Sorcha ran two different marathons: the Windy Marathon and the Sunny Marathon - Leaving Cert Mathematics - Question e - 2022

To find the z-score for the Windy Marathon:

$z = \frac{S - \mu}{\sigma}$ where $S$ is the finishing time, $\mu = 254$, and $\sigma = 38$.

Replacing with the proportion we found: $z = \frac{0.8775 - 1}{1.17} = 0.116$

Using the z-score relation in the Sunny Marathon, with mean finishing time $\mu = 247$ and $\sigma = 29$:

We can express Sorcha's position similarly: $z = \frac{S - 247}{29}$ And we previously found the $z$ score for the Windy Marathon to be about $1.16$. Thus: $1.16 = \frac{S - 247}{29}$

Solving for $S$, we have: $S = 1.16 \cdot 29 + 247 = 298.08$ Next, we convert this back into position in the Sunny Marathon: Using the z-value calculated, we transform it to position: $position = z \cdot 2000 = 1921.6$ Rounding gives approximately 1922, indicating Sorcha's finishing position as either 1921st or 1922nd.