Chapter 6.2, Problem 69E

To determine

Probability for the total team time in the 400 − meter freestyle relay for a randomly selected race is less than 220 seconds.

Answer to Problem 69E

The probability for the total team time in the 400 − meter freestyle relay is less than 220 seconds is 0.2236.

Explanation of Solution

Given information:

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:

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

Calculations:

If 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,

Mean of total time is the sum of mean of times for each swimmer:

  ${\mu }_{X_1+X_2+X_3+X_4}={\mu }_{X_1}+{\mu }_{X_2}+{\mu }_{X_3}+{\mu }_{X_4}=55.2+58.0+56.3+54.7=224.2\text{seconds}$

We know that

Standard deviation is the square root of Variance:

  ${\sigma }_{X_1+X_2+X_3+X_4}=\sqrt{{\sigma }_{X_1}^2+{\sigma }_{X_2}^2+{\sigma }_{X_3}^2+{\sigma }_{X_4}^2}=\sqrt{{\left(2.8\right)}^2+{\left(3.0\right)}^2+{\left(2.6\right)}^2+{\left(2.7\right)}^2}\approx 5.5579\text{seconds}$

Calculate the z − score:

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

Use Table − A to find the corresponding probability:

  $P\left(X<220\right)=P\left(z<-0.76\right)=0.2236$

Thus,

For a randomly selected race, theprobability for the total team time in the 400 − meter freestyle relay is less than 220 seconds is 0.2236.