Times for a surgical procedure are normally distributed. There are two methods. Method A has a mean of 28 Minutes and a standard deviation of 4 minutes, while B has a mean of 32 minutes and a standard deviation of 2 minutes.

a)which procedure is preferred if the procedure must be completed within 28 minutes?
b) Thirty-eight minutes?
c) Thirty-six minutes?
Explain your reasoning?

Z = (score-mean)/SD

Compare the Z scores for both methods for each question.

a. Method A, Z = 0, Method B, Z = -2

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.

Use a similar process for b and c.

To determine which procedure is preferred given a specific time constraint, we need to compare the probability of completing the procedure within that time for each method.

a) If the procedure must be completed within 28 minutes, we need to find the probability that Method A or B will take less than or equal to 28 minutes.

For Method A:
Z-score = (28 - 28) / 4 = 0 (since the mean is 28 and the standard deviation is 4)
Probability (Z ≤ 0) = 0.5 (from the standard normal distribution table)

For Method B:
Z-score = (28 - 32) / 2 = -2 (since the mean is 32 and the standard deviation is 2)
Probability (Z ≤ -2) is approximately 0.0228 (from the standard normal distribution table)

Since the probability of completing the procedure within 28 minutes is higher for Method A (0.5) compared to Method B (0.0228), Method A is preferred in this case.

b) If the time constraint is 38 minutes, we need to find the probability of completing the procedure within that time for each method.

For Method A:
Z-score = (38 - 28) / 4 = 2.5
Probability (Z ≤ 2.5) is approximately 0.9938

For Method B:
Z-score = (38 - 32) / 2 = 3
Probability (Z ≤ 3) is approximately 0.9987

Since the probability of completing the procedure within 38 minutes is higher for Method B (0.9987) compared to Method A (0.9938), Method B is preferred in this case.

c) If the time constraint is 36 minutes, we need to find the probability of completing the procedure within that time for each method.

For Method A:
Z-score = (36 - 28) / 4 = 2
Probability (Z ≤ 2) is approximately 0.9772

For Method B:
Z-score = (36 - 32) / 2 = 2
Probability (Z ≤ 2) is approximately 0.9772

Since both methods have the same probability of completing the procedure within 36 minutes (0.9772), the preference between the two methods would depend on other factors such as cost, risk, or efficiency.

In summary:
a) Method A is preferred if the procedure must be completed within 28 minutes.
b) Method B is preferred if the procedure must be completed within 38 minutes.
c) Both methods have the same probability of completing the procedure within 36 minutes, so the preference would depend on other factors.

To determine which surgical procedure is preferred based on time constraints, we'll compare the probabilities of completing the procedure within specific time limits for both methods.

a) If the procedure must be completed within 28 minutes, we need to calculate the probability of completing the procedure within this time frame for each method.

For Method A:
Using the mean (28 minutes) and standard deviation (4 minutes) of Method A, we can calculate the z-score to find the probability.

z-score = (x - mean) / standard deviation
z-score = (28 - 28) / 4
z-score = 0 / 4
z-score = 0

Now, we can use the z-table or software to find the probability corresponding to a z-score of 0. The probability of completing the procedure within 28 minutes using Method A is the probability of having a z-score of 0, which is 0.5 or 50%.

For Method B:
Using the mean (32 minutes) and standard deviation (2 minutes) of Method B, we calculate the z-score.

z-score = (x - mean) / standard deviation
z-score = (28 - 32) / 2
z-score = -4 / 2
z-score = -2

We find the probability corresponding to a z-score of -2. According to the z-table or software, the probability is approximately 0.0228 or 2.28%.

Since the probability of completing the procedure within 28 minutes is higher for Method A (50%) compared to Method B (2.28%), Method A is preferred.

b) To find the preferred method for completing the procedure within 38 minutes, we repeat the same calculations as above.

Method A:
z-score = (38 - 28) / 4
z-score = 10 / 4
z-score = 2.5

Method B:
z-score = (38 - 32) / 2
z-score = 6 / 2
z-score = 3

Using the z-table or software, we find that the probability for Method A is approximately 0.9938 or 99.38%, while the probability for Method B is approximately 0.9987 or 99.87%.

Based on these probabilities, Method B is preferred for completing the procedure within 38 minutes.

c) For completing the procedure within 36 minutes:

Method A:
z-score = (36 - 28) / 4
z-score = 8 / 4
z-score = 2

Method B:
z-score = (36 - 32) / 2
z-score = 4 / 2
z-score = 2

Using the z-table or software, the probability for both methods is approximately 0.9772 or 97.72%.

Since both methods have the same probability of completing the procedure within 36 minutes, there is no preference between Method A and Method B.

In summary:
a) For completing the procedure within 28 minutes, Method A is preferred.
b) For completing the procedure within 38 minutes, Method B is preferred.
c) For completing the procedure within 36 minutes, there is no preference between Method A and Method B.