Q. 11 T2.11. As part of the President'... [FREE SOLUTION]

91影视

The Practice of Statistics for AP

David Moore,Daren Starnes,Dan Yates

$Math Studyset 91影视 Explanations$ Math

4th Edition 路 809 pages

ISBN: 9781319113339

Chapter 2: Q. 11 (page 139)

T2.11. As part of the President's Challenge, students can attempt to earn the Presidential Physical Fitness Award or the National Physical Fitness Award by meeting qualifying standards in five events: curl-ups, shuttle run, sit and reach, one-mile run, and pull-ups. The qualifying standards are based on the $1985$ School Population Fitness Survey. For the Presidential award, the standard for each event is the $85 t h$ percentile of the results for a specific age group and gender among students who participated in the $1985$ survey. For the National award, the standard is the $50 t h$ percentile. To win either award, a student must meet the qualifying standard for all five events.
Jane, who is $9$ years old, did $40$ curl-ups in one minute. Matt, who is $12$ years old, also did $40$ curl-ups in one minute. The qualifying standard for the Presidential award is $39$ curl-ups for Jane and $50$ curl-ups for Matt. For the National award, the standards are $30$ and $40$ , respectively.
(a) Compare Jane's and Matt's performances using percentiles. Explain in language simple enough for someone who knows little statistics to understand.
(b) Who has the higher standardized value ( $z$ -score), Jane or Matt? Justify your answer.

Short Answer

Expert verified

(a) Jane's performance was better.

(b) Jane has the highest standardized score.

Step by step solution

Part (a) Step 1: Given information

Jane, is $9$ years old, did $40$ curl-ups in one minute. Matt, is $12$ years old, also did $40$ curl-ups in one minute. The qualifying standard for the Presidential award is $39$ curl-ups for Jane and $50$ curl-ups for Matt.

Part(a) Step 2: Explanation

The President's award falls into the $85^{t h}$ percentile. In addition, the National award is in the $50^{t h}$ percentile. For Jane, $40$ curl-ups is higher than $39$ curl-ups (the normal Presidential award), so she achieved greater than the $85^{t h}$ percentile, which means she outperformed $85$ percent of all $9$ -year-olds.

For Matt, $40$ curl-ups is at the $50^{t h}$ percentile (the requirement for the National award is $40$ ), therefore he scored at the 50th percentile, which means he outperformed $50 %$ of all children aged $12$ years old.

Jane outperformed Matt (in comparison to other youngsters) because she scored above the $85^{t h}$ percentile, whilst Matt scored below the $50^{t h}$ percentile.

Part (b) Step 1: Given information

The standard for each event is the $85 t h$ percentile of the results for a specific age group and gender among students who participated in the $1985$ survey. And the National award, the standard is the $50 t h$ percentile.

Part (b) Step 2: Explanation

Jane had completed more curl-ups than the required number of curl-ups for the Presidential Award for 9-year-olds. Jane, on the other hand, had more curl-ups than the 85th percentile.

Matt had the same amount of curl-ups as the National Award for 12 year old's qualifying level. As a result, Matt scored in the 50th percentile.

The 85th percentile is significantly higher than the 50th percentile. As a result, the standardized score that corresponds to the 85th percentile is also higher than the standardized score that corresponds to the 50th percentile.

As a result, Jane scores better (higher) than the 85th percentile, and since Matt scored in the 50th percentile, Jane's standardized score must be higher than Matt's.