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 $85th$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 $50th$ 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.

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.

The President's award falls into the ${85}^{th}$percentile. In addition, the National award is in the ${50}^{th}$ percentile. For Jane, $40$curl-ups is higher than $39$curl-ups (the normal Presidential award), so she achieved greater than the ${85}^{th}$percentile, which means she outperformed $85$percent of all $9$-year-olds.

For Matt, $40$curl-ups is at the ${50}^{th}$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 ${85}^{th}$ percentile, whilst Matt scored below the ${50}^{th}$percentile.

The standard for each event is the $85th$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 $50th$ percentile.

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.