Question

at a large regional collegiate women’s swim meet, an official records the time it takes each swimmer to swim 100 meters for all swimmers who compete in only one stroke category. the following table shows the mean times and corresponding standard deviations for the collegiate women at the swim meet for each of the four stroke categories.

stroke category	mean 100 meter time	standard deviation
breaststroke	63.3 seconds	0.92 seconds
butterfly	54.4 seconds	0.94 seconds
freestyle	50.2 seconds	0.76 seconds

for each of the 4 stroke categories, consider a random variable representing the time of a randomly selected swimmer in that category. what is the standard deviation of the sum of the 4 random variables?

Explanation:

Step1: Recall variance addition rule

For independent random variables \(X_1, X_2, X_3, X_4\), the variance of their sum is the sum of their variances:
\(Var(X_1+X_2+X_3+X_4) = Var(X_1) + Var(X_2) + Var(X_3) + Var(X_4)\)

Step2: Calculate each variable's variance

Variance is the square of standard deviation:

• Backstroke: \(Var(X_1) = 0.70^2 = 0.49\)
• Breaststroke: \(Var(X_2) = 0.92^2 = 0.8464\)
• Butterfly: \(Var(X_3) = 0.94^2 = 0.8836\)
• Freestyle: \(Var(X_4) = 0.76^2 = 0.5776\)

Step3: Sum all variances

\(Var(\text{Total}) = 0.49 + 0.8464 + 0.8836 + 0.5776 = 2.7976\)

Step4: Find standard deviation of sum

Standard deviation is the square root of total variance:
\(\sigma_{\text{Total}} = \sqrt{2.7976}\)

Answer:

1.673 seconds (rounded to 3 decimal places)