T-stat vs. Z-stat Calculation - Sampling

Q

QCAce

#1
I thought this would be a good question for the stats experts at the Cove. :agree1:

We've tested 20 samples of slow turning motors that we buy for fourteen days, and we've come up with an average # of degrees per day of rotation for each motor.

What I would like to do is present to Engineering a 99% confidence interval range of where I feel these motors run, based on these 20 samples, (see attached data).

When I perform the calculations using a t value of 2.861, (.995 @ 19 DOF), I get a 99% confidence range from 25.70 to 25.75 deg/day.
If I assume the stdev of my sample is equal to the population stdev, however, and I use a Z table I get a 99% confidence range from 25.62 to 25.83 deg /day.

I feel much more comfortable with the Z calculated range than I do the t calculation. Can anyone tell me why? and am I justified to report the Z calculated range?

Thanks in advance.........
 

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J

JAltmann

#2
Re: T-stat vs Z-stat Calculation

You should be using the "t" values, since you are using a small sample size, sample size<30. The Z values so a greater range because the Z-calcualtion is meant for use in populations or large sample groups. The t valuse are for smaller sample sizes and hence offer a lesser confidence interval to help in aiding to make proper decisions about the population.
 

Steve Prevette

Deming Disciple
Staff member
Super Moderator
#3
Re: T-stat vs Z-stat Calculation

JAltman is correct - usually 25 to 30 is the cutoff to use t vs. z. What happens is that you are using the data to estimate both the mean and the standard deviation. The z distribution does a good job of capturing the uncertainty in estimating the mean, but not the standard deviation. "Student" developed the t distribution because it was seen at small sample sizes that the estimation of the variation (size of the confidence interval) was too small. Thus the "t" curve is a little fatter in its shape than the normal distribution.
 

Bev D

Heretical Statistician
Staff member
Super Moderator
#4
Re: T-stat vs Z-stat Calculation

Exactly

I would pose one question tho: are you sure you want the confidence interval of the mean? This is an estimate of the accuracy of the estimate of the mean. It says nothing about the individual values. I ask because I've seen a lot of people who wanted to predict the behavior of the individual values by their average disregrading what could be fairly large individual variation. You also have a reference for the upper and lower confidence limits as "USpec" and "LSpec". You also have one extreme value in your data set...
 
Q

QCAce

#5
Re: T-stat vs Z-stat Calculation

I think where I may get confused is in my use (or possibly improper use) of the equation for the t statistic:

t = ( avg - spec ) / ( s / sqrt(n) ) or (spec - avg) / ( s / sqrt(n) )

where
t = 2.861
avg = sample average (25.72)
spec = spec limit I'm looking to calculate
s = sample std deviation (.039316)
n = sample size (20)

Solving for spec I get a 99% confidence range of 25.70 - 25.75.
But my actual motor readings range from 25.68 - 25.85 ?

So what is the confidence range telling me? That if I continue to sample 20 motors the average of those 20 motors will fall withing 25.70-25.75?
How would I properly estimate the confidence interval for the total population of motors?

Thanks again.....
 

Bev D

Heretical Statistician
Staff member
Super Moderator
#6
clarifying question: are you looking for a spec on the INDIVIDUAL MOTORS or on the average of any given set of motors????

Please note that at some point I - and no doubt others - will come back and tell you that spec limits don't come from the parts they come from the design.
 
S

stefanhg

#7
Hi Andy,

The resulting distribution is not normal, more likely asymmetrical distribution – see the file.

You need to use distribution fitting software to estimate the confidence interval for the total population.
 

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Bev D

Heretical Statistician
Staff member
Super Moderator
#8
The distribution of this sample is not Normal, but since we don't know the pedigree of the sample (was it really random - homogenous and representative - from the population) WE can make no inferences about the population.

The real question at this point - and the one I typically ask at work when someone comes asking me about how to perfrom a specific statistical analysis or what the results mean: what are trying to do? What larger question are you trying to answer with the analysis....

Let's start with a simple premise: spec limits come from the Design's intent to meet the Customer requirements.

IF, Andy's engineers are attempting to use an off the shelf motor (for which they will have no control over the quality) and they intend to design their product around the motor's actual performance, then Andy's question and folowing comments makes sense. ...and his analysis may be flawed

Andy: what are attempting to do?
 
Q

QCAce

#9
The distribution of this sample is not Normal, but since we don't know the pedigree of the sample (was it really random - homogenous and representative - from the population) WE can make no inferences about the population.

The real question at this point - and the one I typically ask at work when someone comes asking me about how to perfrom a specific statistical analysis or what the results mean: what are trying to do? What larger question are you trying to answer with the analysis....

Let's start with a simple premise: spec limits come from the Design's intent to meet the Customer requirements.

IF, Andy's engineers are attempting to use an off the shelf motor (for which they will have no control over the quality) and they intend to design their product around the motor's actual performance, then Andy's question and folowing comments makes sense. ...and his analysis may be flawed

Andy: what are attempting to do?
Bev,

Your assumption is correct. These are off-the-shelf motors we are purchasing. Engineering is telling me that they are fine with 25.66 +/- 1.0 deg/day as a specification. Based on the motors I've tested I would say we appear to be well within that specification.

But there would be value in knowing, with some level of confidence, what we could expect to receiving as far as performance (rate in deg/day) of the motors. So based on the 20 samples I've measured, are there any inferences I can make about the population of motors we've received?
 

Bev D

Heretical Statistician
Staff member
Super Moderator
#10
Sometimes statistics doesn't add value to what you see in the graph.

The 20 motors are all bunched at just above the spec level you gave and are no where near the upper or lower limit. even the 1 'extreme' value is uninteresting given the spec limits.

Of course, not knowing the pedigree of the motors (all from one lot?) we can't make any inferences about lot to lot or time to variation at the supplier making the overall perfromance variation larger than what you've seen. That would be my caveat to any presentation of theis data.

For those peopel who insist on soem statistics - you could give them the average and standard deviation as well as the +/- 3 stanadard deviation limits about the mean. The data isn't really Normal but it's spread is so small compared to the spec limits there is no statistical formula that would change your interpretation of the data at this point.

That said I wouldn't give a false indication that the motors will always behave like these 20 - don't let go and set their design based on a tighter variation than the supplier is publishing (I would assume that there is some justification for those limits being that wide and that you got a tight little distribution in your sample of 20 becase they were all made at the same time from the same raw material / component batches. I would expect that you will see greater variation in the future...
 
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