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88 3 Estimating Data Parameters
approximated by the normal distribution. (This assumption should have to be
checked with the methods described in section 5.1.) In these conditions the
confidence interval is:
x ± t 19,0.975 ×SE = x ± 2.09×SE ⇒ [312, 390].
If the 95% confidence interval were computed with the z percentile, one would
wrongly obtain a narrower interval: [315, 387].
Example 3.3
Q: How many cases should one have of the PRT data in order to be able to
establish a 95% confidence interval for its mean, with a tolerance of 3%?
A: Since the tolerance is smaller than the one previously obtained in Example 3.1,
we are clearly in a large sample situation. We have:
2
z 1−α 2 / s ≤ ε ⇒ n ≥ z 1−α 2 / s . 3.13
x n ε x
Using the previous sample mean and sample standard deviation and with
z 0.975 =1.96, one obtains:
n ≥ 558.
Note the growth of n with the square of 1/ε .
The solutions of all the previous examples can be easily computed using
Tools.xls (see Appendix F).
An often used tool in Statistical Quality Control is the control chart for the
sample mean, the so-called x-bar chart. The x-bar chart displays means, e.g. of
measurements performed on equal-sized samples of manufactured items, randomly
drawn along the time. The chart also shows the centre line (CL), corresponding to
the nominal value or the grand mean in a large sequence of samples, and lines of
the upper control limit (UCL) and lower control limit (LCL), computed as a ks
deviation from the mean, usually with k = 3 and s the sample standard deviation.
Items above UCL or below LCL are said to be out of control. Sometimes, lines
corresponding to a smaller deviation of the grand mean, e.g. with k = 2, are also
drawn, corresponding to the so-called upper warning line (UWL) and lower
warning line (LWL).
Example 3.4
Q: Consider the first 48 measurements of total area of defects, for the first class of
the Cork Stoppers dataset, as constituting 16 samples of 3 cork stoppers
randomly drawn at successive times. Draw the respective x-bar chart with 3-sigma
control lines and 2-sigma warning lines.
