EXAMPLE 6.14 Is It Right now?

CASE 6.1 In Example 6.13, sample evidence suggested that the mean fill amount was not at the desired target of 473 ml. In particular, it appeared that the process was overfilling the bottles on average. In response, Bestea’s production staff made adjustments to the process and collected a sample of 20 bottles from the “corrected” process. From this sample, we find . (We assume that the standard deviation is the same, .) In this case, the sample mean is less than 473 ml—to be exact, 0.44 ml less than 473 ml.

Did the production staff overreact and adjust the mean level too low? We need to ask a similar question as in Example 6.13. In particular, what is the probability that the mean of a sample of size from a Normal population with mean and standard deviation is as far away or farther away from 473 ml as 0.44 ml? The answer is 0.328. A sample result this far from 473 ml would happen just by chance in 32.8% of samples from a population having a true mean of 473 ml. An outcome that could so easily happen just by chance is not convincing evidence that the population mean differs from 473 ml.

At this moment, Bestea does not have strong evidence to further tamper with the process settings. But, with this said, no decision is static or necessarily correct. Considering the cost of underfilling in terms of disgruntled customers is potentially greater than the waste cost of overfilling, Bestea personnel might be well served to gather more data if there is any suspicion that the process mean fill amount is too low. In Section 6.5, we discuss sample size considerations for detecting departures from the null hypothesis that are considered important given a specified probability of detection.