Question

Suppose 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. suppose that to qualify for the nationals, a woman must complete the 200-meter backstroke in less than 128 seconds. what proportion of competitive female swimmers will qualify for the nationals? give your answer to four (4) decimal places.

Answer 1

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:


`P(X\ \textless \ x)=P\left(z\ \textless \ \frac{x-\bar{x}}{\sigma} \right)`

Thus, the probability that a woman completes the 200-meter backstroke in less than 128 seconds is given by:


`P(X\ \textless \ 128)=P\left(z\ \textless \ (128-141)/(7) \right)=P(z\ \textless \ -1.857)=0.03165`

Therefore, the proportion of competitive female swimmers who will qualify for the nationals is 0.0317 to 4 decimal places.