Assessing a claim about a mean
In this example, we ask whether a sample mean is consistent with the underlying population mean having a target value.
Quality control for cornflake packets
In a factory producing packets of cornflakes, the weight of cornflakes that a filling machine places in each packet varies from packet to packet. From extensive previous monitoring of the operation of the machine, it is known that the net weight of '500 gm' packets is approximately normal with standard deviation σ = 10 gm.
The mean net weight of cornflakes in the packets is controlled by a single knob. The target is for a mean of µ = 520 gm to ensure that few packets will contain less than 500 gm. Samples are regularly taken to assess whether the machine needs to be adjusted. A sample of 10 packets was weighed and contained an average of 529 gm. Does this indicate that the underlying mean has drifted from µ = 520 and that the machine needs to be adjusted?
A simulation
If the filling machine is working to specifications, each packet would contain a weight that is sampled from a normal distribution with µ = 520 and σ = 10.
How unlikely is it to get the mean of a sample of size 10 that is as far from 520 as 529 if the machine is working correctly?
A simulation helps to answer this question.
Click Simulate to randomly generate the weights of 10 packets from a normal (µ = 520, σ = 10) distribution. Click Accumulate then run the simulation between 100 and 200 times. (Hold down the Simulate button to speed up the process.)
Observe that although many of the individual cornflake packets weigh more than 529 gm, it is rare for the mean weight to be as far from the target as 529 gm (i.e. either ≥529 gm or ≤511 gm).
There is therefore strong evidence that the machine is no longer filling packets with a mean weight of 520 gm and needs adjusting — a sample mean of 529 gm would be unlikely if the machine was filling packets to specifications.
We will return to this example later.