Question 1172782: An X-bar control chart monitors the mean of a process by checking that the average stays between control.pngand control2.png. When the process is under control,
(a) What is the probability that five consecutive sample means of n cases stay within these limits?
(b) What is the probability that all of the means for 100 days falls within the control limits?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem. We're dealing with an X-bar control chart, which monitors the mean of a process.
**Understanding X-bar Control Charts**
* When a process is "under control," it means the process is operating as expected, and the sample means should fall within the control limits (upper and lower).
* We'll assume that each sample mean is independent of the others.
**a. Probability of Five Consecutive Sample Means Within Limits**
1. **Probability of One Sample Mean Within Limits:**
* When the process is under control, the probability of a single sample mean falling within the control limits is 1 (or 100%). This is because, by definition, the control limits are set to capture the expected variation when the process is in control.
2. **Probability of Five Consecutive Sample Means:**
* Since the sample means are independent, we multiply the probabilities together.
* Probability = (Probability of one sample mean within limits)^5
* Probability = 1^5 = 1
* Therefore, the probability that five consecutive sample means stay within the control limits is 1 (or 100%).
**b. Probability of All Means for 100 Days Within Control Limits**
1. **Probability of One Day's Mean Within Limits:**
* As in part (a), when the process is under control, the probability of a single day's sample mean falling within the control limits is 1.
2. **Probability of 100 Days' Means Within Limits:**
* Again, since the sample means are independent, we multiply the probabilities together.
* Probability = (Probability of one day's mean within limits)^100
* Probability = 1^100 = 1
* Therefore, the probability that all of the means for 100 days fall within the control limits is 1 (or 100%).
**Important Note:**
* These results assume that the process remains perfectly in control for the entire period. In real-world scenarios, there's always a small chance of variations that could cause sample means to fall outside the control limits, even when the process is generally stable.
* The control limits are set to capture the natural variation of the process, therefore when the process is in control, all points should fall within the limits.
|
|
|
