lower percentile of times, that is, runs faster.

Kenji:

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 8.5 minutes.

So Kenji's z-score is found when . So

Nedda:

A high school class ran one mile with a mean of seven minutes and a standard deviation of four minutes. Nedda, a student in the class, ran one mile in eight minutes.

So Nedda's z-score is found when . So

Since Kenji's time has a lower Z-score's than Nedda, he is considered a better runner relative to his competition.

b. Who is the fastest runner with respect to his or her class? Explain why.

Whoever has the lower z-score.

We have the z-scores for Kenji and Nedda, and we have to find Rachel's z-score.

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 seven minutes.

So Rachel's z-score is found when

Since her time has the lowest Z-score, Rachel is the fastest runner with respect to her class.

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step-by-step explanation: