Question

Swim team Hanover High School has the best women’s swimming team in the region. The 400-meter freestyle relay team is undefeated this year. In the 400-meter freestyle relay, each swimmer swims 100 meters. The times, in seconds, for the four swimmers this season are approximately Normally distributed with means and standard deviations as shown. Assume that the swimmer’s individual times are independent. Find the probability that the total team time in the 400-meter freestyle relay for a randomly selected race is less than 220 seconds.

Short answer

The probability of a total team time of fewer than 220 seconds in the 400-meter freestyle relay is $0.2236.$

Step-by-step solution

Given Information

The Mean and Standard deviation,

For Wendy:

${\mu }_{X_1}=55.2\text{seconds,}{\sigma }_{X_1}=2.8\text{seconds}$

For Jill:

${\mu }_{X_2}=58.0\text{seconds,}{\sigma }_{X_2}=3.0\text{seconds}$

For Carmen:

${\mu }_{X_3}=56.3\text{seconds,}{\sigma }_{X_3}=2.6\text{seconds}$

For Latrice:

${\mu }_{X_4}=54.7\text{seconds,}{\sigma }_{X_4}=2.7\text{seconds}$

Calculation

When Both X and Y are independent,

Property mean:

${\mu }_{aX+bY}=a{\mu }_X+b{\mu }_Y$

Property variance:

${\sigma }_{aX+bY}^2=a^2{\mu }_X^2+b^2{\mu }_Y^2$

Now,

The sum of each swimmer's mean times is the mean of overall time:

$\begin{array}{rcl}{\mu }_{X_1+X_2+X_3+X_4}& =& {\mu }_{X_1}+{\mu }_{X_2}+{\mu }_{X_3}+{\mu }_{X_4}\\ & =& 55.2+58.0+56.3+54.7\\ & =& 224.2\text{seconds}\end{array}$

We are aware of this.

The square root of variance is standard deviation:

$\sigma X_1+X_2+X_3+X_4=\sqrt{{\sigma }_{X_1}^2+{\sigma }_{X_2}^2+{\sigma }_{X_3}^2+{\sigma }_{X_4}^2}=\sqrt{(2.8{)}^2+(3.0{)}^2+(2.6{)}^2+(2.7{)}^2}$

$\approx 5.5579$seconds

To Calculate the z - score:

$z=\frac{x-\mu }{\sigma }=\frac{220-224.2}{5.5579}\approx -0.76$

To find the relevant probability, use Table - A:

$\begin{array}{rcl}P(X& <& 220)=P(z<-0.76)\\ & =& 0.2236\end{array}$

Thus,

The likelihood that the total team time in the 400-meter freestyle relay is fewer than 220 seconds in a randomly selected race is $0.2236$.