A company has a new machine designed to fill bags with, on average, 1 kg of granulated sugar. The production manager wishes to investigate if the machine is adjusted correctly. He intends to take a sample of bags and carry out a hypothesis test.
State null and alternative hypotheses for the manager’s test, defining any parameters you use.[2]
The production manager decides to take the first bag of sugar produced each morning and the first bag of sugar produced each afternoon, in a 5-day working week, to form a sample of 10 bags for the test.
Give two reasons why the production manager’s sample is not suitable for a \(z\)-test.[2]
The company has a different machine which fills larger bags with, on average, \(2\text{ kg}\) of granulated sugar. One of the company’s sales representatives has reported that some customers suspect the machine is no longer set correctly, and that the average mass of sugar in the bags may in fact be less than \(2\text{ kg}\). The production manager decides to carry out a hypothesis test at the \(2.5\%\) level of significance with a suitable sample of \(40\) bags of sugar. Summary data for the mass, \(x\text{ kg}\), of sugar in these bags is as follows. \[n=40\qquad \sum x=78.88\qquad \sum x^2=155.6746\]
State the hypotheses and find the critical region for this test.[5]
State the conclusion of the test in the context of the question.[2]