Page 25 - Mechanical Engineers' Handbook (Volume 2)

14 Instrument Statics

3.6 Amount of Data to Take
Exactly what data to take and how much data to take are two important questions to be
answered in any experiment. Assuming that the correct variables have been measured, the
amount of data to be obtained can be determined by using the relation

x W R
5
ˆ
x
10 m (28)
where it is presumed that several sample sets may exist for estimation of and that the
mean of means of the sample sets is denoted by . This equation can be rewritten using Eqs.
x
(13) and (14) (assuming random sampling):
x k( , )ˆ R

5
x
10
m
ˆ R
5
x k( , ) (29)
n 10 m
The value of n to achieve the difference in x within a stated percent of can be
determined from
n k(v, )ˆ 2
m
(%/100) ˆ R (5/10 ) (30)
This equation can only yield valid values of n once valid estimates of ˆ ,ˆ , k, R, and m are
available. This means that the most correct values of n can only be obtained once the
measurement system and data-taking procedure have been specified so that R and m are
known. Furthermore, either a preliminary experiment or a portion of the actual experiment
ˆ
should be performed to obtain good estimates of ˆ and . Because k depends not only on
the type of distribution the data follows but also on the sample size n, the solution is iterative.
Thus, the most valid estimates of the amount of data to take can only be obtained after the
experiment has begun. However, the equation can be quite useful for prediction purposes if
one wishes to estimate values of ˆ ,ˆ , k, R, and m. This is especially important in experi-
ments for which the cost of a single run may be relatively high.

Example 4 Amount of Data to Take. The life for a certain type of automotive tire is to
be established. The mean and standard deviation of the life estimated for these tires are
84,000 and 7,230 km, respectively, from a sample of nine tires. On the basis of the sample,
how much data are required to establish the life of this type of tire to within 10% with
90% confidence and a resolution of 5 km?

Solution: Confidence limits are as follows:

ˆ tˆ R
5
x
10 m
5 ˆ 5
ˆ (0.10)x tˆ R t x R
x
10 m n 10 m
m
t (0.10)x R (5/10 ) (0.10)(84000) 5
1.6
n ˆ x 7230