Question

the amount of time it takes an athlete to complete a ninja warrior course is normally distributed with mean 8.25 minutes and standard deviation 0.4 minutes.
75% of athletes finish the course by what time?
7.98 minutes
8.52 minutes
8.25 minutes
8.66 minutes

Explanation:

Step1: Find the z - score

We know that if \(X\sim N(\mu,\sigma^{2})\), and we want to find \(x\) such that \(P(X\leq x)=0.75\). Using the standard normal distribution \(Z=\frac{X - \mu}{\sigma}\sim N(0,1)\). Looking up in the standard normal table (or using a calculator with a normal - distribution function), the \(z\) - score corresponding to a cumulative probability of \(0.75\) is \(z = 0.6745\).

Step2: Use the z - score formula to find \(x\)

The z - score formula is \(z=\frac{x-\mu}{\sigma}\). We are given \(\mu = 8.25\), \(\sigma=0.4\) and \(z = 0.6745\).
Substitute the values into the formula: \(0.6745=\frac{x - 8.25}{0.4}\).
Multiply both sides by \(0.4\): \(x-8.25=0.6745\times0.4\).
\(x-8.25 = 0.2698\).
Add \(8.25\) to both sides: \(x=8.25 + 0.2698=8.5198\approx8.52\).

Answer:

8.52 minutes