Given that the
time to complete a 200-meter backstroke swim for female competitive
swimmers is normally distributed with a mean μ = 141 seconds and a
standard deviation σ = 7 seconds.
The probability of a normally distributed data is given by:
[tex]P(X\ \textless \ x)=P\left(z\ \textless \ \frac{x-\bar{x}}{\sigma} \right)[/tex]
Thus, the probability that a woman completes the 200-meter backstroke in less than 128 seconds is given by:
[tex]P(X\ \textless \ 128)=P\left(z\ \textless \ \frac{128-141}{7} \right)=P(z\ \textless \ -1.857)=0.03165[/tex]
Therefore, the proportion of competitive female swimmers who will qualify for the nationals is 0.0317 to 4 decimal places.
