Upper and lower control limites

Question

An automatic filling machine is used to fill 1 litter booltes of cola. the machine's output is approx. normal with a mean of 1.0 liter and a standard deviation of .01 liter. output is monitored using means of samples of 25 obersvation.

Determine upper and lower control limits that will include roughly 97 percent f the sample means when the process is in control.

Given these sampel means: 1.005,1.001,.998.1.002, .995 and .999 is the process in control?

Answer 1

If the pop SD is 0.01 l then the sample SD is 0.01/√25=0.01/5=0.002 l.

From the normal distribution tables a Z value of 2.17 corresponds to an upper limit of 98.5% and a Z value of -2.17 corresponds to a lower limit of 1.5%. So, over the whole range 97% of the values should be between Z=-2.17 and +2.17. The quantity 2.17 represents the number of SD's from the mean, so the "safe" range is 1.0±2.17*0.002=1.0±0.00434 approximately. The lower limit is 1-0.00434=0.99566 and the upper limit is 1.00434. We'll settle for 0.996 and 1.004 as the limits.

So:

1.005 not in control (> 1.004)

1.001 in control

0.998  in control

1.002 in control

0.995 not in control (< 0.996)

0.999 in control

Comments

You can use the tables in two ways. The usual way is to take a Z value you've calculated and then look up the number (between 0 and 1) corresponding to the Z value. The area under the normal curve is standardised as 1, which of course is 100%. When you look up Z=2.17 you see the number 0.9850 meaning that 98.5% of the area is to the left of Z=2.17. If you look up Z=-2.17 you get 0.0150 or 1.5%. The mean cuts the normal curve into two equal halves. On the right of the mean we have all the positive Z values while on the left we have all the negative Z values. Because of symmetry 98.5% of the area on the left takes you up to Z=-2.17. Outside this range we have 1.5% (Z=-2.17) and 1.5% (Z=2.17, 100-98.5%), making the 3%=100-97 asked for in the question.

The other way to use the table is to look in the body of the table, as I did, and work back to the Z value. If we start with a different percentage we will arrive at a different Z value. All the Z value really is is a measure of how many standard deviations we are away from the mean. So Z=2.17 corresponds only to the number 0.9850. Take a look at the table (also available online) and you'll see what I mean.

I think you probably knew most of what I just mentioned, but I hope I've included the answer to your comment. And I have to say I'm no expert on statistics, so do go for a second opinion on my answer, won't you?