EXAMPLE 6.13 Are the Bottles Being Filled as Advertised?

bestea1

CASE 6.1 The filling process is not new to Bestea. Data on past production shows that the distribution of the contents is close to Normal, with standard deviation . To assess the state of the bottling process, 20 bottles were randomly selected from the streaming high volume production line. The sample mean content () is found to be 474.54 ml. Is a sample mean of 474.54 ml convincing evidence that the mean fill of all bottles produced by the current process differs from the desired level of 473 ml?

If we lack proper statistical thinking, this is a juncture to knee-jerk one of two possible conclusions:

  • Conclude that “The mean of the bottles sampled is not 473 ml so the process is not filling the bottles at a mean level of 473 ml.”
  • Conclude that “The difference of 1.54 ml is small relative to the 473 ml baseline so there is nothing unusual going on here.”

Both responses fail to consider the underlying variability of the population, which ultimately implies a failure to consider the sampling variability of the mean statistic.

So, what is the conclusion? One way to answer this question is to compute the probability of observing a sample mean at least as far from 473 ml as 1.54 ml, assuming, in fact, the underlying process mean is equal to 473 ml. Taking into account sampling variability, the answer is 0.00058. (You learn how to find this probability in Example 6.18.) Because this probability is so small, we see that the sample mean is incompatible with a population mean of . With this evidence, we are led to the conclusion that the underlying bottling process does not have mean of . The estimated average overfilling amount of 1.54 ml per bottle may seem fairly inconsequential. But, when it is put in the context of the high-volume production bottling environment and the potential cumulative waste across many bottles, then correcting the potential overfilling is of great practical importance.