To determine who had the more convincing victory, we need to calculate the z-scores for both Alex's and Sally's finishing times in their respective races.

Formula for Z-Score:

$z = \frac{x - \mu}{\sigma}$ Where:

• $x$ is the individual score (finishing time in this case),
• $\mu$ is the mean,
• $\sigma$ is the standard deviation.

1. Alex's Z-Score:

For Alex, we have:

• $x = 184.98$ minutes,
• $\mu = 185.84$ minutes,
• $\sigma = 0.377$ minute.

Now, calculating Alex's z-score: $z_A = \frac{184.98 - 185.84}{0.377} = \frac{-0.86}{0.377} \approx -2.28$

2. Sally's Z-Score:

For Sally, we have:

• $x = 111.24$ minutes,
• $\mu = 111.6$ minutes,
• $\sigma = 0.146$ minute.

Now, calculating Sally's z-score: $z_S = \frac{111.24 - 111.6}{0.146} = \frac{-0.36}{0.146} \approx -2.47$

Interpretation:

• Z-scores indicate how many standard deviations away a score is from the mean.
• The lower the z-score, the further away from the mean (in this case, faster relative to the average) the individual is.

Since Sally's z-score (-2.47) is lower than Alex's z-score (-2.28), Sally's performance was more statistically exceptional, meaning Sally had the more convincing victory.

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Would you like any further details or have any questions?

Related Questions:

1. What does a negative z-score signify in this context?
2. How do z-scores help in comparing performances across different distributions?
3. Can we use z-scores to compare athletes from different sports or competitions?
4. What would a z-score of 0 indicate in a race?
5. How does the standard deviation affect the interpretation of z-scores?

Tip:

Z-scores are particularly useful when comparing performances across different events with varying averages and spreads, as they standardize different datasets.