SOLUTION: 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 pr

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.

RELATED QUESTIONS
Please help me solve this. An X-bar control chart monitors the mean of a process by... (answered by CPhill)

A truck manufacturer monitors the width of the door seam as vehicles come off its... (answered by CPhill)

Please help me solve this. A truck manufacturer monitors the width of the door seam (answered by CPhill)

Claim: The mean time between uses of a TV remote control by males during commercials... (answered by stanbon)

An automatic filling machine is used to fill 1-liter bottles of cola. The machine�s... (answered by stanbon)

A process is in control when the average amount of instant coffee that is packed in a... (answered by Boreal)

� Suppose you are interested in how long it takes to get your food at a restaurant. Now,... (answered by Boreal)

Assume that a two-sigma limit control chart with the UCL and LCL is designed. Mean is 18... (answered by stanbon)

A recent quality control test at a local chocolate bar factory finds that the mass of the (answered by Boreal)