Question

Ace! Professional tennis player Novak Djokovic hits the ball extremely hard. His first-

serve speeds follow an approximately Normal distribution with mean $115miles$per hour

(mph) and standard deviation $6mph$. Choose one of Djokovic’s first serves at random. Let

Y = its speed, measured in miles per hour.

a. Write the event “speed is between 100 and 120 miles per hour” in terms of Y. Then find

this probability.

b. Find the value of c such that $P(Y\le c)=0.15$

Short answer

a) The probability for the event “speed is between 100 and 120 miles per hour” in terms of Y is $79.05\%$.

b) The units of the data values are miles per hour and thus$c=108.76$.

Step-by-step solution

Step 1. Given information. 

We have given values of mean$\mu =115$ standard deviation $\sigma =6$and $x=100or120$

Let $Y$represents the speed.

 a) Step 2. To find the probability. 

The z- score is the value decreased by the mean, divided by the standard deviation .

$$\begin{gathered} z=\frac{x-\mu }{\sigma }=\frac{100-115}{6}=-2.50 \\ z=\frac{x-\mu }{\sigma }=\frac{120-115}{6}\approx 0.83 \end{gathered}$$

Now,

$$\begin{gathered} P(100<Y<120)=P(-2.50<Z<0.83) \\ =P(Z<0.83)-P(Z<-2.50) \\ =0.7967-0.0062 \\ =0.7905 \\ =79.05\% \end{gathered}$$

b) Step 3.To find the value of c. 

We have,

$$\begin{gathered} P(Y\le c)=0.15 \\ z=-1.0+.04 \\ z=-1.04 \end{gathered}$$

Now,

$$\begin{gathered} z=\frac{x-\mu }{\sigma }=\frac{c-115}{6} \\ \frac{c-115}{6}=-1.04 \\ c-115=-1.04\left(6\right) \\ c=115-1.04\left(6\right) \\ c=108.76 \end{gathered}$$