7. Product and Process Comparisons 7.2. Comparisons based on data from one process 7.2.4. Does the proportion of defectives meet requirements? 7.2.4.1. Confidence intervals |
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Confidence intervals using the method of Agresti and Coull |
The Wilson method for calculating confidence intervals for proportions
(introduced by Wilson (1927), recommended by
Brown, Cai and DasGupta (2001)
and Agresti and Coull (1998))
is based on inverting the hypothesis test given in
Section 7.2.4.
That is, solve for the two values of |
Formulas for the confidence intervals |
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Procedure does not strongly depend on values of |
This approach can be substantiated on the grounds that it is the
exact algebraic counterpart to the (large-sample) hypothesis test
given in section 7.2.4 and is also supported by the research of
Agresti and Coull. One advantage of this procedure is that its
worth does not strongly depend upon the value of |
Another advantage is that the lower limit cannot be negative |
Another advantage is that the lower limit cannot be negative. That is
not true for the confidence expression most frequently used:
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One-sided confidence intervals |
A one-sided confidence interval can also be constructed simply by
replacing each |
Example |
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Conclusion from the example | Since the lower bound does not exceed 0.10, in which case it would exceed the hypothesized value, the null hypothesis that the proportion defective is at most 0.10, which was given in the preceding section, would not be rejected if we used the confidence interval to test the hypothesis. Of course a confidence interval has value in its own right and does not have to be used for hypothesis testing. |
Exact Intervals for Small Numbers of Failures and/or Small Sample Sizes | |
Constrution of exact two-sided confidence intervals based on the binomial distribution |
If the number of failures is very small or if the sample size
|
Note |
The interval |
Binomial confidence interval example |
The equations above that determine
1. Initalize constants. alpha = 0.10 Nd = 4 N = 20 2. Define a function for upper limit (fu) and a function for the lower limit (fl). fu = F(Nd,pu,20) - alpha/2 fl = F(Nd-1,pl,20) - (1-alpha/2) F is the cumulative density function for the binominal distribution. 3. Find the value of pu that corresponds to fu = 0 and the value of pl that corresponds to fl = 0 using software to find the roots of a function.The values of pu = 0.401029 pl = 0.071354
Thus, a 90 % confidence interval for the proportion defective, The calculations used in this example can be performed using both Dataplot code and R code. |
Terminology Note | Previous versions of the Handbook referred to the method described here as the Agresti-Coull method. However, common practice in the statistics literature is to refer to the method given here as the Wilson method and a similar, but different, method described in Brown, Cai, and DasGupta as the Agresti-Coull method (the Agresti-Coull paper refers to this as the "adjusted Wald" method). We have modified our terminology to be consistent with common practice in the statistical literature. |