91影视

Given in the question that the details of the swimming team members from Hanover High School.

Let $X_i,i=1,2,3,4$be the random variables showing the times in seconds for the four swimmers this season.

Given the mean and standard deviation of $X_i$are

$\begin{array}{r}{渭}_{X_1}=55.2,{蟽}_{X_1}=2.8\\ {渭}_{X_2}=58.0,{蟽}_{X_X}=3.0\\ {渭}_{X_3}=56.3,{蟽}_{X_3}=2.6\\ {渭}_{X_4}=54.7,{蟽}_{X_4}=2.7\end{array}$

The probability that the total team time in the $400$-meter freestyle relay is less than $220$seconds must be determined.

Consider the variable $T=X_1+X_2+X_3+X_4$

It showing the total times for the four swimmers in the season.

Let's find$P(T鈮�220)$

Given that the swimmer's individual times are independent.

Mean of $T$

$$\begin{gathered} {渭}_T={渭}_{X_1}+X{渭}_{X_2}+{渭}_{X_3}+{渭}_{X_4} \\ =55.2+58.0+56.3+54.7 \\ =224.2 \end{gathered}$$

Standard deviation of $T$

$$\begin{gathered} {蟽}_T=\sqrt{{{蟽}_{X_1}}^2+{{蟽}_{X_2}}^2+{{蟽}_{X_3}}^2+{{蟽}_{X_4}}^2} \\ =\sqrt{{2.8}^2+{3.0}^2+{2.6}^2+{2.7}^2} \\ =5.56 \end{gathered}$$

The probability that the total team time in the $400$-meter freestyle relay is less than $220$seconds must be determined.

Let's standardize $T=220$

$$\begin{gathered} z=\frac{x鈭�{渭}_T}{{蟽}_T} \\ =\frac{220鈭�224.2}{5.56} \\ =鈭�0.76 \end{gathered}$$

Using a table of typical normal probabilities as a guide,

$$\begin{gathered} P(T鈮�220)=P(z鈮�鈭�0.76) \\ =P(z鈮�0.76) \\ =1鈭�P(z鈮�0.76) \\ =1鈭�0.7764 \\ =0.2236 \end{gathered}$$