Questions: Times for a surgical procedure are normally distributed. There are two methods. Method A has a mean of 21 minutes and a standard deviation of 5 minutes, while method B has a mean of 25 minutes and a standard deviation of 2.5 minutes (a) Which procedure is preferred if the procedure must be completed within 23 minutes? Method A Method B Either Method (b) Which procedure is preferred if the procedure must be completed within 32.0 minutes? Method A Method B Either Method (c) Which procedure is preferred if the procedure must be completed within 29 minutes? Method A Method B Either Method

Solution Steps

For Method A, the probability of completing the procedure within 23 minutes is given by:

\[ P_A = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.4) - \Phi(-\infty) = 0.6554 \]

For Method B, the probability is:

\[ P_B = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-0.8) - \Phi(-\infty) = 0.2119 \]

Comparing the probabilities:

\[ P_A = 0.6554 > P_B = 0.2119 \]

Thus, the preferred method for completion within 23 minutes is Method A.

For Method A, the probability of completing the procedure within 32 minutes is:

\[ P_A = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(2.2) - \Phi(-\infty) = 0.9861 \]

For Method B, the probability is:

\[ P_B = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(2.8) - \Phi(-\infty) = 0.9974 \]

Comparing the probabilities:

\[ P_A = 0.9861 < P_B = 0.9974 \]

Thus, the preferred method for completion within 32 minutes is Method B.

For Method A, the probability of completing the procedure within 29 minutes is:

\[ P_A = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.6) - \Phi(-\infty) = 0.9452 \]

For Method B, the probability is:

\[ P_B = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.6) - \Phi(-\infty) = 0.9452 \]

Since both probabilities are equal:

\[ P_A = P_B = 0.9452 \]

Thus, the preferred method for completion within 29 minutes is either method.

Final Answer