Chapter 6.2, Problem 63E

To determine

To calculate; the probability that the total team time in the 400 metre freestyle relay is less than 220 seconds.

Answer to Problem 63E

  $\text{0}\text{.2236}$

Explanation of Solution

Given:

  $\begin{array}{l}{\mu }_X=24\\ {\text{σ}}_X\text{ = 2}\end{array}$

  $\begin{array}{l}{\text{μ}}_{\text{aX + bY}}{\text{ = a(μ}}_{\text{X}})+b({\text{μ}}_Y)\\ {\text{σ}}^{\text{2}}{}_{\text{aX + bY}}{\text{ = a}}^{\text{2}}{\text{(σ}}^2{}_{\text{X}})+{\text{b}}^{\text{2}}{\text{(σ}}^2{}_{\text{Y}})\end{array}$

  $\begin{array}{l}{\mu }_{X_1+X_2+X_3+X_4}={\mu }_{X_1}+{\mu }_{X_2}+{\mu }_{X_3}+{\mu }_{X_4}=55.2+58.0+56.3+54.7=224.2\\ {\text{σ}}_{{}_{X_1+X_2+X_3+X_4}}\text{= }\sqrt{{\text{σ}}^2{}_{X_1}+{\text{σ}}^2{}_{X_2}+{\text{σ}}^2{}_{X_3}+{\text{σ}}^2{}_{X_4}}=\sqrt{{2.8}^2+3^2+{2.6}^2+{2.7}^2}\approx 5.5579\end{array}$

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

  $\text{P( X < 220) = P (Z < - 0}\text{.76) = 0}\text{.2236}$