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.
Now suppose the fastest 6% of female swimmers in the nation are offered college scholarships. In order to be given a scholarship, a swimmer must complete the 200-meter backstroke in no more than how many seconds?
Give your answer in whole numbers.

Answer 1

Answer:

The swimmer must complete the 200-meter backstroke in no more than 130 seconds.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 141, \sigma = 7$

Fastest 6%

At most in the 6th percentile, that is, at most a value of X when Z has a pvalue of 0.07. So we have to find X when Z = -1.555.

$Z = \frac{X - \mu}{\sigma}$

$-1.555 = \frac{X - 141}{7}$

$X - 141 = -1.555*7$

$X = 130$

The swimmer must complete the 200-meter backstroke in no more than 130 seconds.