EXAMPLE 5.15 Checking for Binomial Compatibility

Consider an application in which manufactured fuel injectors are sampled periodically to check for compliance to specifications. Figure 5.8 shows the counts of defective injectors found in 40 consecutive samples. The counts appear to be behaving randomly over time. Summing over the 40 samples, we find the total number of observed defects to be 210 out of the 8000 total number of injectors inspected. This is associated with a proportion defective of 0.02625. Assuming that the random variable of the defect counts for each sample follows the , the standard deviation of will have a value around

In terms of variance, the variance of the counts is expected to be around or 5.11. Computing the sample variance on the observed counts, we would find a variance of 9.47. The observed variance of the counts is nearly twice of what is expected if the counts were truly following the binomial distribution. It appears that the binomial model does not fully account for the overall variation of the counts.

The statistical software JMP provides a nice option of superimposing a binomial distribution fit on the observed counts. Figure 5.9 shows the distribution overlaid on the histogram of the count data. The mismatch between the binomial distribution fit and the observed counts is clear. The observed counts are spread out more than expected by the binomial distribution, with a greater number of counts found both at the lower and upper ends of the histogram.