We are in effect testing the null hypothesis
on the assumption that 
, using a sample size of 50. Suppose for
the moment that this hypothesis is true (`innocent until proved guilty').
What then is the probability of drawing a sample 
with a sample
mean of 996 (or less)?
Formally, what we want is
Convert the problem to a z-score question (as in question 4
above). In this case the deviation between 
and the hypothesized
is 
, while the standard error is
The z-score is therefore 
. And 
is found from the normal table as
That is, if it were the case that 
and 
,
then there would be only a 7.9 per cent chance (`p-value') of drawing a
sample of 50 bulbs having a sample mean life of 996 hours or less. The
sample therefore looks somewhat unlikely on the null hypothesis, which in
turn casts doubt on that hypothesis. If the quality control rule has set a
`significance level' of 10 per cent for this kind of testing, then the sample
with mean 996 (and p-value 7.9 per cent) does call for inspection of the
production process. (You might verify that if we had found a sample mean of
997 this would not have called for inspection: What would be the p-value in
this case?)
Here you may return to the
question.