7.2.4.1. Confidence intervals

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

Confidence intervals using the method of Agresti and Coull

The method recommended by Agresti and Coull (1998) and also by Brown, Cai and DasGupta (2001) is to use the form of the confidence interval that corresponds to the hypothesis test given in Section 7.2.4. That is, solve for the two values of p₀ (say, p_upper and p_lower) that result from setting z = z(alpha/2

and solving for p₀ = p_upper, and then setting z = - z(alpha/2

and solving for p₀ = p_lower. (Here, as in Section 7.2.4, z(alpha/2

denotes the variate value from the standard normal distribution such that the area to the right of the value is alpha

/2.) Although solving for the two values of p₀ might sound complicated, the appropriate expressions can be obtained by straightforward but slightly tedious algebra. Such algebraic manipulation isn't necessary, however, as the appropriate expressions are given in various sources. Specifically, we have

Formulas for the confidence intervals

$U. L. = {phat + z(alpha/2)^(2)/(2n) + z(alpha/2/(sqrt[(phat*(1-phat))/n + z(alpha/2)^(2)/(4n^2)]}/ {1 + z(alpha/2)^(2)/n}$

$L. L. = {phat + z(alpha/2)^(2)/(2n) - z(alpha/2/(sqrt[(phat*(1-phat))/n + z(alpha/2)^(2)/(4n^2)]}/ {1 + z(alpha/2)^(2)/n}$

Procedure does not strongly depend on values of p and n

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 n and/or p, and indeed was recommended by Agesti and Coull for virtually all combinations of n and p.

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:

phat +/- z(alpha/2)*SQRT[(phat*(1-phat))/n]

A confidence limit approach that produces a lower limit which is an impossible value for the parameter for which the interval is constructed is an inferior approach. This also applies to limits for the control charts that are discussed in Chapter 6.

One-sided confidence intervals

A one-sided confidence interval can also be constructed simply by replacing each z(alpha/2)

in the expression for the lower or upper limit, whichever is desired. The 95% one-sided interval for p for the example in the preceding section is:

Example

$p >= {phat + z(alpha)^(2)/(2n) - z(alpha)*sqrt[(phat*(1-phat))/n + z(alpha)^(2)/(4n^2)]}/ {1 + z(alpha)^(2)/n}$

$p >= {0.13 + 1.645^2/(2*200) - 1.645*sqrt[(0.13*0.8713)/200 + 1.645^2/(4*200^2)]}/ {1 + z(alpha)^(2)/n}$

p 0.09577

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 .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.