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 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

To be given a scholarship, a female swimmer must complete the 200-meter backstroke in no more than 130 seconds.

Step-by-step explanation:

To find the time a female swimmer must complete the 200-meter backstroke in order to be given a scholarship, we need to find the z-score corresponding to the fastest 6% of female swimmers. Using the z-score formula, z = (x - μ) / σ, we can rearrange it to solve for x. Plugging in the given values, we have z = (x - 141) / 7. To find the z-score corresponding to the 6th percentile, we need to find the z-score that separates the lowest 6% from the rest of the distribution. This z-score can be found using standard normal distribution tables or a z-score calculator. The z-score corresponding to the 6th percentile is approximately -1.55. Plugging this value back into the z-score formula and solving for x, we get (x - 141) / 7 = -1.55. Solving for x gives us x ≈ 130.15 seconds. Therefore, a female swimmer must complete the 200-meter backstroke in no more than 130 seconds to be given a scholarship.

Answer 2

In order to be given a scholarship, a swimmer must complete the 200-meter backstroke in no more than 144 seconds.

Explanation

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 fastest 6% of female swimmers in the nation are offered college scholarships. So, the probability will be 0.06

According to the normal distribution table, the z-score for probability of 0.06 is 0.5239

Now using z-score formula z= (x- μ)/σ , we will get...

`0.5239= (x-141)/(7)\\ \\ x-141= 3.6673\\ \\ x= 3.6673+141 \\ \\ x= 144.6673`

That means,
`x < 144` (in whole number)

So, in order to be given a scholarship, a swimmer must complete the 200-meter backstroke in no more than 144 seconds.