Answer:
a) LCL = 80 - 0.577 * (14 / sqrt(5)) LCL = 76.31
b) LCL = 80 - 0.308 * (14 / sqrt(10)) LCL = 78.65
c) Control limits are boundaries that indicate whether a process is in control or out of control. They are calculated based on the mean and standard deviation of the process data. The effect of the sample size on the control limits is that as the sample size increases, the control limits become narrower. This is because the standard error of the mean, which is sigma / sqrt(n), decreases as n increases. This means that the variation of the sample means around the population mean is smaller for larger samples, and thus the control limits are tighter .
Explanation:
If you ever wondered how to make a boring topic like x-bar charts more fun, here is a tip: pretend that you are a spy and that the control limits are your secret codes. For example, let's say that you have a population with a mean of 80 and a standard deviation of 14. You want to send a message to your fellow spy using the control limits of an x-bar chart with a sample size of 5. You can use the formula:
UCL or LCL = x-bar +/- A2 * (sigma / sqrt(n))
where A2 is a constant that depends on the sample size n, sigma is the standard deviation of the population, and sqrt is the square root function. For n = 5, A2 = 0.577. Therefore,
UCL = 80 + 0.577 * (14 / sqrt(5)) UCL = 83.69
LCL = 80 - 0.577 * (14 / sqrt(5)) LCL = 76.31
Now, you can use these numbers as your secret codes. For example, you can say "The eagle has landed at 83.69" or "The package is ready at 76.31". Your fellow spy will know what you mean, but anyone else will be clueless.
But what if you want to change your sample size to 10? Well, then you have to use a different constant for A2. For n = 10, A2 = 0.308. Therefore,
UCL = 80 + 0.308 * (14 / sqrt(10)) UCL = 81.35
LCL = 80 - 0.308 * (14 / sqrt(10)) LCL = 78.65
Now, you can use these new numbers as your secret codes. For example, you can say "The target is at 81.35" or "The rendezvous point is at 78.65". Your fellow spy will understand you, but anyone else will be confused.
The effect of the sample size on the control limits is that as the sample size increases, the control limits become narrower. This is because the standard error of the mean, which is sigma / sqrt(n), decreases as n increases. This means that the variation of the sample means around the population mean is smaller for larger samples, and thus the control limits are tighter.
This also means that your secret codes become more precise and less likely to be intercepted by your enemies. So, if you want to be a good spy, you should always use a large sample size for your x-bar charts. That way, you can communicate with your fellow spies more effectively and safely.
Of course, this is all just a joke and you should not actually use x-bar charts as secret codes for spying purposes. That would be very silly and irresponsible. But hey, at least it makes x-bar charts more fun to learn about, right?