A manufacturer of pain relievers claims that it takes an average of
12.75 min. for a person to be relieved of headache pain after taking
it's pain reliever.  The time it takes to get relief is normally
distributed with a standard deviation of 0.5 min.  A sample of (12)
people is taken and the data arae shown below:  12.9, 13.2, 12.7,
13.1, 13.0, 13.1, 13.0, 12.6, 13.1, 13.0, 13.1, 12.8.  (a) Find the
sample mean.  (b) Find the standard error of X.  If the manufacturer
claims that the mean is 12.75 min. find the Z-score of the sample
mean.  (d) What do you think of the manufacturer's claim based on the
Z-score?

Hi thepilk,

Here are your answers, with short explanations.

a) Sample mean:
This is the sum of all of the observations divided by the number of
observations. In this case, that is:
155.6/12 = 12.97 or about 13 (rounded to significant digits)

b) Standard deviation (error):
This is the square root of the variance. The sample variance is:
s^2 = sum(x - M)^2/(N-1) where M is the mean and N is the number of
observations.
So this one is: s^2 =
(.01+.04+.09+.01+0+.01+0+.16+.01+0+.01+.04)/(12-1) = .0345
Hence s = 0.186 is the standard deviation.

c) z-score:
Z-score is an indication of how far an observation lies from the mean,
in terms of the variance.
z = (x-M)/s
z = (12.75-12.97)/0.186 = -1.184

d) Since about 68% of the observations should be within 1 standard
deviation away from the mean, and this z-score is more than one
standard deviation away from the mean, the manufacturer claim is
unlikely to be true. However, the sample size of this data set is
small, so one cannot be very confident in the results.

---

Clarification of Answer by jasonm1-ga on 11 Aug 2002 16:57 PDT

I should include a reference of some sort. There are good books on the
subject, but here is an online guide that's handy:

http://www.sysurvey.com/tips/statistics/contents.htm

And another:

http://davidmlane.com/hyperstat/index.html

It was a 4 part question, and 2 of his answers were challenged by a
later Researcher.  The original guy agreed those 2 answers were wrong.

I think that Jasonm1-ga has made an error here.  The standard error of
the mean is not the same as the standard deviation.  The standard
error is given by SD/sqrt(N).  The standard error is therefore equal
to 0.18257/sqrt(12) = 0.053.

The error arises partly because people use +/- to mean SE or SD, and
they don't specify.

The equations are given at:
http://davidmlane.com/hyperstat/A103735.html

Search term: standard error of the mean 
://www.google.com/search?q=standard+error+of+the+mean


I would also cast a little doubt on the interpretation of the
conclusion to part D.  If a maufacturer's claim is deemed to be
'unlikely to be true' because it has a 32% (i.e. about 1 in 3) chance
of being wrong, we are going to say that people are wrong an awful lot
of the time.  And whilst they may well be wrong, we are probably going
to be embarrassed if we claim that 1 in 3 correct claims is wrong. 
(Although this sort of decision becomes rapidly complex.)

My final point is that the confidence of the probability judgment,
when the null hypothesis is rejected, is not dependent upon sample
size.  If the null hypothesis (that the manufacturer is correct) is
rejected, the sample size does not come into it.  Indeed, we might
state with more confidence that the manufacturer is wrong, because we
managed to find that they were wrong, even though we didn't try very
hard (i.e. we had a small sample).


(Incidentally, the analysis also shows the problem that not using
sufficient digits might cause - when this sort of calculation has
several steps, which follow one another, rounding errors tend to add
up.  The error here was small, but could ave been large had the SD
been used in more steps.)

You are absolutely correct on the Standard Error part. Short lapse of judgement.