The Central Limit Theorem
Example (Identically Distributed Fixture Gaps)
A forging process is generating parts whose maximum distortion from nominal is the critical quality characteristic, X. From experience, one believes X has average 5.2 mm and standard deviation 2.1. Let Xbar5 denote the average characteristic value of five parts selected and measured each hour. Which is correct and most complete?
a. Xbar5 is normally distributed with mean 5.2 and standard deviation 0.94.
b. Xbar5 is likely approximately normally distributed with mean 5.2 and standard deviation 0.94.
c. The credibility of assuming a normal distribution for Xbar5 can be evaluated by studying the properties of several Xbar5 numbers.
d. Training issues are assignable causes because local authority can fix them.
e. All of the above are correct except a.
f. All of the above answers are correct except d.
Answer: The central limit theorem only guarantees approximate normality in the limit that n =∞.Therefore, since there is no reason to believe that X is normally distributed, e.g., it cannot be negative, there is no reason to assume that Xbar5 is exactly normally distributed. Yet, often with n = 5 approximate normality of averages holds with standard deviation approximately equal toσ0 ÷ sqrt[n] = 2.1 ÷ sqrt[5] = 0.94. Also, the credibility of this distribution assumption can be evaluated by studying many values of Xbar5, e.g., using a normal probability plot Therefore, the most complete of the correct answers is (d).
Example (Monitoring Hospital Waiting Times)
The time between the arrival of patients in an emergency room (ER) and when they meet with doctors, X, can be a critical characteristic. Assume that times are typically 20 minutes with standard deviation 10 minutes. Suppose that the average of seven consecutive patient times was 35 minutes. Which is correct and most complete?
a. A rough estimate for the probability that this would happen without assignable causes is 0.000004.
b. This data constitutes a signal that something unusual is happening.
c. It might be reasonable to assign additional resources to the ER.
d. It is possible that no assignable causes are present.
e. All of the above are correct.
Answer:
It has not been established that the averages of seven consecutive times, Xbar7, are normally distributed to a good approximation under usual circumstances. Still, it is reasonable to assume this for rough predictions. Then, the central limit theorem gives that Xbar7, under usual circumstances, has mean 20 minutes and standard deviation 10 ÷ sqrt[7] = 3.8 minutes. The chance that Xbar7 would be greater than 35 minutes is estimated to be
Pr{Z > (35 – 20) ÷ 3.8} = Pr{Z < –4.49} = 0.000004
Therefore, it might constitute a good reason to send in additional medical resources if they are available.
The answer is (e), all of the above are correct.