Question

An article in knee surgery sports traumatol arthrosc ["effect of provider volume on resource utilization for surgical procedures" (2005, vol. 13, pp. 273–279)] showed a mean time of 129 minutes and a standard deviation of 14 minutes for acl reconstruction surgery at high-volume hospitals (with more than 300 such surgeries per year).

(a) what is the probability that your acl surgery at a highvolume hospital requires a time more than two standard deviations above the mean?
(b) what is the probability that your acl surgery at a highvolume hospital is completed in less than 100 minutes?
(c) the probability of a completed acl surgery at a highvolume hospital is equal to 95% at what time?
(d) if your surgery requires 199 minutes, what do you conclude about the volume of such surgeries at your hospital? explain.

Answer 1

To find the probability of surgery time more than two standard deviations above the mean, calculate the z-score and use a standard normal distribution table. To find the probability of surgery time less than 100 minutes, calculate the z-score and use the standard normal distribution table. To find the time at which surgery is completed with a 95% probability, find the z-score associated with a 95% probability and calculate the time. If the surgery time is significantly different from the mean, it suggests a difference in the volume of ACL surgeries at the hospital.

Step-by-step explanation:

(a) Probability of ACL surgery time more than two standard deviations above the mean:

1. Calculate the z-score using the formula: z = (x - mean) / standard deviation.
2. Calculate the z-score for two standard deviations above the mean: z = (2 * standard deviation) + mean.
3. Use a standard normal distribution table or calculator to find the probability associated with the z-score obtained in step 2.

(b) Probability of ACL surgery time less than 100 minutes:

1. Calculate the z-score using the formula: z = (x - mean) / standard deviation.
2. Use a standard normal distribution table or calculator to find the probability associated with the z-score obtained in step 1.

(c) Time at which ACL surgery is completed with a 95% probability:

1. Find the z-score associated with a 95% probability using a standard normal distribution table or calculator.
2. Calculate the time using the formula: time = (z-score * standard deviation) + mean.

(d) Conclusion about the volume of ACL surgeries:

If your surgery requires 199 minutes, it is beyond two standard deviations from the mean. This suggests that the volume of ACL surgeries at your hospital may be lower than high-volume hospitals, as the mean time and standard deviation are different.