Question

professional tennis player novak djokovic hits the ball extremely hard. his first - serve speeds can be modeled by a normal distribution with mean 112 miles per hour (mph) and standard deviation 5 mph. find the 85th percentile of djokovics first - serve speeds. 121.1 mph 106.8 mph 117.2 mph 116.25 mph

Explanation:

Step1: Find the z - score for the 85th percentile

We know that if \(X\sim N(\mu,\sigma^{2})\), and we want to find \(x\) such that \(P(X\leq x)=0.85\). Using a standard normal table (or a calculator with a normal - distribution function, e.g., the invNorm function on a TI - 84 Plus: invNorm\((0.85,0,1)\)), the \(z\) - score \(z\) corresponding to a cumulative probability of \(0.85\) is approximately \(z = 1.04\).

Step2: Use the z - score formula \(z=\frac{x-\mu}{\sigma}\) to solve for \(x\)

We are given that \(\mu = 112\) (mean) and \(\sigma=5\) (standard deviation). Substitute \(z = 1.04\), \(\mu = 112\), and \(\sigma = 5\) into the formula \(z=\frac{x-\mu}{\sigma}\).

$$1.04=\frac{x - 112}{5}$$

Multiply both sides of the equation by \(5\): \(1.04\times5=x - 112\).

$$5.2=x - 112$$

Add \(112\) to both sides of the equation: \(x=112 + 5.2\).

Answer:

\(117.2\) mph (the third option)