Question

question 3 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. question 4

Explanation:

Step1: Find the z - score

We know that the fastest 6% of swimmers get scholarships. So we need to find the z - score corresponding to the area of \(1 - 0.06=0.94\) in the standard normal distribution table. Looking up the value in the standard - normal table, the z - score \(z\) corresponding to an area of 0.94 is approximately \(z = 1.56\).

Step2: Use the z - score formula

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(x\) is the value we want to find, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. We know \(\mu = 141\), \(\sigma = 7\), and \(z = 1.56\). Rearranging the formula for \(x\) gives \(x=\mu+z\sigma\).

Step3: Calculate \(x\)

Substitute the values into the formula: \(x = 141+1.56\times7\). First, calculate \(1.56\times7 = 10.92\). Then, \(x=141 + 10.92=151.92\). Rounding to the nearest whole number, \(x = 152\).

Answer:

152