Question

Elite Gymnastics, Men - After the 2004 Olympic games, the scoring system for gymnastics was overhauled. Rather than rank performances from 0 points to 10 points as the old system did, the new system judges routines based on a combination of the difficulty of the skills attempted in the performance and the gymnast's execution of the skills.

Male gymnasts compete on six different apparatus: Vault, Floor, Rings, Pommel Horse, Parallel Bars, and Horizontal Bar. The values given in this problem were calculated from the reported results of international competitions of male elite gymnasts between the years 1999 and 2019.

Under the old scoring system (last used in 2005), scores on the Rings apparatus had a mean of 9.146 and a standard deviation of 0.505. Under the new scoring system (first used in 2006), scores on the Rings apparatus had a mean of 14.342 and a standard deviation of 0.695.

Round all calculated answers to 4 decimal places.

At the 2005 Gymnastics World Championships under the old scoring system, Sami Aalto from Finland earned a score of 8.55 on Rings. At the 2019 Gymnastics World Championships under the new scoring system, Robert Tvorogal from Lithuania earned a score of 13.166 on Rings.

1. What is the z-score for Sami Aalto on Rings in 2005?
2. What is the z-score for Robert Tvorogal on Rings in 2019?
3. Based on the z-scores, which gymnast had a higher score, relative to the other gymnasts who competed under the scoring system used in their competition?
A. Sami Aalto
B. Robert Tvorogal
C. The gymnasts scored equally well

Answer 1

The z-score for Sami Aalto in 2005 on the Rings apparatus is -1.175. The z-score for Robert Tvorogal in 2019 on the Rings apparatus is -1.6941. Based on the z-scores, Robert Tvorogal had a higher score compared to the other gymnasts who competed under the scoring system used in their competition.

Step-by-step explanation:

1. What is the z-score for Sami Aalto on Rings in 2005?

A z-score measures how many standard deviations an individual score is from the mean. To calculate the z-score for Sami Aalto, we subtract the mean of scores on the Rings apparatus in 2005 from his score, then divide by the standard deviation. The formula is:

z = (x - mean) / standard deviation

Using the given values, the calculation is:

z = (8.55 - 9.146) / 0.505 = -1.175

2. What is the z-score for Robert Tvorogal on Rings in 2019?

Using the same formula, the calculation for Robert Tvorogal's z-score is:

z = (13.166 - 14.342) / 0.695 = -1.6941

3. Based on the z-scores, which gymnast had a higher score, relative to the other gymnasts who competed under the scoring system used in their competition?

To compare the z-scores, we look at the absolute values. Since a z-score measures how far a score deviates from the mean, a higher absolute value indicates a larger deviation. In this case, Robert Tvorogal's z-score (-1.6941) has a higher absolute value than Sami Aalto's z-score (-1.175), indicating that Robert had a higher score relative to the other gymnasts compared to Sami's score in their respective competitions.